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    Intuitive transformation geometry and frieze patterns : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics Education at Massey University
    (Massey University, 1992) Kelly, Daniel Joseph
    This study on intuitive frieze pattern construction and description was set up as an attempt to answer part of a general question: "Do students bring intuitive transformation geometry concepts with them into the classroom and, if so, what is the character of those concepts?" The motivation to explore this topic arose, in part, from the particular relevance that transformation geometry has to New Zealand: kowhaiwhai (Maori rafter patterns) are examples of frieze patterns and are suggested by recent mathematics curriculum documents as a way for Form 3 and 4 students to explore transformations. When very few restrictions were put on the subjects, frieze patterns made by Standard 3 and 4 students displayed evidence of the use of transformations such as translation, vertical reflection, and half-turn. Transformations, such as horizontal reflection and glide reflection, were very rarely used by themselves. However, from the frieze group analysis alone, no strong conclusions could be drawn about the frieze patterns featuring a combination of two or more different symmetry types (besides translation). The Form 4 class surveyed showed similar results, with an increase in the proportion of students using half-turn by itself. Another contrast between the two age groups was the production of disjoint and connected patterns: the Primary students' patterns were mostly disjoint, whereas the Secondary students made almost equal numbers of disjoint and connected designs. In a restricted frieze construction activity, which required the subjects to use asymmetric objects (right-angled scalene triangles), the use of non-translation transformations reduced considerably from the first exercise, although vertical reflection was still popular amongst 70% of the Primary students. However, the results of a small survey of 10 children suggested that if the strips to be filled in are aligned vertically, the rarer symmetries such as glide reflection may be used more easily than in the horizontal case. The style analysis revealed that the Primary (pre-formal) and Tertiary (post-formal) groups were quite similar in the patterns they drew under the restricted conditions, and therefore in the probable construction methods used to produce them. The Form 4's patterns differed in several ways, especially by their extensive use of half turn and tilings. It seems that the Fourth Form students were affected by the formal transformation geometry framework to which they had been recently exposed. Interviews of 10 Primary students provided information about the intentions and methods used to construct the frieze patterns under both restricted and unrestricted conditions. The case studies revealed that several standard approaches to frieze pattern construction were employed, none of which corresponded with the mathematical structure of a symmetry group. It was also found that a number of methods could be used to make the same pattern. The qualitative analysis highlighted some shortfalls of the quantitative approach. For example, some students used transformations not detected by the frieze group analysis, and some symmetries present in the children's patterns were incidental (a spin-off of another motivation) or accidental. Ambiguities in pattern classification also arose. The Primary children's descriptions of the seven different frieze groups (which were discrete examples) displayed several characteristic features. For instance, they often used a form of simile or metaphor, comparing a pattern pan to a real world object with the same set of symmetries. In addition, many children considered a pattern's translation unit to be 'the pattern'. In this case, the interviews suggested that the repetition (translation) was obvious to the students. Also interesting was the tendency of these subjects to write down orientation or direction judgements, omitting the relationships between adjacent congruent figures within a pattern. However, the Primary children did use more explicit transformation terminology when able to describe the patterns orally. A peculiar feature of these explanations was that the symmetry described was often not differentiable from another symmetry. For example, to a child, the phrase "turn upside down" can mean a half-turn or a horizontal reflection or both; the result is identical in many cases. Secondary and Tertiary students tended not to use implicit phrases in their pattern descriptions but were more explicit and precise, using a wider range of criteria in their descriptions. The results from this activity also indicated that the Primary and older students alike did not perceive the patterns to extend infinitely beyond the confines of the the page, highlighting another difference between the mathematical structure of a symmetry group and the intuitive cognitive processes of the students. An additional matching activity was conducted in the interviews, requiring the subjects to match various pairs of frieze patterns and discuss the similarities they saw. It appeared that transformation criteria were not verbalized predominantly over other criteria such as orientation or direction judgements, although many matches were made between patterns with the same underlying frieze group. Finally, educational implications for mathematics were indicated and areas for further research were suggested.
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    On the geometry of generalized linear models : submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Luo, Dongwen
    The perspective afforded by Euclidean geometry led to the rapid development of linear models in the early stages of the twentieth century: Fisher saw the data as a point in finite-dimensional Euclidean space, the model as a subspace and least squares fitting as projection of the observation vector onto the model space. From the late 1960s to early 1970s, Fienberg revealed geometry underlying loglinear models for two-way tables, while Haberman discussed geometry for the log-transformed case. Generalized linear models, however, have largely eluded geometers until recently. In 1997 an extension of Fisher's view to generalized linear models was given by Kass and Vos, using the language of differential geometry. The aim of this work is to develop a simple, general geometric framework for generalized linear models, closely related to the thinking of Fienberg and Haberman. Whereas Kass and Vos developed a geometric view which leads to the usual scoring method, we develop geometry which leads to a new algorithm. A linearization of this new algorithm yields the scoring method. The geometry discussed by Kass and Vos is based on the log-likelihood function whereas the geometry developed here depends on sufficiency. In the geometry of generalized linear models, developed through chapters 1 to 3, an observation with n values is viewed as a vector in Euclidian space Rn. This Euclidian space Rn is partitioned into two orthogonal spaces, the sufficiency space S and the auxiliary space A, with respect to a new basis. We focus on two mean sets relating to generalized linear models, one for the untransformed model space and another for the link-transformed model space. There are two critical properties of the maximum likelihood estimate of the parameters of a generalized linear model with canonical link. The first property is that the coefficients of the basis of the sufficiency space, the sufficient statistics, are preserved in the untransformed model space in the fitting process. The second property is that the coefficients of the basis of the auxiliary space are zeroed in the link-transformed model space in the fitting process. Linear models and loglinear models serve as special cases of generalized linear models with identity and log link respectively. Based on the geometric framework discussed in the thesis, a new algorithm is constructed for fitting generalized linear models with canonical link in Chapter 4. This algorithm, which relies on sufficient statistics for the parameters in the model rather than the likelihood function, takes two projections alternately, orthogonal projection onto a sufficiency affine plane and non-orthogonal projection onto the transformed model space. In the process, we match the model space and sufficient statistics iteratively until convergence. Linearization of the new algorithm induces the scoring method. In Chapter 5 we pay special attention to a subset of loglinear models, graphical loglinear models, those which are the intersection of a finite set of conditional independence statements. The model space of one conditional independence statement is described through the notions of "corresponding point convex hull" and "set convex hull". The fitting of one conditional independence statement is considered geometrically using a direct fitting method and the familiar iterative proportional fitting method.