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Item Congruences in racks and quandles : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū Campus, Palmerston North, New Zealand(Massey University, 2022) Burrows, Wayne JohnRacks and quandles are related algebraic structures based on axioms of invertibility and self-distributivity, and in the case of quandles, an additional idempotence axiom — thus every quandle is a rack. They have practical application as the three quandle axioms algebraically encode the Reidemeister moves of knot theory. However, racks and quandles are interesting and worthy of study in their own right and that is what we do here. Congruences are a means of distilling patterns of behaviour within algebraic structures. They allow us to form a quotient that gives us a coarser view of the structure from which we can discern interesting properties. Congruences need to respect the operations in the algebraic structure. Racks, although often defined in terms of only one binary operation, necessarily, as a result of the invertibility axiom, have two binary operations — a primary rack operation and an inverse rack operation. We have a rack in the quotient only when the congruence respects both operations. A congruence that respects both operations we call a rack congruence or a quandle congruence. Congruences defined in terms of only one of the binary operations may not preserve the rack structure in the quotient. This raises the question of whether congruences that respect only one rack operation — half congruences — can exist. We show they can by constructing examples of half congruences that do not induce a rack in the quotient. For weighted average quandles on Q we completely characterise congruences in terms of certain subgroups of Q. Depending on the weight, congruences can exhibit one of three possible behaviours. Weighted average quandles are a special case of the more general Alexander quandle. For Alexander quandles, we characterise when a congruence induces an Alexander quandle in the quotient. In weighted average quandles every congruence comes from a subgroup of Q. In Alexander quandles, there are additional congruences that do not come from a subgroup. We give examples of congruences that exhibit that more complex behaviour.Item Word problems in teaching and learning algebra : a thesis presented in partial fulfilment of the requirements for the degree of Master of Educational Studies in Mathematics at Massey University, Palmerston North, New Zealand(Massey University, 2002) Bennett, Philip RThis research seeks to examine student understanding of algebra and how teachers facilitate algebraic learning for the purpose of improving learning outcomes. Based on the earlier work of Nathan and Koedinger (2000a; 2000c; 2001), the role of word problems in particular is investigated in relation to student development of algebraic understanding and technique. The year 10 students surveyed displayed particularly low levels of algebraic thinking and poor algebraic skill. The results show that as the structural complexity of problems increased, student understanding diminished and there was a clear shift in student choice of strategy. The use of calculators showed a significant increase in algebraic proficiency, supporting the view that beginning algebra students find it difficult to focus simultaneously on the algebraic and arithmetical aspects of problems. Story problems with result unknown and start unknown complexity solicited a greater proportion of informal strategies than equation problem counterparts. When students chose to use algebra, it was predominantly for problems in an equation format. The results indicate a disparity between what is being taught and what is being learned. This may be explained in part by the apparent philosophical conflict in teacher beliefs, where importance is placed both on achieving success in algebraic technique, and also on encouraging student driven solution methods. In order to capture student interest, teachers endorse the use of informal strategies by students through advocating word problems as applications of the real world and promoting a goal oriented approach to problem solving. Findings from this study suggest that in order to promote algebraic thinking teachers should present problems for which algebraic means of finding a solution is both preferred and optimal. Students should be made explicitly aware of the purpose for a particular set task, such as word problems, and monitored carefully in their choice of strategyItem Teaching and learning algebra word problems : a thesis presented in partial fulfilment of the requirements for the degree of Master of Educational Studies in Mathematics, Massey University, Palmerston North, New Zealand(Massey University, 2007) Lawrence, AnneThis study reports on a classroom design experiment into the teaching and learning of algebra word problems. The study was set in the mathematics department of a coeducational secondary school, and involved two teachers and 30 Year 12 students. The teachers and the researcher worked collaboratively to design and implement an intervention that focused explicitly on translation between word problems and algebra. Two issues were considered: the impact of the intervention on students, and the impact of the study on teachers. Students' responses to classroom activities, supported by individual student interviews, were used to examine their approaches to solving algebra word problems. Video-stimulated focus group interviews explored students' responses to classroom activities, and informed the ongoing planning and implementation of classroom activities. Data about the impact on teachers' understandings, beliefs and practices was gathered through individual interviews and classroom observations as well as the ongoing dialogue of the research team. The most significant impact on students related to their understandings of algebra as a tool. Some students were able to combine their new-found translation skills with algebraic manipulation skills to solve word problems algebraically. However, other students had difficulties at various stages of the translation process. Factors identified as supporting student learning included explicit objectives and clarity around what was to be learnt, the opportunity for students to engage in conversations about their thinking and to practise translating between verbal and symbolic forms, structured progression of learning tasks, time to consolidate understandings, and, a heuristic for problem solving. Participation in the project impacted on teachers in two ways: firstly, with regards to the immediate intervention of teaching algebra; and secondly, with regards to teaching strategies for mathematics in general. Translation activities provided a tool for teachers to engage students in mathematical discussion, enabling them to elicit and build on student thinking. As teachers developed new understandings about how their students approached word problems they gained insight into the importance of selecting problems for which students needed to use algebra. However, teachers experienced difficulty designing quality instructional activities, including algebra word problems, that pressed for algebraic thinking. The focus on translation within the study encouraged a shift in teacher practice away from a skills-focus toward a problem-focus. Whilst it was apparent that instructional focus on translation shifted teachers and students away from an emphasis on procedure, it was equally clear that translation alone is insufficient as an intervention. Students need both procedural and relational understandings to develop an understanding of the use of algebra as a tool to solve word problems. Students also need to develop fluency with a range of strategies, including algebra, in order to be able to select appropriate strategies to solve particular problems. This study affirmed for teachers that teaching with a focus on understanding can provide an effective and efficient method for increasing students' motivation, interest and success.Item On two problems of arithmetic degree theory : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University(Massey University, 1996) Compton, Alistair AllanThe reader of this thesis should already have a basic understanding of ideal theory. For this reason it is recommended that a good introduction to this subject would be gained from reading D. G. Northcott's book "Ideal Theory", paying special attention to chapters one and three. This thesis consists of three chapters, with chapter one providing the definitions and theorems which will be used throughout. Then I will be considering two problems on the arithmetic degree of an ideal, one posed by Sturmfels, Trung and Vogel and the other by Renschuch. These problems will be described in the introductions to chapters two and three.Item Developing early algebraic understanding in an inquiry classroom : a thesis presented in partial fulfilment of the requirements for the degree of Master of Education at Massey University, Palmerston North, New Zealand(Massey University, 2007) Hunter, Jodie Margaret RobertaThis study explores Year 5 and 6 students’ construction of early algebraic concepts within an inquiry classroom context. Also under consideration are the tools—the instructional tasks and models, the forms of notation and symbolisation, the discourse and interaction, and the teacher’s pedagogical actions—which mediate student development of early algebraic reasoning. An emergent theoretical perspective which brings together social and constructivist theories of learning underpins the focus of the study. Relevant literature is drawn on to illustrate the need for student focus to shift from a procedural perspective of number operations and relations to understanding their structural aspects. Comprehensive evidence in the literature is provided of the significant role of the teacher in developing the students’ early algebraic reasoning through facilitating their participation in making conjectures, generalising, justifying and formalising. A classroom-based qualitative research approach—teaching experiment—matched the emergent theoretical frame taken in the study. The teaching experiment approach supported a collaborative teacher-researcher partnership. Student interviews, participant and video recorded observations, and classroom artefacts formed the data collection. On-going and retrospective data analysis was used to develop the findings as one classroom case study. Important changes in student reasoning were revealed in the findings as the teacher guided development of productive discourse and facilitated extended time and space for student discussion and exploration within an inquiry context. Students were provided with many rich opportunities to engage with tasks and models which explicitly focused on developing relational thinking, understanding of algebraic notation, the exploration of the properties and relationships of numbers, and functional patterns. Evidence is provided that through engaging with the tasks and models, the students learnt to make conjectures, represent, justify, generalise and formalise their observations. Of significance in deepening student understanding of early algebraic concepts were the repeated challenges to their partial understandings. The research findings provide insights into ways teachers can assist students to use their implicit understanding of number relations and properties as a foundation for the construction of early algebraic reasoning. The results of this study suggest that student participation in mathematical activity which included explanation, argumentation and justification supported their development of rich algebraic reasoning.Item The quotient between length and multiplicity : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University(Massey University, 2000) Allsop, Nicholas FrederickThis dissertation examines the finiteness of the algebraic invariants nA(M) and θA(M). These invariants, based on the ratio of length and multiplicity and the ratio of Loewy length and multiplicity respectively, are studied in general and under certain conditions. The finiteness of θA(M) is established for a large class of algebraic structures. nA(M) is shown to be finite in the low dimensional case as well as when we restrict our attention to special sets of ideals. Also considered in this dissertation are equivalent conditions for the local case to be bounded by the graded case when evaluating nA(M).