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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. [ABSTRACT FROM AUTHOR]

Direct numerical simulations (DNS) at bulk Reynolds number Re = 10 4 and bulk Richardson number Ri = 0.25 of plane Couette flow are performed with the results used to analyze the structure and mixing intensity in strongly stable boundary-layer flows. The Couette flow set-up is used as a proxy for a real-world stable boundary layer flow with surface thermal heterogeneity. Along the upper and lower walls, the temperature is either homogeneous or varies sinusoidally, but the horizontal-mean surface temperature is the same in all cases. Over homogeneous surfaces, the strong stratification always quenches turbulence resulting in linear velocity and temperature profiles. However, over a heterogeneous surface turbulence survives. Molecular diffusion and turbulence contribute to down-gradient momentum transfer. The total (diffusive plus turbulent) heat flux is directed downward, but its turbulent contribution is positive, i.e., up the mean temperature gradient. Analysis of covariances of velocity and temperature, their skewness, and the flow structure suggests that counter-gradient heat transport is due to quasi-organized cell-like vortical motions generated by surface thermal heterogeneity. These motions transfer heat upwards similar to their counterparts in highly convective boundary layers. Thus, the flow over heterogeneous surface features local convective instabilities and upward eddy heat transport, although the overall stratification remains stable with downward mean heat transfer. The DNS results are compared to the results from large-eddy simulations of weakly stable boundary layers (Mironov and Sullivan in J Atmos Sci 73:449–464, 2016). The DNS findings corroborate the key role of temperature variance in setting the structure and transport properties of stably stratified flow over heterogeneous surfaces, and the importance of third-order transport of the temperature variance. [ABSTRACT FROM AUTHOR]

A persistent spatial organization of eddies is identified in the lowest portion of the stably stratified planetary boundary layer. The analysis uses flow realizations from published large-eddy simulations (Sullivan et al. in J Atmos Sci 73(4):1815–1840, 2016) ranging in stability from near-neutral to almost z-less stratification. The coherent turbulent structure is well approximated as a series of uniform momentum zones (UMZs) and uniform temperature zones (UTZs) separated by thin layers of intense gradients that are significantly greater than the mean. This pattern yields stairstep-like instantaneous flow profiles whose shape is distinct from the mean profiles that emerge from long-term averaging. However, the scaling of the stairstep organization is closely related to the resulting mean profiles. The differences in velocity and temperature across the thin gradient layers remain proportional to the surface momentum and heat flux conditions regardless of stratification. The vertical thickness of UMZs and UTZs is proportional to height above the surface for near-neutral and weak stratification, but becomes thinner and less dependent on height as the stability increases. Deviations from the logarithmic mean profiles for velocity and temperature observed under neutral conditions are therefore predominately due to the reduction in eddy size with increasing stratification, which is empirically captured by existing Monin–Obukhov similarity relations for momentum and heat. The zone properties are additionally used to explain trends in the turbulent Prandtl number, thus providing a connection between the eddy organization, mean profiles, and turbulent diffusivity in stably stratified conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This is the report of an investigation in partnership with a team of junior secondary classroom teachers of the potential of posing more challenging tasks to their students. A range of data sources are used to represent the experience of teachers and students in a unit of work that incorporated a range of challenging tasks. Students learned the intended mathematics and they reacted positively to the challenges. The teachers identified both opportunities and constraints in such teaching, especially those related to changes to their pedagogies and assessment of student work. [ABSTRACT FROM AUTHOR]

Insights into teachers’ planning of mathematics reported here were gathered as part of a broader project examining aspects of the implementation of the Australian curriculum in mathematics (and English). In particular, the responses of primary and secondary teachers to a survey of various aspects of decisions that inform their use of curriculum documents and assessment processes to plan their teaching are discussed. Teachers appear to have a clear idea of the overall topic as the focus of their planning, but they are less clear when asked to articulate the important ideas in that topic. While there is considerable diversity in the processes that teachers use for planning and in the ways that assessment information informs that planning, a consistent theme was that teachers make active decisions at all stages in the planning process. Teachers use a variety of assessment data in various ways, but these are not typically data extracted from external assessments. This research has important implications for those responsible for supporting teachers in the transition to the Australian Curriculum: Mathematics. [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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A large-eddy simulation is modified to include multiple scalars emitted by a plant canopy. Each of these scalars is subjected to varying rates of chemical loss. Presented is a detailed comparison between conserved species and species undergoing first- and second-order chemical loss. Profiles of mean mixing ratio, mixing-ratio variance and vertical mixing-ratio flux reveal the influence of chemical reactivity. Distribution of the scalar source through the depth of the canopy is shown to locally reduce the reaction rate for second-order species. Transport efficiencies, diffusion coefficients, and mean source heights also exhibit chemical dependencies. Budgets of mixing-ratio variance and flux elucidate the mechanisms through which chemistry modifies each. Instantaneous fields show the existence of intermittently occurring coherent structures that are thought to enhance species segregation. [ABSTRACT FROM AUTHOR]

In this work, we study the characteristics of a stably stratified atmospheric boundary layer using large-eddy simulation (LES). In order to simulate the stable planetary boundary layer, we developed a modified version of the two-part subgrid-scale model of Sullivan et al. This improved version of the model is used to simulate a highly cooled yet fairly windy stable boundary layer with a surface heat flux of < wθ >o = -0.05 m K s-1 and a geostrophic wind speed of Ug = 15 m s-1. Flow visualization and evaluation of the turbulence statistics from this case reveal the development of a continuously turbulent boundary layer with small-scale structures. The stability of the boundary layer coupled with the presence of a strong capping inversion results in the development of a dominant gravity wave at the top of the stable boundary layer that appears to be related to the most unstable wave predicted by the Taylor–Goldstein equation. As a result of the decay of turbulence aloft, a strong-low level jet forms above the boundary layer. The time dependent behaviour of the jet is compared with Blackadar's inertial oscillation analysis. [ABSTRACT FROM AUTHOR]

Turbulent statistics of neutrally stratified shear-driven flow within and above a sparse forest canopy are presented from a large-eddy simulation (LES) and compared with those from observations within and above a deciduous forest with similar height and foliage density. First- and second-order moments from the LES agree with observations quite well. Third-order moments from the LES have the same sign and similar vertical patterns as those from the observations, but the LES yields smaller magnitudes of such higher-order moments. Turbulent spectra and cospectra from the LES agree well with observations above the forest. However, at the highest frequencies, the LES spectra have steeper slopes than observations. Quadrant and conditional analyses of the LES resolved-scale flow fields also agree with observations. For example, both LES and observation find that sweeps are more important than ejections for the transport of momentum within the forest, while inward and outward interaction contributions are both small, except near the forest floor. The intermittency of the transport of momentum and scalar increases with depth into the forest. Finally, ramp structures in the time series of a passive scalar at multiple levels within and above the forest show similar features to those measured from field towers. Two-dimensional (height-time cross-section) contours of the passive scalar and wind vectors show sweeps and ejections, and the characteristics of the static pressure perturbation near the ground resemble those deduced from field tower-based measurements. In spite of the limited grid resolution (2 m × 2 m × 2 m) and domain size (192 m × 192 m × 60 m) used in this LES, we demonstrate that the LES is capable of resolving the most important characteristics of the turbulent flow within and above a forest canopy. [ABSTRACT FROM AUTHOR]