California State University (CSU): Open Journal Systems
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Polynomial Generalizations of Knot Colorings
Get PDFIn the field of knot theory, knot invariants are properties preserved across all embeddings and projections of the same knot. Fox n-coloring is a classical knot invariant which associates to each knot projection a system of linear equations. We generalize Fox’s n-coloring by using two, not necessarily distinct, polynomials over a field F, which we say form a (g,f)F coloring. We introduce a sufficient condition, called strong, for a pair of polynomials to form a (g,f)F coloring. We confirm a family of pairs of linear polynomials each of which form a (g,f)F coloring. We prove that there are no strong pairs containing an irreducible quadratic polynomial over a field F not of characteristic two. Furthermore, we find a method to produce polynomials with unbounded degree that form colorings over suitable fields
Pathways and Pitfalls on the Road to Equity in Mathematics Classrooms
Get PDFThe article describes the foundational principles and central design features of the MathPathways & Pitfalls (MPP) intervention curriculum for elementary grades and illustrates challenges increating MPP-like middle school algebra readiness materials. We describe key design features and discusshow they support language, discourse, and equity development. The article offers vignettes created fromrecent middle school classroom observations and teacher interviews to illustrate the particular instructionalpathways and pitfalls emerging in the pilot of MPP-like algebra readiness student and teacher materials
Dense Orbits of a Two Point Algebra
Get PDFFinite dimensional algebras over a field can be classified into three classes by representation type. These are finite, tame, and wild. Wild algebras contain copies of the representations of all finite dimensional algebras so classifying the indecomposable representations is not feasible. We then look at representations whose orbit is dense in their associated irreducible component of their representation variety. With this geometric perspective, we can approximate our representations. We give an example of a two point dense orbit algebra of wild representation type and show that it has a dense orbit for dimension vectors of certain types
An IFS for the Stretched Level-n Sierpinski Gasket
Get PDFBroadly, fractals are sets that exhibit a repeating pattern at multiple scales. One important fractal set is the Sierpinski Gasket (SG) which is made up of nested equilateral triangles. A variation of the classic Sierpinski gasket is to create n-levels with the equilateral triangles. Another variation is to stretch the points of intersection for the triangles in SG into line segments of length 0 < α < 1/3. When one combines these variations, one arrives at the stretched level-n Sierpinski gaskets (SSGn) which are the focus of this work. We give an introduction to iterated function systems (IFS) and determine an IFS which generates SSG3. We then describe how one can acquire the IFS for SSGn in general, and conclude with a theorem which determines the Hausdorff dimension for SSGn
Blowup of Solutions to a Damped Euler Equation with Homogeneous Three-Point Boundary Condition
Get PDFIt is known that solutions to the inviscid Proudman-Johnson equation subject to a homogeneous three-point boundary condition can develop singularities in finite time. Given the regularizing effect that damping can have, in this paper we consider the possibility of finite-time singularity formation in solutions of the generalized inviscid Proudman-Johnson equation with damping subject to the same homogeneous three-point boundary condition. In particular, we will derive criteria the initial condition must satisfy in order for solutions to blowup in finite time under the effect of a smooth time-dependent damping term.  
Experiential Learning Theory as a Guide for Experiential Educators in Higher Education
No full textCore concepts of Experiential Learning Theory—the learning cycle,learning style, and learning space—have been widely used by experiential educatorsin higher education for nearly half a century. We examine the latest thinkingabout these three concepts and highlight some exemplary applications from themany disciplinary applications of experiential learning in higher education.
I think that only slight acquaintance with the history of education is needed to provethat educational reformers and innovators alone have felt the need for a philosophyof education. Those who adhered to the established system needed merely a few fine soundingwords to justify existing practices. The real work was done by habits whichwere so fixed as to be institutional. The lesson for progressive education is that itrequires in an urgent degree, a degree more pressing than was incumbent upon formerinnovators, a philosophy of education based on a philosophy of experience.
John Dewey, Experience and Educatio
Reframing Experiential Education : A Broader Perspective of Community Engagement
No full textThis article invites the reader to reframe the traditional perspective of experiential education to a broader conceptualization of community engagement in which various stakeholders, in addition to students, are the beneficiaries of the learning experience. In addition to acknowledging and celebrating the pedagogical approach, this narrative also provides a friendly critique of our traditional and perhaps somewhat limited perspective ofexperiential education. Challenges and potential detrimental impact are considered, coupled with approaches on how to minimize those issues
