Timothy Fukawa-Connelly

Using Toulmin analysis to analyse an instructor's proof presentation in abstract algebra

This paper provides a method for analysing undergraduate teaching of proof-based courses using Toulmins model (1969) of argumentation. It presents a case study of one instructors presentation of proofs. The analysis shows that the instructor presents different levels of detail in different proofs; thus, the students have an inconsistent set of written models for their work. Similarly, the analysis shows that the details the instructor says aloud differ from what she writes down. Although her verbal commentary provides additional detail and appears to have pedagogical value, for instance, by modelling thinking that supports proof writing, this value might be better realized if she were to change her teaching practices.

Making Real Analysis Relevant to Secondary Teachers: Building Up from and Stepping Down to Practice

Abstract Future teachers often claim that advanced undergraduate courses, even those that attempt to connect to school mathematics, are not useful for their teaching. This paper proposes a new way of designing advanced undergraduate content courses for secondary teachers. The model involves beginning with an analysis of the curriculum and practices of school mathematics and its teaching, and then using those to build up to the advanced mathematics – in this case, real analysis. After developing definitions, examples, theorems, and proofs, the model then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. To illustrate the model, we provide and discuss two example tasks.

Student Understanding of Symbols in Introductory Statistics Courses

This study explores student understanding of the symbolic representation system in statistics. Furthermore, it attempts to describe the relation between student understanding of the symbolic system and statistical concepts that students develop as the result of an introductory undergraduate statistics course. The theory, drawn from the notion of semantic function that links representations and concepts, seeks to expand the range of representations considered in exploring students’ statistical proficiencies. Results suggest that students experience considerable difficulty in making correct associations between symbols and concepts; that they describe the relationship as seemingly arbitrary; and that they are unlikely to understand statistics as quantities that can vary. Finally, this study describes students’ need for robust knowledge of preliminary concepts in order to understand the construct of a sampling distribution.

Responsibility for proving and defining in abstract algebra class

There is considerable variety in inquiry-oriented instruction, but what is common is that students assume roles in mathematical activity that in a traditional, lecture-based class are either assumed by the teacher (or text) or are not visible at all in traditional math classrooms. This paper is a case study of the teaching of an inquiry-based undergraduate abstract algebra course. In particular, gives a theoretical account of the defining and proving processes. The study examines the intellectual responsibility for the processes of defining and proving that the professor devolved to the students. While the professor wanted the students to engage in all aspects of defining and proving, he was only successful at devolving responsibility for certain aspects and much more successful at devolving responsibility for proving than conjecturing or defining. This study suggests that even a well-intentioned instructor may not be able to devolve responsibility to students for some aspects of mathematical practice without using a research-based curriculum or further professional development.

The Pedagogical Examples of Groups and Rings That Algebraists Think Are Most Important in an Introductory Course

This article reports on an exploratory study designed to investigate the reasoning behind algebraists’ selection of examples. Variation theory provided a lens to analyze their collections of examples. Our findings include the classes of examples of groups and rings that algebraists believe to be most pedagogically useful. Chief among their selection criteria was that these examples illustrate not only the concept at hand but also lay the foundation for more abstract constructions. Additionally, we found that the algebraists, tending to think in terms of classes of examples, used a relatively small number in their own teaching and research.RésuméCet article donne le compte-rendu d’une étude préliminaire conçue dans le but d’analyser le raisonnement qui sous-tend le choix des exemples chez les algébristes. Grâce à la théorie de la variation, nous sommes en mesure d’analyser les séries d’exemples. Nos résultats incluent les classes d’exemples posés en termes d’ensembles et de cercles que les algébristes estiment les plus utiles sur le plan pédagogique. Parmi leurs critères de sélection, le plus important est le fait que ces exemples illustrent non seulement le concept dont il est question, ils constituent également les fondements de constructions plus abstraites. De plus, nous avons constaté que les algébristes, qui ont tendance à penser en termes de classes d’exemples, en citent un nombre relativement peu élevé dans leur propre enseignement et dans leur recherche.

The Incoming Statistical Knowledge of Undergraduate Majors in a Department of Mathematics and Statistics.

Studies have shown that at the end of an introductory statistics course, students struggle with building block concepts, such as mean and standard deviation, and rely on procedural understandings of the concepts. This study aims to investigate the understandings entering freshman of a department of mathematics and statistics (including mathematics education), students who are presumably better prepared in terms of mathematics and statistics than the average university student, have of introductory statistics. This case study found that these students enter college with common statistical misunderstandings, lack of knowledge, and idiosyncratic collections of correct statistical knowledge. Moreover, they also have a wide range of beliefs about their knowledge with some of the students who believe that they have the strongest knowledge also having significant misconceptions. More attention to these statistical building blocks may be required in a university introduction statistics course.