Jennifer S. Thom

The Emergence of 3D Geometry From Children's (Teacher-Guided) Classification Tasks

Geometry, classification, and the classification of geometrical objects are integral aspects of recent curriculum documents in mathematics education. Such curriculum documents, however, leave open how the work of classifying objects according to geometrical properties can be accomplished given that the knowledge of these properties is the planned outcome of the curriculum or lesson. The fundamental question of the present study therefore is this: How can a lesson in which children are asked to participate in a task of classifying regular 3-dimensional objects be a geometry lesson, given that the participating 2nd-grade children do not yet classify according to geometrical properties (predicates)? In our analyses, which are inspired by ethnomethodological studies of work, we focus on the embodied and collective work that leads to the emergence of the geometrical nature of this lesson. Thus, we report both the collective and the individual work by means of which the lesson outcomes—the complete classification of a set of “mystery” objects according to geometrical (shape) rather than other (color, size, “pointy-ness”) properties—are achieved. In the process, our study shows how geometrical work is reproduced by 2nd-grade children who, in a division of labor with their teachers, produce a particular set of geometrical practices (sorting three-dimensional objects according to their geometrical properties) for the 1st time.

Untying a Dreamcatcher: Coming to Understand Possibilities for Teaching Students of Aboriginal Inheritance.

Increasing the number of Aboriginal students graduating from university is a goal of many Canadian universities. Realizing this goal may present challenges to the orientation and methodology of university curricula that have been developed without consideration of the traditional epistemologies of Aboriginal peoples. In this article, three scholars in the Faculty of Education at the University of Victoria take up this issue by dialoguing with each other about the possibilities of incorporating Aboriginal perspectives into their courses. These conversations are woven together into the narrative form of a four-act play in which the authors caricature their personalities to highlight their initial resistances and eventual reconsiderations. As non-Aboriginal instructors from different cultural backgrounds, the authors confront issues of respect, responsibility, and (mis)representation as they struggle with the dilemmas involved in cross-cultural understanding. Through this journey they come to imagine a world where cultural differences, including the traditional epistemologies of Aboriginal peoples, present possibilities for greater understanding of each other and more authentic expressions of our humanity.

Exploring The Issues of Incorporating Cultural Differences in Education: A Curriculum Journey in Playwriting

In response to a mandate to develop a more welcoming university for students, especially those of Aboriginal inheritance, we set out on a journey for ways of accommodating cultural differences in our university classrooms. Over the course of a year, we met regularly and audiotaped our conversations. By talking, transcribing, writing, and rewriting, we carried our understandings forth in a recursive manner. In our efforts to represent the ideas that arose in our conversations, the conceptual movements in our thinking, and the insights that evolved, a play took shape. In the play, three characters evolve their understandings over the course of four acts. In this article, we reflect on the conditions that produced the play; we comment on the process of writing the play; and we share the understandings, insights, and transformations that occurred in our desire to live well amidst differences.

Looking at the Observer: Challenges to the Study of Conceptions and Conceptual Change

In a typical study of students’ conceptions and conceptual change, researchers analyze what a student does or says in a classroom or in an interview and recognizes ideas that match or do not match their own understanding of the topic. Attributing the perspective they recognize in the student, those studies support the idea that a conception is the way by means of which an individual intrinsically conceives (of) a given phenomenon. They then hypothesize the existence of some mental structures that can be theoretically and objectively re-constructed based on what is observed in a student’s performance. Thus, researchers studying conceptions commonly assume that the observer and the observed are separate entities. However, even in the most theoretical and hardest of all sciences, physics, the independence of the measured object and the measuring subject is not taken for granted: Light, for example, will present itself as waves or as particles depending on how we examine it. The artificial sense of separation from the object(s) of study found in many accounts on students’ conceptions makes irrelevant the relationship that exists between the observer and the observed: an interdependence and co-emergence of the observer and the observed. This tight relation exists because each participant not only reacts upon what others say but also acts upon the reactions that his/her own actions give rise to. With this situation come epistemological, practical, and ethical implications for those researching in mathematics and science education. Positing or questioning the existence of an objective reality mediates how we accept or reject another human being and the worldviews s/he develops. It provides a rationale that guides our actions. This is especially important when it comes to teaching and learning at a time where the ability to deal with the plurality and diversity of human culture have emerged as significant referents for our social behavior.

(Re)(con)figuring Space: Three Children’s Geometric Reasonings

Despite decades of research revealing the importance of and need for developing students’ spatial reasoning skills, geometry receives the least attention in North American K-12 mathematics classrooms. This chapter focuses on three grade one children as they worked on a spatial-geometric task. The study as part of a larger research project inquired into the actual forms, activities and processes that constituted the children’s reasonings and geometry during the three episodes. The findings contribute to current early years research by further explicating the body’s role in the children’s spatial-geometric reasonings, the impact of these on their conceptions, and how geometry emerged as an ongoing creative process of (re)(con)figuring space. Key implications are considered regarding young children’s spatial-geometric reasoning in the mathematics classroom.

Curriculum and Complex Systems Theory

Educators frequently use ideas from complex systems (complexity) theories as metaphors for talking about educational practice. To assist educators in making appropriate use of such ideas, we provide a nonspecialist introduction to some key concepts in this article. We use examples from classrooms to exemplify how the ideas from the complex systems theory can be used to appropriately model phenomena of interest to educators. We caution readers to not use ideas from complex systems theories in inappropriate ways, which would destroy the metaphoric relations that are to be established between theory and educational practice.