Capturing congruence with a turtle

This note compares turtle geometry and Euclidean geometry with respect to their treatment of similarity and difference of plane figures. It is observed that while the Euclidean notion of congruence faithfully captures a common perception of sameness, the turtle expression of this idea is too weak. To deal with this insufficiency we add a new turtle operation, FLIP, which turns the turtle upside down. This brings the turtle's power to express invariance of shape up to Euclid's.The problem and its solution are viewed briefly from the perspectives of mathematics, computer science and education. The mathematical discussion compares the turtle group and the Euclidean group. The computational discussion focuses on the issue of expressive power of a language and how it may be enhanced. The educational discussion suggests a classroom implementation of the above ideas.     

Learning group isomorphism: A crossroads of many concepts

This article is concerned with how undergraduate students in their first abstract algebra course learn the concept of group isomorphism. To probe the students' thinking, we interviewed them while they were working on tasks involving various aspects of isomorphism. Here are some of the observations that emerged from analysis of the interviews. First, students show a strong need for canonical, unique, step-by-step procedures and tend to get stuck of having to deal with some degrees of freedom in their choices. Second, students exhibit various degrees of personification and localization in their language, as in I can find a function that takes every element of G to every element of G vs. there exists a function from G to G. Third, when having to deal with a list of properties, students choose first the properties they perceive as simpler; however, it turns out that their choice depends on the type of the task and the type of complexity involved. That is, in tasks involving groups in general, students mostly prefer to work with properties which aresyntactically simple, whereas in tasks involving specific groups, students prefer properties which arecomputationally simpler.     

Interweaving mathematics and pedagogy in task design: a tale of one task

In this article we introduce a usage-goal framework within which task design can be guided and analyzed. We tell a tale of one task, the Pentomino Problem, and its evolution through predictive analysis, trial, reflective analysis, and adjustment. In describing several iterations of the task implementation, we focus on mathematical affordances embedded in the design and also briefly touch upon pedagogical affordances.     

On learning fundamental concepts of group theory

The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of understanding, and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.Work on this project has been partially supported by the National Science Foundation.     

Research mathematicians as teacher educators: focusing on mathematics for secondary mathematics teachers

We believe that professional mathematicians who teach undergraduate mathematics courses to prospective teachers play an important role in the education of secondary school mathematics teachers. Thus, we explored the views of research mathematicians on the mathematics that should be taught to prospective mathematics teachers, on how the courses they teach can serve teachers in their work with school students, and on the changes they would implement if their courses were designed specifically for prospective teachers. We constructed profiles of the four mathematicians based on their responses to a clinical interview. We employed the construct of mathematics teacher-educators’ triad in the reflective analysis of our findings and extended the construct based on the results of this study. In conclusion, we commented on potential ways to draw stronger connections between university mathematics and the mathematics taught in schools.     

On fractions and non-standard representations: Pre-service teachers' concepts

The present study examined the reasoning strategies and arguments given by pre-service school teachers as they solved two problems regarding fractions in different symbolic representations. In the first problem, the pre-service school teachers were asked to compare between two different fractions having the same numerical representation, and in the second problem, they were asked to compare between different notational representations of the same fraction. Numeration systems in bases other than ten were used to generate various representations of fractions. All students were asked to provide justifications to their responses. Strategies and arguments relative to pre-service teachers' concepts of fractions and place value were identified and analyzed based on results of 38 individual clinical interviews, and written responses of 124 students. It was found that the majority of students believe that fractions change their numerical value under different symbolic representations.     

Stuck on convention: a story of derivative relationships

In this article, we explore the responses of a group of undergraduate mathematics students to tasks that deal with areas, perimeters, volumes, and derivatives. The tasks challenge the conventional representations of formulas that students are used to from their schooling. Our analysis attends to the specific mathematical ideas and ways of reasoning raised by students, which supported or hindered their appreciation of an unconventional representation. We identify themes that emerged in these responses and analyze those via different theoretical lenses—the lens of transfer and the lens of aesthetics. We conclude with pedagogical recommendations to help learners appreciate the structure of mathematics and challenge the resilience of certain conventions.