Views of cognition: different lenses for ‘looking-in’ on classrooms

The evolution of different theories of cognition over the years has given mathematics education researchers new tools for highlighting particular characteristics of classroom issues to enable more detailed investigation of how students learn mathematics. In the past there has been a strong dominance of the dualistic view separating body and mind. In recent times, however, the body has been given a more central role in shaping the mind. This has led to the situation where some long-standing conundrums of mathematics education have become more tractable to researchers. Similarly, using older ideas in new ways, light is being shone on what is possible at different stages of development or different levels of schooling. In this issue different views of cognition have been mined by the authors of different articles to frame studies and analyses or invent and apply new tools. They bring new lenses for looking-in on classrooms, a fresh view with old lenses and new methodological tools to the fore. Using a small selection of the articles of the issue the empowerment that differing views of cognition have enabled for mathematics education research is demonstrated.     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

Resonant Hopf–Hopf Interactions in Delay Differential Equations

A second-order delay differential equation (DDE) which models certain mechanical and neuromechanical regulatory systems is analyzed. We show that there are points in parameter space for which 1:2 resonant Hopf–Hopf interaction occurs at a steady state of the system. Using a singularity theoretic classification scheme [as presented by LeBlanc (1995) and LeBlanc and Langford (1996)], we then give the bifurcation diagrams for periodic solutions in two cases: variation of the delay and variation of the feedback gain near the resonance point. In both cases, period-doubling bifurcations of periodic solutions occur, and it is argued that two tori can bifurcate from these periodic solutions near the period doubling point. These results are then compared to numerical simulations of the DDE.     

Nonpremixed bluff-body burner flow and flame imaging study

A nonpremixed bluff-body burner flow and flame have been studied using planar flow visualization and species concentration imaging techniques. The burner consists of a central jet of CH ₄ in a cylindrical bluff-body and an outer coflowing-air stream. Planar flow visualization, using Mie scattering from seed particles added to the fuel jet, Raman scattering from CH ₄ and laser-induced fluorescence of CH combined with Raman scattering of CH ₄ provided information on turbulent flow, mixing and combustion. The CH ₄ imaging system utilized two cameras, which enhanced the dynamic range of the diagnostic system by a factor of 10 over a single-camera system. It was observed that the fuel jet stagnated on the axis due to interaction with the high velocity air flow. The flow and mixing were found to have significant coherent and noncoherent, large-scale, time-varying structures. The detailed CH ₄ Raman and CH fluorescence measurements of an air-dominated bluff-body flame revealed that the stagnation zone governs mixing and flame stability. Through large-scale mixing, the stagnated jet feeds the recirculation zone and also creates a favorable condition to stabilize the flame detached from the bluff-body. The instantaneous flame zone, as defined by CH, was found to be narrow and concentrated in an envelope around the stagnation zone. This narrow flame characteristic is consistent with that observed for jet flames. Although the internal structure of the flame envelops have not yet been defined, these results suggest that this bluff-body flame can be modeled by a flame sheet type approach, where the reaction front is captured by the large-scale structures. This should simplify the development of modeling approaches for these flows since molecular mixing and chemical reaction, which occur within the flame sheet, can be separated from the large-scale mixing process.     

One-step and multi-step linear equations: a content analysis of five textbook series

Being able to solve one-step and multi-step linear equations is a hallmark of algebraic proficiency in the United States and abroad. The purpose of this paper is to report on a textbook comparison study regarding the treatment of linear equations in five textbook series that are popular in the United States: Center for Mathematics Education Project Algebra 1 and Algebra 2 (CME), Core-Plus Mathematics Program Courses 1–3 (CPMP), Glencoe Algebra 1 and Algebra 2, Interactive Mathematics Program Years 1–3 (IMP), and University of Chicago School Mathematics Project Algebra and Advanced Algebra (UCSMP). Data are reported for the following curriculum variables: content, cognitive behavior, real-world context, and tools (technology and manipulatives). Sequencing of the content, and links between symbolic and functions-based approaches are discussed. Based on the data presented, what students experience relative to setting up and solving one-step and multi-step linear equations is likely to be markedly different, depending on which textbook is used in their classrooms. Implications for practice and ideas for further research are discussed.