The Codevelopment of Mathematical Concepts and the Practice of Defining

Abstract

We examined the codevelopment of mathematical concepts and the mathematical practice of defining within a sixth-grade class investigating space and geometry. Drawing upon existing literature, we present a framework for describing forms of participation in defining, what we term aspects of definitional practice. Analysis of classroom interactions during 16 episodes spanning earlier and later phases of instruction illustrate how student participation in aspects of definitional practice influenced their emerging conceptions of the geometry of shape and form and how emerging conceptions of shape and form provided opportunities to develop and elaborate aspects of definitional practice. Several forms of teacher discourse appeared to support students' participation and students' increasing agency over time. These included: (a) requesting that members of the class participate in various aspects of practice, (b) asking questions that serve to expand the mathematical system, (c) modeling participation in aspects of practice, (d) proposing examples that create contest (i.e., monsters), and (e) explicitly stating expectations of and purposes for participating in the practice.

Analyzing Pedagogical Routines in the Upper Secondary School Teacher’s Discourse Using the Commognitive Aproach

“…In the case of processes, the discourse expresses actions and is personalized, whilst in the case of objects, it reveals states and has an impersonal character. In our case, we can see that Anna starts the lesson with procedural definitions (i.e., describing how to construct an object) instead of structural definitions (i.e., describing properties that characterize an object) (Kobiela & Lehrer, 2015). These types of procedural definitions with generic examples justify the use of the remaining types of examples (concrete, worked-out examples, and exercises) to consolidate the use of the procedures or become familiar with the notation.…”

Exploring prospective student teacher's question on mathematics teaching practice

“…(National Council of Teachers of Mathematics, n.d.). Concepts can be reconstructed through the following activities: 1) proposing concept definitions, 2) constructing and evaluating examples rather than examples, 3) definition of nature and relations, 4) construction of definitions by explanation or argument, 5) revision of definitions, 6) proving and reasoning about the relationship system, 7) asking the definition of the question, and 8) negotiating the criteria of adequacy and acceptance of the definition, so that the concept can be taught through various learning activities (Kobiela & Lehrer, 2015).…”

El discurso de estudiantes para maestro cuando describen y definen cuerpos geométricos

“…Alvarado Monroy y González Astudillo (2016) estudiaron cómo construyen definiciones los estudiantes universitarios cuando hacen demostraciones e identificaron distintos tipos de definiciones. Por otro lado, Kobiela y Lehrer (2015) analizaron las interacciones que se producen entre estudiantes de Educación Primaria cuando estos construyen definiciones geométricas. Más recientemente, Gilboa, Kidron y Dreyfus (2019) han investigado a estudiantes de Educación Secundaria que trabajaban por parejas mientras construían definiciones matemáticas formales vinculadas a la idea de recta tangente.…”