Defining a rectangle under a social and practical setting by two seventh graders

Abstract

Abstract:Regarding defining as a mathematical activity bridging informal to formal proof, two seventh graders will reinvent the definition of rectangles under a social and practical setting based on their informal argumentation. Their apprehensions of figures, implicit concepts/theorems and the cognitive architecture of defining are discussed in this paper. Kurzreferat

The Codevelopment of Mathematical Concepts and the Practice of Defining

“…Monsters often provoked explosions of contributions, suggesting that students were invested in the practice and eager to participate. Our study confirms similar findings pointing to the importance of examples in helping students make sense of mathematical concepts (e.g., Ambrose & Kenehan, 2009;Dahlberg & Houseman, 1997;Roth & Thom, 2009) and, in particular, of nonexamples (e.g., de Villiers, 1998;Lehrer & Curtis, 2000;Lin & Yang, 2002;Zaslavsky & Shir, 2005) or unfamiliar examples (e.g., Lehrer et al, 1999;Zandieh & Rasmussen, 2010) in provoking students to expand their definitions. However, we extend this work by highlighting how nonexamples in particular should be designed, given students' starting points, and when it might be productive to present nonexamples.…”

“…As illustrated in the previous example, definitional arguments often motivate students to engage in the practice of revising definitions to include additional properties or relations, to eliminate unnecessary or inaccurate properties (e.g., Borasi, 1992;de Villiers, 1998;Lin & Yang, 2002), or to better conform to features or roles that definitions should play (Zaslavsky & Shir, 2005). Historically, revisions often occur when existing definitions are not precise enough to distinguish examples from nonexamples (Lakatos, 1976).…”

“…Defining also appears to support students' description of objects, moving them away from holistic descriptions toward more mathematical descriptions that focus on relevant parts and properties (Ambrose & Kenehan, 2009;Lehrer et al, 1999). In other cases in which students attended to mathematical features, they learned to reason about which features to include in the definition and which were not relevant to the definition (e.g., Borasi, 1992;de Villiers, 1998;Herbst, et al, 2005;Keiser, 2000;Lehrer & Curtis, 2000;Lehrer et al, 1999;Lin & Yang, 2002;Zaslavsky & Shir, 2005). For example, when defining "perfect" solids, the students in Lehrer and Curtis's (2000) study continuously revised their conjectured rules as they evaluated new examples and nonexamples.…”

“…One conjecture, that three faces needed to meet at each vertex, was eliminated once a student noticed that the octahedron had four faces coming together at each vertex. In a few cases, students also improved in writing economic definitions (Borasi, 1992;de Villiers, 1998;Lin & Yang, 2002). Considerations of economy of definition may provoke investigation of relations among relevant features and, hence, what needs to be stated explicitly and what is implied by explicit statement.…”

Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles

“…So, linguistically, a square is seen as a "shape with exact sides" and a rectangle as a "(same) shape with long sides". In this case language makes explicit that square and rectangle are two kinds of a same thing, deeply related to each other and not separated into distinct categories (also see Lin and Yang 2002). In fact this is typical of Taoism in which a central idea is the evolution of events as a process of change and the ideology of "grasping ways beyond categories" or to "categorize in order to unite categories" ( ) .The rationale behind the design of the study we present is that there can be means for constructing the meaning of squares and rectangles, generalizing the perception of square and not-square rectangular shapes, other than everyday language (that reinforces the conditions for an obstacle against an inclusive definition).…”