Browsing by Subject "Algebra"

Now showing 1 - 3 of 3
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    Multiple representations and rate of change: the nature of diverse Students' Initial Understanding.
    (2010-02) Whitney, Stephanie Rae
    Access to quality mathematics content and instruction has been equated as a civil right (Moses & Cobb, 2001), as it is a gatekeeper to higher education and lucrative careers. Unfortunately, access to empowering mathematics is not a reality for many urban youth. Data show an achievement gap between Black, Hispanic and American Indian students and their peers which, in turn, result in unequal access to education and economic opportunities (NAEP, 2007; U.S. Department of Education, 1997). This evaluative case study focused on eight racially diverse high school students from an urban charter school who were studying Algebra 1 in classroom contexts that were student-centered and discourse-based. The study took place after a sequence of six lessons that introduced representations of linear functions through geometric patterns. Pairs of students participated in a series of interview tasks, largely modeled from problems in their curriculum Algebra Connections (Dietiker, Kysh, Sallee, Hoey, 2005), that focused on their understanding of representations of functions (tables, graphs, equations and context), the embodiment of rate of change in the representations and translating among the representations. The study sought to understand the nature of the students’ initial understandings of representations and translations including their natural language when describing observations, the perspectives from which they approached the tasks (explicit or recursive) and how their thinking compares to literature in the field. The students were able to translate linear functions directly between any two representations and many were able to extend their thinking to novel problems involving non-linear functions.
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    Parametrizations of Irreducible Rational Representations of Coxeter Groups
    (2024-04) Corsi, Craig
    We develop two closely related methods for parametrizing the rational irreducible characters of an arbitrary finite Coxeter group W. The goal is to provide a uniform approach to such a parametrization, independent of Coxeter type. The two methods generalize two approaches to describing the irreducible representations of the symmetric groups, which coincide in type A but do not coincide generally. Our methods associate characters to pairs of reflection subgroups, in one case by considering common constituents of permutation and signed permutation modules, and in the other case by a generalization of the Specht modules. We ask whether, using either method, we can identify a set of subgroup pairs for which the matrix of multiplicities of rational irreducibles in the characters associated to these subgroup pairs is unitriangular. Such a unitriangular matrix provides a parametrization of the irreducible rational characters. For the noncrystallographic types H and I, we give a positive answer to this question. In type H, we show computationally that we can parametrize the irreducible rational characters of H4 using generalized Specht modules, and we can parametrize the irreducible characters of H3 using both methods. Moreover, we give an explicit decomposition of the generalized common constituents for the dihedral groups I2(n) for all n, and we prove that we can always exhibit a unitriangular multiplicity matrix using generalized common constituents. In type A our theory coincides with the classical theory of Specht modules and does not give any new information. In type B the approach we take is closely related to an existing parametrization of the irreducible characters, but it appears to have some novel elements.
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    A Proposed Algebra Problem-Analysis Model
    (2015-08) Walick, Christopher
    The National Mathematics Advisory Panel (2008) states that algebra is a gateway to high school graduation and college success. While existing research emphasizes the importance of quality algebra instruction, the current body of research on algebra problem-analysis for struggling secondary students is small. This paper proposes a problem-solving model to help support those students struggling with algebra. The model integrates the recommendations from math policy boards and research. It is composed of five core sections, each section focusing on a specific critical component of school algebra. The study examines the relationship between the five skills within the model to an established measure of algebra, as well as the validity of the measures being used to assess the different skill areas The results indicate that there is a significant relationship between the five sections of the model and algebra proficiency, and that the model is able to identify non-proficiency students with a high degree of accuracy.