Multiple Visions of Teachers\u27 Understandings of Mathematics

In this dialog, the notion of mathematical understanding as might be needed by classroom teachers is critically examined by mathematics educators, mathematicians, and a classroom teacher, based on the outcomes of recent work with expert classroom teachers. Terminology, assumptions and examples are discussed and analysed from a number of points of view. Ultimately, the goal is to construct common ground from which appropriate mathematics courses for future teachers might be developed and taught. The need for common terminology and a unifying framework from which to work becomes apparent as multiple interpretations and visions are discussed

Using iron oxide nanoparticles to diagnose CNS inflammatory diseases and PCNSL

Abstract OBJECTIVE: The study goal was to assess the benefits and potential limitations in the use of ultrasmall superparamagnetic iron oxide (USPIO) nanoparticles in the MRI diagnosis of CNS inflammatory diseases and primary CNS lymphoma. METHODS: Twenty patients with presumptive or known CNS lesions underwent MRI study. Eighteen patients received both gadolinium-based contrast agents (GBCAs) and 1 of 2 USPIO contrast agents (ferumoxytol and ferumoxtran-10) 24 hours apart, which allowed direct comparative analysis. The remaining 2 patients had only USPIO-enhanced MRI because of a renal contraindication to GBCA. Conventional T1- and T2-weighted MRI were acquired before and after contrast administration in all patients, and perfusion MRI for relative cerebral blood volume (rCBV) assessment was obtained in all 9 patients receiving ferumoxytol. RESULTS: USPIO-enhanced MRI showed an equal number of enhancing brain lesions in 9 of 18 patients (50%), more enhancing lesions in 2 of 18 patients (11%), and fewer enhancing lesions in 3 of 18 patients (17%) compared with GBCA-enhanced MRI. Four of 18 patients (22%) showed no MRI enhancement. Dynamic susceptibility-weighted contrast-enhanced perfusion MRI using ferumoxytol showed low rCBV (ratio <1.0) in 3 cases of demyelination or inflammation, modestly elevated rCBV in 5 cases of CNS lymphoma or lymphoproliferative disorder (range: 1.3-4.1), and no measurable disease in one case. CONCLUSIONS: This study showed that USPIO-enhanced brain MRI can be useful in the diagnosis of CNS inflammatory disorders and lymphoma, and is also useful for patients with renal compromise at risk of nephrogenic systemic fibrosis who are unable to receive GBCA

Indigenous Maths, Global Math, and Indigenizing Mathematics

Current Truth and Reconciliation efforts around mathematics generally begin at the wrong place, with the nearly universal and conventional mathematics (Global Math) commonly taught in schools and universities. To truly succeed in Reconciliation, we must begin with Indigenous Maths, with Creation even, and we must carefully and critically explore the relationships between the many Indigenous Maths, the one Global math, and Indigenizing mathematics

Panel: Looking in

Panelists discuss what they see and hope for in university mathematics instruction, from their particular â outsider-insiderâ perspectives. Ed Doolittle, Kari Marken, Rina ZazkisNon UBCUnreviewedAuthor affiliation: First Nations University of CanadaFacult

A parametrix for stable step two hypoelliptic partial differential operators

grantor: University of TorontoThe purpose of this thesis is to calculate a parametrix for step two hypoelliptic partial differential operators which are defined on a manifold and satisfy a stability hypothesis. The calculations are based on the classical Hadamard-Kodaira method for elliptic operators which involves reducing the problem to solving a first order nonlinear Hamilton-Jacobi equation and first order linear transport equations. However, for the method to succeed in the step two hypoelliptic case, two major modifications must be made: first, the parametrix must be expressed as an integral over the characteristic variety in which case the integrand rather than the parametrix itself is reduced: second, in order to solve the Hamilton-Jacobi equation by the classical action integral method, complex initial conditions must be supplied to the Hamiltonian system. These two variations on the classical procedure together introduce numerous technical difficulties. The procedure is first applied to find a fundamental solution for the left-invariant model case. The results for the model case are then perturbed to give a parametrix for operators satisfying the hypotheses which are sums of squares of vector fields. Finally, the results are extended to operators of the above type with lower order terms.Ph.D