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

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Massey University
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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.
Transformations (Mathematics), Geometry, Study and teaching