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Traditionally Australian primary school teachers have been viewed as generalists responsible for instruction across all content areas. Adopting self-determination theory as a lens, the aim of the study was to explore the extent to which generalist primary school teachers are interested in becoming subject matter specialists. Questionnaire data were collected from 104 early years primary school teachers. Findings suggest that two-thirds of these generalist teachers expressed an interest in specialising in either English, mathematics, and to a far lesser extent, science, such that they would be responsible for exclusively teaching this subject. Preferences for specialisation were based on teachers' self-perceived content and pedagogical expertise and/ or their enjoyment of teaching in this content area. By contrast, the one-third of teachers who would choose to remain generalists referred to the value in a variety of teaching experiences, teaching from a whole child perspective and content integration. Implications for educational policy are discussed.

Just as students experience productive struggle or spend time in the 'zone of confusion' when engaging with challenging tasks, teachers also experience similar difficulties and periods of confusion when engaging with new pedagogical approaches. Prior to a 19-week lockdown due to Coronavirus (COVID-19) during 2020, two Foundation teachers implemented a student-centred pedagogical approach when teaching with challenging tasks. While they had some initial success implementing the pedagogical approach and a three-phase lesson structure, they struggled to do so online during the lockdown. It is the experiences of these teachers, in particular their experience of confusion relating to aspects of the pedagogical approach, and how this confusion was overcome, that is reported in this paper. Central to our findings is the importance of teachers reflecting on their own experiences of struggle and the impact this had on their professional learning, as well as the notion that adversity can be a catalyst for change.

In this article, educational and psychological perspectives are used to examine how anxiety and uncertainty in the mathematics classroom can be reframed to benefit mathematical teaching and learning. Links between anxiety and uncertainty are discussed and from this discussion, two methods are proposed for reducing the negative impact of mathematics anxiety on learning. Firstly, it is argued that initiatives designed to teach students to use emotion regulation skills can help to reframe anxiety and improve students' ability to regulate negative emotion. In particular, the skill of reappraisal is discussed. Secondly, it is proposed that pedagogical approaches that encourage students to embrace challenge will help to normalise uncertainty in classrooms and promote more positive affect and engagement with mathematics. These approaches include choosing suitable tasks, differentiating those tasks, orchestrating classroom discussions that include all students and consolidating the learning with further tasks suitably varied. Rather than teachers seeking to remove anxiety, students can be encouraged to manage their anxiety in order to promote more positive mathematical learning and to develop mathematical resilience.

Given what is known about the importance of productive struggle for supporting student learning of mathematics at all levels, the current study sought to examine teacher attitudes towards student struggle when students learn mathematics in remote learning settings compared with classroom settings. Eighty-two Australian early years primary teachers involved in a professional learning initiative focused on teaching mathematics through sequences of challenging tasks completed a questionnaire inviting them to compare the two settings. Drawing on a mixed-methods approach, we found that teachers were more positive about the value of student struggle in classroom-based settings compared with remote learning settings. Qualitative analysis of open-ended responses revealed four themes capturing why teachers viewed efforts to support productive struggle in a remote learning setting as potentially problematic: absence of a teacher-facilitated, synchronous, learning environment; parents' negative attitudes towards struggle when learning mathematics; lack of social connection and peer-to-peer collaboration; and difficulties accessing learning materials. Suggestions for mitigating some of these challenges in the future are put forward.

Despite teacher enjoyment of teaching being associated with a range of benefits for teachers and their students, little research exploring the characteristics of teachers who enjoy teaching mathematics exists. Semi-structured interviews were conducted with ten teachers who scored in the top third in terms of their enjoyment of teaching mathematics from a sample of 118 generalist, early-year primary school teachers. Thematic analysis of interview data revealed that negative-to-positive turning point experiences were commonplace amongst high-enjoyment primary teachers, but so too were reports of consistently positive feelings towards mathematics throughout their entire lives. While most articulated a passion for teaching mathematics, all teachers considered instances when students experienced 'light bulb' moments in their mathematical understanding that made a mathematics lesson enjoyable to teach. An important implication of our research is that one need not be a 'maths person' to come to love teaching mathematics. Rather, positive experiences in one's pre-service teacher education, or in-service teacher professional learning, have the potential to transform one's attitude towards teaching mathematics.

This article is based upon Peter Sullivan's (2021) structured inquiry approach to teaching mathematics, in which the authors take a deep dive into the technique of "Spotlighting" in the "Explore" phase of this model to assist in optimal learning experiences for students. Working from the premise that students are best placed to connect with instruction after a powerful and sometimes confusing learning experience, Sullivan et al.'s (2021) model provides three distinct phases to structure a mathematics lesson. They are: (1) The Launch phase--poses the task to students without any further instruction about the strategies required to solve the task; (2) The Explore phase--where collaboration and peer learning is encouraged as a support for students to consider multiple solutions and solution pathways; and (3) The Summarise/Review phase--provides an opportunity for the teacher to orchestrate class discussions by presenting various student solutions that reflect the mathematical focus of the lesson. The authors conclude they have been pleasantly surprised at how particular student explanations, derived from real time spotlight moments, were critical in shaping successful learning opportunities for the rest of the class.

This paper outlines a tutoring initiative intended to re-engage students who have become alienated from mathematics class at school. The initiative is focused on priming students for their subsequent mathematics lessons, and is referred to as "Getting Ready in Numeracy" (G.R.I.N.). Following a discussion outlining the principles of G.R.I.N., data from interviews with G.R.I.N. tutors and classroom teachers are presented. Findings suggest that participating in G.R.I.N. enhanced student engagement in mathematics and helped foster a growth mindset. Analysis of interviews also revealed some of the challenges associated with the initiative, in particular, issues relating to communication and coordination, such as: understanding the purpose of G.R.I.N., timetabling, planning and having sufficient students in the target group. In addition, in some instances, tensions between the professional and pedagogical expertise of the G.R.I.N. tutor and classroom teacher required navigation. Recommendations for mitigating these challenges and tensions are put forward, all of which are reliant on strong leadership and support for the program at the school level.

Teaching assistants are not always utilised effectively in mathematics classrooms. Moreover, there is limited research examining instructional models that might more meaningfully incorporate teaching assistants into the teaching and learning of mathematics. To address this gap in the literature, the current study explored three distinct teacher-tutor dyads involved in a particular intervention program, Getting Ready in Numeracy (G.R.I.N.), from a single Australian primary school. All three tutors currently operated as teaching assistants in the classrooms of their G.R.I.N. teacher counterparts. The professional working relationships between G.R.I.N. teachers and their respective tutors were investigated through interviews. We outline common themes between the different dyads' approaches to G.R.I.N., and argue that all three dyads' practices are consistent with principles outlined in the literature around the effective utilisation of teaching assistants. We conclude by synthesising our findings, and presenting an embedded tutor model for the G.R.I.N. program. This model is presented as a prototype of good practice in a context where G.R.I.N. has been implemented successfully, whilst also serving as a more general model for how teaching assistants can be meaningfully incorporated into mathematics intervention programs.

In this article, three of David Clarke's long time collaborators offer perspectives on how David shaped and continued to influence projects in which they were involved, over a 30 year period. The three sections of the article focus on David's contributions to original research on high stakes assessment, research on the development and use of assessment alternatives to traditional pen and paper assessments, and research in three smaller projects involving the use of open-ended questions, the implementation of the "Australian Curriculum: Mathematics," and the complexity of the mathematics classroom, respectively. Through discussion of these various national and international projects, the authors provide examples of David's capacity to synthesise theoretical ideas, to appreciate the implications for classroom practice, and to communicate both perspectives to researchers, teacher educators and teachers. The authors argue that David's capacity to distil crucially important ideas into brief, highly insightful statements, was fundamental to the enactment and effectiveness of both research studies and projects for the teaching profession.

Peter Sullivan and colleagues present a report on an initiative at a Victorian primary school which embraced opportunities offered by purposeful collaborative professional learning communities. They utilised a form of Japanese Lesson Study as part of their approach. There are three parts to the story of this initiative: (1) a description of the approach to professional learning; (2) an explanation of the model of instruction being explored; and (3) a summary of the 'polished' lesson on middle primary mathematics (fractions).

Consumer, economic and financial literacy education at school is central to active and informed citizenship. Over the past decade, the Australian Securities and Investments Commission has led various policy initiatives and influenced curriculum and resource development in this area. However, there remains a paucity of research exploring how Australian teachers make sense of and approach their work as financial literacy educators or their professional learning needs and interests in this interdisciplinary field. This article reports research exploring practising teachers' perceptions of the opportunities for financial literacy teaching and learning. Data were collected from 35 teachers in 16 Victorian primary schools. The findings suggest a need to educate teachers to: reflect upon the knowledge, skills and capabilities required to make informed financial decisions; identify and interpret the possibilities for financial literacy teaching and learning in the Australian Curriculum; and enact sophisticated pedagogical practice.

In this article, the authors from Monash University and the University of Sydney have collaborated to present a research-informed model to support the planning and teaching of mathematics, using a student centred structured inquiry approach.

Improved representation of turbulent processes in numerical models of tropical cyclones (TCs) is expected to improve intensity forecasts. To this end, the authors use a large-eddy simulation (with 31-m horizontal grid spacing) of an idealized category 5 TC to understand the role of turbulent processes in the inner core of TCs and their role on the mean intensity. Azimuthally and temporally averaged budgets of the momentum fields show that TC turbulence acts to weaken the maximum tangential velocity, diminish the strength of radial inflow into the eye, and suppress the magnitude of the mean eyewall updraft. Turbulent flux divergences in both the vertical and radial directions are shown to influence the TC mean wind field, with the vertical being dominant in most of the inflowing boundary layer and the eyewall (analogous to traditional atmospheric boundary layer flows), while the radial becomes important only in the eyewall. The validity of the downgradient eddy viscosity hypothesis is largely confirmed for mean velocity fields, except in narrow regions which generally correspond to weak gradients of the mean fields, as well as a narrow region in the eye. This study also provides guidance for values of effective eddy viscosities and vertical mixing length in the most turbulent regions of intense TCs, which have rarely been measured observationally. A generalized formulation of effective eddy viscosity (including the Reynolds normal stresses) is presented. Significance Statement: This study uses a turbulence-resolving simulation of a category 5 tropical cyclone to understand the role of turbulence in intense storms. Results show that turbulence clearly modulates storm structure and intensity. This study provides guidance for the values of turbulent quantities (which are usually parameterized in comparatively coarse operational TC forecast models) in scarcely observed regions of intense storms. Furthermore, a complete formulation of the effective eddy viscosities is proposed, incorporating contributions from typically ignored Reynolds normal stress terms. [ABSTRACT FROM AUTHOR]

This paper reports on a New Zealand iteration of the Encouraging Persistence, Maintaining Challenge (EPMC) project, which proposes that students learn mathematics best when they build connections between mathematical ideas for themselves. This iteration explores the actions, perceptions and learning of 12 primary teachers and their 281 students during the implementation of a set of challenging tasks related to geometric reasoning. The teachers launched the suggested tasks, ensuring that the challenge was maintained. The students explored these tasks with minimal input from the teacher, and learning was summarised and extended. The teachers were positive about the intervention. The challenging task approach enabled students' thinking became visible and, at times, the teachers' prior perceptions of their students' ability were challenged. A highly significant difference between the students' pre- and post-assessment scores was found. The students were supported to have autonomy in their learning and make mathematical connections themselves. The students became less reliant on their teachers' help and were positive about their involvement in the project.

Australian schools, like schools elsewhere, have been through a period of closure. The closure creates both threats and opportunities for teachers and students. In the context of a project exploring approaches to teaching in early years, we outline some considerations and offer advice to teachers and educators on strategies for welcoming students back to school.

The fear of making mistakes and avoidance of risks can create disabling anxiety and limit learning opportunities. However, it is possible to teach mathematics in ways that not only reduce student agency but also have the effect of excluding some students. This article proposes approaches that are intended to both foster student decision making and a sense of inclusion in classroom learning. It describes some characteristics of classrooms that may go some way to minimising student anxiety without reducing the opportunities for cognitive activation. There are three parts that are discussed: features of tasks posed, pedagogical approaches, and classroom culture.

In this article, the authors emphasise the importance of purposeful mathematical discussion and using sequences of connected, cumulative, and challenging tasks in advancing students' spatial reasoning in the early years of schooling.

This article reports on a project which involved introducing challenging tasks in the early years' classrooms of twenty Victorian primary schools. The obstacles teachers encountered and how they overcame those obstacles are discussed.

The authors describe a mathematics lesson designed to allow students to use their own thinking to solve elapsed time problems. Their approach emphasises student-centered structured inquiry, and follows a 'Launch, Explore and Summarise' format.

This article reports the findings of classroom research exploring the potential of posing challenging mathematical problems situated in real-world financial contexts to activate mathematical knowledge, skills and reasoning. The Organisation for Economic Cooperation and Development (OECD) Programme for International Student Assessment (PISA) 2012 mathematical literacy assessment framework (OECD, 2013) provided theoretical framing for the study, which examined the use of a "financial dilemma" titled "Shopping for shoes" in a Year 5/6 composite class in a suburban Australian primary school. The social and mathematical dimensions of the task, together with a particular lesson structure, successfully engaged 10 to 12-year-old students in exploring mathematics through a financial problem-solving experience. The findings reveal that students' notions of friendship and fairness guided the way they "formulated" the problem and "employed" mathematics. Strategies intended to encourage students to "interpret" the alignment and reasonableness of their social and mathematical thinking against the problem context were critical to the lesson.

Advances in coastal modeling and computation provide the opportunity to examine non-hydrostatic and compressible fluid effects at very small scales, but the cost of these new capabilities and the accuracy of these models versus trusted non-hydrostatic codes has yet to be determined. Here the Coastal and Regional Ocean COmmunity model (CROCO, v1.2) and the National Center for Atmospheric Research large-eddy simulation (NCAR-LES) model are compared, with a focus on their simulation accuracy and computational efficiency. These models differ significantly in numerics and capabilities, so they are run on common classic problems of surface-forced, boundary-layer turbulence. In terms of accuracy, we compare turbulence statistics, including the effect of the explicit subgrid-scale (SGS) parameterization, the effect of the second (dilatational) viscosity, and the sensitivity to the speed of sound, which is used as part of the CROCO compressible turbulence formulation. To gauge how far CROCO is from the NCAR-LES, we first compare the NCAR-LES with two other non-hydrostatic Boussinesq approximation LES codes (PALM and Oceananigans), defining the notion and magnitude of accuracy for the LES and CROCO comparison. To judge efficiency of CROCO, strong and weak scaling simulation sets vary different problem sizes and workloads per processor, respectively. Additionally, the effects of 2D decomposition of CROCO and NCAR-LES and supercomputer settings are tested. In summary, the accuracy comparison between CROCO and the NCAR-LES is similar to the NCAR-LES compared to other LES codes. However, the additional capabilities of CROCO (e.g., nesting, non-uniform grid, and realism of ocean configuration in general) and its weakly compressible formulation come with roughly an order of magnitude of additional costs, despite efforts to reduce them by adjusting the second viscosity and speed of sound as far as accuracy allows. However, a new variant of the non-hydrostatic CROCO formulation is currently undergoing prototype testing and should enable faster simulations by releasing the stability constrain by the free surface. Overall, when the additional features of CROCO are needed (nesting, complex topography, etc.) additional costs are justified, while in idealized settings (a rectangular domain with periodic boundary conditions) the NCAR-LES is faster in arriving at nearly the same result. [ABSTRACT FROM AUTHOR]

Recent field observations suggest that the air‐sea momentum flux (or the drag coefficient) is significantly reduced when the dominant wind‐forced surface waves are misaligned from local wind. Such conditions may occur under rapidly changing strong winds (such as under tropical cyclones) or in coastal shallow waters where waves are refracted by bottom topography. A recent Large Eddy Simulation (LES) study also shows that the drag coefficient is reduced by a misaligned strongly forced wave train (with a small wave age of 1.37). In order to investigate more realistic field conditions, this study employs LES to examine the effect of a misaligned (up to 90°) surface wave train over a wide range of wave age up to 10.95. For all wave ages examined, the drag coefficient is reduced compared to the flat surface condition when the misalignment angle exceeds around 22.5°–45°. The drag reduction may occur even if the form drag of the wave train is positive. Plain Language Summary: This study uses computer simulations to investigate turbulent wind blowing over ocean surface waves that are misaligned (propagate in different directions) from the wind. The study focuses on frictional wind forcing (wind stress) imparted on the ocean surface, because the wind stress is an important boundary condition for ocean, atmospheric, and surface wave models. Surface waves can be misaligned from wind when the wind field rapidly changes such as during tropical storms or in shallow water where wave directions are modified by varying water depth. When waves propagate slower than wind in the wind direction, the waves increase the wind stress compared to flat surface (no wave) conditions. However, this study finds that the wind stress is reduced, that is, the ocean surface becomes smoother, when the misalignment angle between wind and waves exceeds about 22.5°–45°. Key Points: Large eddy simulations are preformed to study wind over misaligned surface wavesDrag coefficient is reduced due to misaligned surface waves for all wave ages under wind‐driven wave conditionsDrag reduction can occur even when wave form drag is positive [ABSTRACT FROM AUTHOR]

As part of a research and professional development project that focused on the opportunities and constraints provided by different kinds of mathematical tasks, a group of 67 primary and 40 secondary practising teachers of mathematics were asked to complete a survey focusing on their use of tasks. In this article, we discuss responses to one particular item which sought teachers' ideas on taking a fraction comparison task (which is larger: 2/3 or 201/301?) and converting it into a mathematics lesson in the middle years of schooling. Drawing upon a number of components of "mathematical knowledge for teaching" as a framework, we attempt to examine those aspects of mathematical knowledge which are involved in making such a conversion. Our recommendation following this analysis is that greater emphasis is necessary in professional development settings on taking a potentially useful task and converting it into a worthwhile mathematics learning experience for students. Knowing the relevant mathematics also seems necessary even if not sufficient to make this conversion. (Contains 4 tables and 1 note.)

Middle school students have been pervasively described in the research literature as exhibiting disaffection, disengagement, and a lack of interest in mathematics classrooms. This study investigated this notion empirically using students' own voice on their wishes for mathematics learning to see if they characterise themselves in this way in their self-reports. Over 3500 Australian middle school students responded to a free-format prompt and reported diverse but overwhelmingly positive aspirations related to intrinsic motivational features inherent in their disposition and to extrinsic factors in their environment, such as the learning context and activities provided. Evidence of all four goal orientations was found in the students' responses but most noticeably for the productive mastery-approach orientation. When these students described what they wish for in their mathematics classrooms, they mentioned challenging work, interesting work, and opportunities to collaborate.

Productive struggle leads to productive classrooms where students work on complex problems, are encouraged to take risks, can struggle and fail yet still feel good about working on hard problems (Boaler, 2016). Teachers can foster a classroom culture that values and promotes productive struggle by providing students with challenging tasks. These tasks are designed in such a way that they are accessible to all students and the expectation is that everyone will persist when solving challenging mathematical tasks. This article looks at examples of challenging mathematical tasks designed to enable students of all abilities to experience productive struggle, and examines the lesson structure and teacher's role when implementing a challenging task, along with students' responses to being challenged through the tasks. It adds to existing research on challenging tasks, including in previous editions of "Australian Primary Mathematics Classroom" (APMC) (e.g., Cheeseman, Clarke, Roche, & Walker, 2016; Roche & Clarke, 2014; Russo, 2016) through challenging tasks being used to evoke productive struggle, and illustrative examples of what productive struggle 'looks like.'

Schematic models have been used extensively in educational research to represent relationships between variables diagrammatically, including the interrelationships between factors associated with teachers' beliefs and practices. A review of such models informed the development of a new model that was used to plan an investigation into primary school teachers' problem-solving beliefs and practices. On the basis of the findings from the research, the model was revised to include the important variable of prior mathematics learning, as well as a repositioning of the influence of teaching experiences in classrooms. (Contains 9 figures and 5 tables.)

Upper-ocean turbulence results from a complex set of interactions between submesoscale turbulence and local boundary layer processes. The interaction between larger-scale currents and turbulent fluctuations is two-way: large-scale shearing motions generate turbulence, and the resulting coherent turbulent fluxes of momentum and buoyancy feed back onto the larger flow. Here we examine the evolution and role of turbulence in the intensification, instability, arrest, and decay (i.e., the life cycle) of a dense filament undergoing frontogenesis in the upper-ocean boundary layer, i.e., cold filament frontogenesis (CFF). This phenomenon is examined in large-eddy simulations (LES) with resolved turbulent motions in large horizontal domains using 109 grid points. The boundary layer turbulence is generated by surface buoyancy loss (cooling flux) and is allowed to freely interact with an initially imposed cold filament, and the evolution is followed through the frontal life cycle. Two control parameters are explored: the initial frontal strength M2 = ∂xb and the surface flux Q * . The former is more consequent: initially weaker fronts sharpen more slowly and become arrested at a later time with a larger width. This reflects a competition between the frontogenetic rate induced by the secondary circulation associated with vertical momentum mixing by the turbulence and the instability rate for the along-filament shear flow. The frontal turbulence is energized by the shear production of the latter, is nonlocally transported away from the primary production zone at the filament centerline, and cascades to dissipation in a broad region surrounding the filament. The turbulent momentum fluxes arresting the frontogenesis are supported across a wide range of horizontal scales. [ABSTRACT FROM AUTHOR]

Through interactions between the modes of "waves" and "vortices", stably stratified turbulence exhibits characteristic features both in 2D and 3D. Using DNS of the Navier–Stokes equations coupled to an equation for temperature (the Bousinessq system), we investigate dynamical properties of stably stratified turbulence. After reviewing the importance of three characteristic length scales and their relations in stably stratified turbulence, we present numerical results showing how structures and spectra develop with the growth of the length scales. It is demonstrated that the temperature fluctuations make sharp fronts vertically which result in non-symmetric PDFs of the vertical derivative ∂ z θ . [ABSTRACT FROM AUTHOR]

This article from Carly Sawatzki and Peter Sullivan encourages schools and teachers to consider ways to embed financial literacy tasks in mathematics lessons more often, and includes an authentic task to try with your students.

There is widespread international interest in parental education as a means of promoting educational equality through improving educational outcomes for young children. The research in this area suggests an association between the home learning environment and children's educational outcomes and highlights the importance of parental education for supporting young children's learning through play. This article reviews the international literature around parental education initiatives (or "interventions") in early childhood and then considers playgroups as potential sites for parental education. The article identifies the universal features of playgroups that make these sites appealing for the implementation of parental education initiatives and discusses the complexities associated with the design of interventions aimed at meeting the diverse needs of parents attending playgroups. It concludes by providing a case for community playgroups as cultural contexts, to be considered sites for parental education through curriculum aimed at supporting parents to actively engage in their children's learning and development through play.

While the general planning advice offered to mathematics teachers seems to be to start with simple examples and build complexity progressively, the research reported in this article is a contribution to the body of literature that argues the reverse. That is, posing of appropriately complex tasks may actually prompt the use of more sophisticated strategies. Results are presented from a detailed study of young children working on tasks that prompt multiplicative thinking. It was found that the tasks involving more complex number triples prompted the use of more sophisticated multiplicative thinking.

The following is a report on an investigation into ways of supporting teachers in converting challenging mathematics tasks into classroom lessons and supporting students in engaging with those tasks. Groups of primary and secondary teachers, respectively, were provided with documentation of ten lessons built around challenging tasks. Teachers responded to survey items in both Likert and free format style after teaching the ten lessons. The responses of the teacher participants indicate that the lesson structure we proposed was helpful, and the elements of the lessons suggested to teachers were both necessary and sufficient for supporting students in engaging with the challenging tasks. Implications for teacher educators and curriculum developers are offered.

The content and purpose of written unit plans in mathematics is an under-researched area. In this article, we provide a brief overview of research on teachers' planning processes and the place of mental and written plans. We report on data from a questionnaire completed by 357 teachers from Victorian Catholic primary schools, where we focused on possible elements of written unit plans for primary mathematics, and the relative importance which teachers attributed to these elements. We then discuss 48 written unit plans which were provided to us by primary schools from which the 357 teachers were drawn. There was considerable variation in the length, intended teaching time, and level of specification of key ideas in these plans. We discuss this variation, and some discrepancies between the ratings of teachers of the importance of certain elements and their presence in the plans we examined. We also suggest potentially productive areas of future enquiry.

Challenges for researchers wishing to gain insights into student beliefs about mathematics and learning include the identification of prompts to which young children can respond easily and which have the potential to provide meaningful insights into their beliefs. This article reports on a selection of such prompts used within an exploratory study that investigated beliefs about mathematics, learning and helping factors for learning mathematics of eight children of 8 to 9 years of age. These prompts, described as "creative interviewing" procedures, are classified under the key categories of visual, verbal and text-based. The article provides brief details of the study as background but focuses on discussing the nature of the tasks, and, through the inclusion of a selection of responses, illustrates their potential value as research tools for gaining insights into young children's beliefs about mathematics and learning. It seems that teachers could use the prompts to gain alternate perspectives into children's beliefs which in turn might inform their instruction.