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Description
Research on combinatorics education is sparse when compared with other fields in mathematics education. This research attempted to contribute to the dearth of literature by examining students' reasoning about enumerative combinatorics problems and how students conceptualize the set of elements being counted in such problems, called the solution set. In particular, the focus was on the stable patterns of reasoning, known as ways of thinking, which students applied in a variety of combinatorial situations and tasks. This study catalogued students' ways of thinking about solution sets as they progressed through an instructional sequence. In addition, the relationships between the catalogued ways of thinking were explored. Further, the study investigated the challenges students experienced as they interacted with the tasks and instructional interventions, and how students' ways of thinking evolved as these challenges were overcome. Finally, it examined the role of instruction in guiding students to develop and extend their ways of thinking. Two pairs of undergraduate students with no formal experience with combinatorics participated in one of the two consecutive teaching experiments conducted in Spring 2012. Many ways of thinking emerged through the grounded theory analysis of the data, but only eight were identified as robust. These robust ways of thinking were classified into three categories: Subsets, Odometer, and Problem Posing. The Subsets category encompasses two ways of thinking, both of which ultimately involve envisioning the solution set as the union of subsets. The three ways of thinking in Odometer category involve holding an item or a set of items constant and systematically varying the other items involved in the counting process. The ways of thinking belonging to Problem Posing category involve spontaneously posing new, related combinatorics problems and finding relationships between the solution sets of the original and the new problem. The evolution of students' ways of thinking in the Problem Posing category was analyzed. This entailed examining the perturbation experienced by students and the resulting accommodation of their thinking. It was found that such perturbation and its resolution was often the result of an instructional intervention. Implications for teaching practice are discussed.
ContributorsHalani, Aviva (Author) / Roh, Kyeong Hah (Thesis advisor) / Fishel, Susanna (Committee member) / Saldanha, Luis (Committee member) / Thompson, Patrick (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2013
ContributorsShi, Ge (Performer) / ASU Library. Music Library (Publisher)
Created2018-03-25
ContributorsShatuho, Kristina (Performer) / ASU Library. Music Library (Publisher)
Created2018-03-27
Description
A new algebraic system, Test Algebra (TA), is proposed for identifying faults in combinatorial testing for SaaS (Software-as-a-Service) applications. In the context of cloud computing, SaaS is a new software delivery model, in which mission-critical applications are composed, deployed, and executed on cloud platforms. Testing SaaS applications is challenging because new applications need to be tested once they are composed, and prior to their deployment. A composition of components providing services yields a configuration providing a SaaS application. While individual components
in the configuration may have been thoroughly tested, faults still arise due to interactions among the components composed, making the configuration faulty. When there are k components, combinatorial testing algorithms can be used to identify faulty interactions for t or fewer components, for some threshold 2 <= t <= k on the size of interactions considered. In general these methods do not identify specific faults, but rather indicate the presence or absence of some fault. To identify specific faults, an adaptive testing regime repeatedly constructs and tests configurations in order to determine, for each interaction of interest, whether it is faulty or not. In order to perform such testing in a loosely coupled distributed environment such as
the cloud, it is imperative that testing results can be combined from many different servers. The TA defines rules to permit results to be combined, and to identify the faulty interactions. Using the TA, configurations can be tested concurrently on different servers and in any order. The results, using the TA, remain the same.
in the configuration may have been thoroughly tested, faults still arise due to interactions among the components composed, making the configuration faulty. When there are k components, combinatorial testing algorithms can be used to identify faulty interactions for t or fewer components, for some threshold 2 <= t <= k on the size of interactions considered. In general these methods do not identify specific faults, but rather indicate the presence or absence of some fault. To identify specific faults, an adaptive testing regime repeatedly constructs and tests configurations in order to determine, for each interaction of interest, whether it is faulty or not. In order to perform such testing in a loosely coupled distributed environment such as
the cloud, it is imperative that testing results can be combined from many different servers. The TA defines rules to permit results to be combined, and to identify the faulty interactions. Using the TA, configurations can be tested concurrently on different servers and in any order. The results, using the TA, remain the same.
ContributorsQi, Guanqiu (Author) / Tsai, Wei-Tek (Thesis advisor) / Davulcu, Hasan (Committee member) / Sarjoughian, Hessam S. (Committee member) / Yu, Hongyu (Committee member) / Arizona State University (Publisher)
Created2014
ContributorsCarlisi, Daniel (Performer) / ASU Library. Music Library (Publisher)
Created2018-04-07
Description
Yannis Constantinidis was the last of the handful of composers referred to collectively as the Greek National School. The members of this group strove to create a distinctive national style for Greece, founded upon a synthesis of Western compositional idioms with melodic, rhyhmic, and modal features of their local folk traditions. Constantinidis particularly looked to the folk melodies of his native Asia Minor and the nearby Dodecanese Islands. His musical output includes operettas, musical comedies, orchestral works, chamber and vocal music, and much piano music, all of which draws upon folk repertories for thematic material. The present essay examines how he incorporates this thematic material in his piano compositions, written between 1943 and 1971, with a special focus on the 22 Songs and Dances from the Dodecanese. In general, Constantinidis's pianistic style is expressed through miniature pieces in which the folk tunes are presented mostly intact, but embedded in accompaniment based in early twentieth-century modal harmony. Following the dictates of the founding members of the Greek National School, Manolis Kalomiris and Georgios Lambelet, the modal basis of his harmonic vocabulary is firmly rooted in the characteristics of the most common modes of Greek folk music. A close study of his 22 Songs and Dances from the Dodecanese not only offers a valuable insight into his harmonic imagination, but also demonstrates how he subtly adapts his source melodies. This work also reveals his care in creating a musical expression of the words of the original folk songs, even in purely instrumental compositon.
ContributorsSavvidou, Dina (Author) / Hamilton, Robert (Thesis advisor) / Little, Bliss (Committee member) / Meir, Baruch (Committee member) / Thompson, Janice M (Committee member) / Arizona State University (Publisher)
Created2011
Description
This paper describes six representative works by twentieth-century Chinese composers: Jian-Zhong Wang, Er-Yao Lin, Yi-Qiang Sun, Pei-Xun Chen, Ying-Hai Li, and Yi Chen, which are recorded by the author on the CD. The six pieces selected for the CD all exemplify traits of Nationalism, with or without Western influences. Of the six works on the CD, two are transcriptions of the Han Chinese folk-like songs, one is a composition in the style of the Uyghur folk music, two are transcriptions of traditional Chinese instrumental music dating back to the eighteenth century, and one is an original composition in a contemporary style using folk materials. Two of the composers, who studied in the United States, were strongly influenced by Western compositional style. The other four, who did not study abroad, retained traditional Chinese style in their compositions. The pianistic level of difficulty in these six pieces varies from intermediate to advanced level. This paper includes biographical information for the six composers, background information on the compositions, and a brief analysis of each work. The author was exposed to these six pieces growing up, always believing that they are beautiful and deserve to be appreciated. When the author came to the United States for her studies, she realized that Chinese compositions, including these six pieces, were not sufficiently known to her peers. This recording and paper are offered in the hopes of promoting a wider familiarity with Chinese music and culture.
ContributorsLuo, Yali, D.M.A (Author) / Hamilton, Robert (Thesis advisor) / Campbell, Andrew (Committee member) / Pagano, Caio (Committee member) / Cosand, Walter (Committee member) / Rogers, Rodney (Committee member) / Arizona State University (Publisher)
Created2012
Description
The purpose of this project was to examine the lives and solo piano works of four members of the early generation of female composers in Taiwan. These four women were born between 1950 and 1960, began to appear on the Taiwanese musical scene after 1980, and were still active as composers at the time of this study. They include Fan-Ling Su (b. 1955), Hwei-Lee Chang (b. 1956), Shyh-Ji Pan-Chew (b. 1957), and Kwang-I Ying (b. 1960). Detailed biographical information on the four composers is presented and discussed. In addition, the musical form and features of all solo piano works at all levels by the four composers are analyzed, and the musical characteristics of each composer's work are discussed. The biography of a fifth composer, Wei-Ho Dai (b. 1950), is also discussed but is placed in the Appendices because her piano music could not be located. This research paper is presented in six chapters: (1) Prologue; the life and music of (2) Fan-Ling Su, (3) Hwei-Lee Chang, (4) Shyh-Ji Pan-Chew, and (5) Kwang-I Ying; and (6) Conclusion. The Prologue provides an overview of the development of Western classical music in Taiwan, a review of extant literature on the selected composers and their music, and the development of piano music in Taiwan. The Conclusion is comprised of comparisons of the four composers' music, including their personal interests and preferences as exhibited in their music. For example, all of the composers have used atonality in their music. Two of the composers, Fan-Ling Su and Kwang-I Ying, openly apply Chinese elements in their piano works, while Hwei-Lee Chang tries to avoid direct use of the Chinese pentatonic scale. The piano works of Hwei-Lee Chang and Shyh-Ji Pan-Chew are chromatic and atonal, and show an economical usage of material. Biographical information on Wei-Ho Dai and an overview of Taiwanese history are presented in the Appendices.
ContributorsWang, Jinding (Author) / Pagano, Caio (Thesis advisor) / Campbell, Andrew (Committee member) / Humphreys, Jere T. (Committee member) / Meyer-Thompson, Janice (Committee member) / Norton, Kay (Committee member) / Arizona State University (Publisher)
Created2011
ContributorsShi, Zhan (Performer) / ASU Library. Music Library (Publisher)
Created2018-03-26
Description
Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both in structure and in the types of objects that they list. More specific types of Gray codes are universal cycles and overlap sequences. Universal cycles are Gray codes on a set of strings of length n in which the first n-1 letters of one object are the same as the last n-1 letters of its predecessor in the listing. Overlap sequences allow this overlap to vary between 1 and n-1. Some of our main contributions to the areas of Gray codes and universal cycles include a new Gray code algorithm for fixed weight m-ary words, and results on the existence of universal cycles for weak orders on [n]. Overlap cycles are a relatively new structure with very few published results. We prove the existence of s-overlap cycles for k-permutations of [n], which has been an open research problem for several years, as well as constructing 1- overlap cycles for Steiner triple and quadruple systems of every order. Also included are various other results of a similar nature covering other structures such as binary strings, m-ary strings, subsets, permutations, weak orders, partitions, and designs. These listing structures lend themselves readily to some classes of combinatorial objects, such as binary n-tuples and m-ary n-tuples. Others require more work to find an appropriate structure, such as k-subsets of an n-set, weak orders, and designs. Still more require a modification in the representation of the objects to fit these structures, such as partitions. Determining when and how we can fit these sets of objects into our three listing structures is the focus of this dissertation.
ContributorsHoran, Victoria E (Author) / Hurlbert, Glenn H. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Colbourn, Charles (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2012