What are the fundamental concepts of group theory?

Burn, Bob

doi:10.1007/bf00369154

Cited by 31 publications

(13 citation statements)

References 5 publications

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“…While some have suggested that an over-reliance on concrete examples of groups leading to a lack of skills in proof production, others, such as Burn (1996), recommend reversing the order of presentation, using examples and applications to stimulate the discovery of definitions and theorems through permutation and symmetry. An example of reducing group theory's high levels of abstraction (Hazzan 2001) is to ask students to construct the operation table for low order groups.…”

Section: Abstract Algebramentioning

confidence: 99%

Key Mathematical Concepts in the Transition from Secondary School to University

Thomas

Druck

Huillet

et al. 2015

The Proceedings of the 12th International Congress on Mathematical Education

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This report1 from the ICME12 Survey Team 4 examines issues in the transition from secondary school to university mathematics with a particular focus on mathematical concepts and aspects of mathematical thinking. It comprises a survey of the recent research related to: calculus and analysis; the algebra of generalised arithmetic and abstract algebra; linear algebra; reasoning, argumentation and proof; and modelling, applications and applied mathematics. This revealed a multi-faceted web of cognitive, curricular and pedagogical issues both within and across the mathematical topics above. In addition we conducted an international survey of those engaged in teaching in university mathematics departments. Specifically, we aimed to elicit perspectives on: what topics are taught, and how, in the early parts of university-level mathematical studies; whether the transition should be smooth; student preparedness for university mathematics studies; and, what university departments do to assist those with limited preparedness. We present a summary of the survey results from 79 respondents from 21 countries.

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Section: Abstract Algebramentioning

confidence: 99%

Key Mathematical Concepts in the Transition from Secondary School to University

Thomas

Druck

Huillet

et al. 2015

The Proceedings of the 12th International Congress on Mathematical Education

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“…The researchers identified the coordination of operation and set as an essential aspect of understanding group and subsequently subgroup. Burn (1996) quickly challenged the use of this question noting that "there is something commendable about suggesting that ℤ 3 is a subgroup of ℤ 6 ..., since every cyclic group of order 6 has a cyclic subgroup of order 3" (p. 373). He was referencing the advanced conception of isomorphism where the set {0,2,4} could be thought of as the same as ℤ 3 and therefore ℤ 3 would in fact be a subgroup of ℤ 6 .…”

Section: Replication Results and Discussionmentioning

confidence: 99%

“…In fact, Dubinsky et al, (1994) reported "On Learning the Fundamental Concepts of Group Theory" where they discussed groups, subgroups, cosets, coset products, and normality, sparking a discussion on this matter. Burn (1996) challenged that isomorphism, closure, associative, identity, inverses, sets, functions, and symmetry were all overlooked as fundamental concepts. Dubinsky, et al (1997) conceded that the title should have stated "on some fundamental concepts of group theory" (p. 251) where functions, sets, permutations, symmetries and the four group axioms are also fundamental to group theory.…”

Section: Fundamental Topics In Group Theorymentioning

confidence: 99%

“…A domain analysis can be a rather daunting process depending on how well explored a given area is. In cases like introductory group theory, without consensus on important topics (Burn, 1996;Dubinsky, Dautermann, Leron, & Zazkis, 1997;Suominen, 2014) Currently, the CCI and PCA exist in the of field mathematics education.…”

Section: On Validity and Concept Inventoriesmentioning

confidence: 99%

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The Design and Validation of a Group Theory Concept Inventory

Melhuish¹

2000

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Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated to an additional 476 students over two rounds.Through follow-up interviews intended for validation, and test analysis processes, the questions were refined to best target conceptions and strengthen validity measures. The GCI consists of seventeen questions, each targeting a different concept in introductory group theory. The results from this study are broken into three papers. The first paper reports on the methodology for creating the GCI with the goal of providing a model for building valid concept inventories. The second paper provides replication results and critiques of previous studies by leveraging three GCI questions (on cyclic groups, subgroups, and isomorphism) that have been adapted from prior studies. The final paper introduces the GCI for use by instructors and mathematics departments with emphasis on how it can be leveraged to investigate their students' understanding of group theory concepts.Through careful creation and extensive field-testing, the GCI has been shown to ii be a meaningful instrument with powerful ability to explore student understanding around group theory concepts at the large-scale.iii Dedication To my parents for teaching me to have a strong work ethic, to live life with kindness, and to pursue a career that truly fulfills you.iv Acknowledgements

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“…They further suggest that the sequencing of such a course should not be linear and discuss a spiral curriculum, learning by successive refinement, microworlds and holistic spray (where students are put into an environment where they are surrounded by group theoretical ideas and the teacher's role is to manage student frustration). Burn (1996) offers a critical analysis of the above paper. The criticism arises not so much from APOS theory nor ISETL, though Burn does suggest that the use of the programming language could be improved if, among other things, the two-dimensional nature of the computer screen could be utilised to include spreadsheets, group tables, and so on.…”

Section: Abstract Algebramentioning

confidence: 99%