Construction Of Mathematical Meanings Research Articles

The Semiotic Bundle as a reflective tool in pre-service mathematics teachers’ education

Contemporary discussions in mathematics education have underlined the role of embodied cognition in mathematics thinking and in conceiving teaching-learning processes as multimodal. In this work, we refer to the Semiotic Bundle approach, according to which teaching-learning processes can be described through the production and evolution of different signs. In this perspective, teacher education should help teachers learn how to pay attention to the multimodality of teaching-learning processes. In this paper, we discuss findings from a pre-service teachers’ professional development experience to show how the Semiotic Bundle can be used as a tool for teachers’ reflection on the teaching-learning processes. The results show that, when pre-service teachers analyzed teaching-learning episodes using the Semiotic Bundle tool, they experienced a reflective activity, through which they became aware of the relationship between the produced signs and the construction of mathematical meanings. The study concludes that the Semiotic Bundle may serve as a reflective tool to be used in pre-service teachers’ education.

Constructing shared mathematical meanings in the classroom with digital artifacts that simulate real-world phenomena

This paper describes and examines students’ shared construction of meanings while learning about quadratic functions via digital artifacts that simulate real-world phenomena, like the motion of a ball on an inclined plane, focusing specifically on the role of the teacher in that construction process. The study follows the interactions of twenty 15-year-old students and their teacher during the completion of three sequential digital tasks, analyzing how these interactions promote the students’ ability to construct the mathematical meanings of the quadratic function. The study was guided by the theory of semiotic mediation, which treats artifacts as fundamental to cognition and views learning as the evolution from meanings connected to the use of a certain artifact to those recognizable as mathematical. The data analysis showed students progressing from the description of the real-world phenomenon toward a construction of the meaning of the quadratic function that models the phenomenon. While marking the “critical moments” in this progress, we analyze the communication strategies used by the teacher to facilitate it. Our research findings showed evidence that certain types of questions and the strategy “re-voicing” can be particularly effective in prompting students’ construction of mathematical meanings.

Experts’ Construction of Mathematical Meaning for Derivatives and Integrals of Complex-Valued Functions

We engaged five mathematicians who conduct research in the domain of complex analysis or use significant tools from complex analysis in their research in interviews about basic concepts of differentiation and integration of complex functions. We placed a variety of constructivist, social-constructivist, and embodied theories in mathematics education in conversation with one another to explore the development of the expert participants’ construction of mathematical meanings while moving between varying levels of abstraction from embodied concepts and real-world contexts to symbolic manipulation and formal theories. The mathematicians relied heavily on direct application of concepts and analogies from differentiation of real-valued functions and employed rotation and dilation as a local linear description of the action of a complex differentiable function with attendant repeated mental imagery and physical gestures. They also employed reasoning about real-valued line integrals to interpret contour integrals but acknowledged significant struggle to conceptually interpret what was analogously accumulated in the complex case. Instead, they all developed more personal meanings through a process of reconciling various aspects across their concrete to formal domains of reasoning. Much of the observed construction of meaning was manifested through contextualizing well-understood aspects of formal mathematical theory. We consequently explore implications for characterizing mathematical conceptual development as an interplay between concrete and formal reasoning rather than a development from one to another.

1B1-I1 問題解決を重視した授業における生徒の数学的意味構成 : LPS日本データ(J-1)の分析から(数学科における熟練教師の力量を探る-授業の国際比較研究の成果から-,自主企画課題研究,日本科学教育学会 第30回年会論文集)

1B1-I1 問題解決を重視した授業における生徒の数学的意味構成 : LPS日本データ(J-1)の分析から(数学科における熟練教師の力量を探る-授業の国際比較研究の成果から-,自主企画課題研究,日本科学教育学会 第30回年会論文集)

Hypothetical Learning Trajectories that use Digital Technology to Tackle an Optimization problem

In this paper, we document the results of the implementation of a series of activities related to an optimisation problem. The objective is describing how the construction of mathematical meanings is promoted when the students face the problem through different representations. The activities, addressed towards 15 year old students, surge from a free interpretation of the construct “Hypothetical Learning Trajectory” (HLT) by Simon (1995), considering different representations and utilising digital technologies. With this, we propose a route to tackle variation problems based on HLT similar to the proposed.

Examining Social and Sociomathematical Norms in Different Classroom Microcultures: Mathematics Teacher Education Perspective

Doing mathematics is not only an individual construction activity but also a social one (Bowers, Cobb, & McClain, 1999; Hershkowitz & Schwarz, 1999). Highly complex human interactions occur in mathematics classrooms. Furthermore, the process of teaching and learning mathematics involves a kind of collective and interactive relation (Bauersfeld, 1980). Investigating how math is learned and taught from a sociological perspective by generally analyzing classroom culture in general and mathematical culture in particular, scholars have drawn on some concepts such as classroom microculture and mathematical classroom traditions (i.e., Cobb, 1999; Cobb, Stephan, McClain, & Gravemeijer, 2001; Cobb, Wood, Yackel, & McNeal, 1992). Cultural constitution, which appears in a small group and makes more interactions possible among participants, can be defined as the system of knowledge, belief, behavior, and tradition shared by the members of a group (Fine, 1987, as cited in Fine, 2003). Like every community, each classroom establishes, sustains, modifies or eliminates various patterns such as norms, standards, obligations, rules, and routines (Sekiguchi, 2005). This process is called culture building (Fine, 2003). From this perspective, this study focuses on the social and sociomathematical norms embedded in the culture building process.A norm is an important element of classroom microculture that is established by the teacher and students (Cobb, 1999). Norms can be defined as ideas that determine manners; what is expected to be done by a group member, or a person under prescribed conditions (Homans, 1951, p. 123). Similarly, Cobb et al. (1992) and Cobb and Yackel (1996a) use the concept of norm in the meaning of specifying and meeting the mutual expectations that arise in the classroom through the interaction between teacher and students. Norms characterize regularities in individual or collective classroom activities (Cobb et al., 2001). Norms are established and developed through constant student-teacher interactions and thus may differ significantly from one classroom to another (Cobb & Yackel, 1996b). This study, with prospective teachers as participants, thus aims to investigate which social and sociomathematical norms exist in different classroom microcultures.Theoretical FrameworkClassroom MicrocultureA classroom is defined as a complex environment that accommodates individuals who come together with the aim of constructing a learning community (Levenson, Tirosh, & Tsamir, 2009). Like every community, a classroom constitutes and develops an association of social relations and its own microculture (Gallego, Cole, & The Laboratory of Comparative Human Cognition, 2001; Lopez & Allal, 2007). The microculture of a mathematics classroom contains social interactions and the construction of mathematical meaning (Voigt, 1995). It does not exist separate from the mathematical activities of a classroom community (Cobb et al., 1992). Its characteristics depend on norms, patterns, and regulations that are difficult to change, such as students' attitudes (Voigt, 1995). Social and sociomathematical norms, together with a classroom's mathematical practices, constitute the classroom microculture where individual and collective mathematical learning occurs (Cobb et al., 2001).Social and Sociomathematical NormsCobb and Yackel (1996a), who extended their studies from general classroom norms to the normative aspects of mathematical arguments regarding student activities, distinguished norms as social and sociomathematical. Social norms express the social-interaction aspects of a classroom that become normative (Yackel, Rasmussen, & King, 2000). These norms are common norms that can be enacted in any field (Cobb & Yackel, 1996b). For example, explaining and justifying solutions, identifying and stating agreement, trying to make sense of others' explanations, expressing disagreement on ideas, and so forth are social norms for discussions where the whole class participates (Cobb & Yackel, 1996a). …

Graphic calculators and connectivity software to be a community of mathematics practitioners

In a teaching experiment carried out at the secondary school level, we observe the students’ processes in modelling activities, where the use of graphic calculators and connectivity software gives a common working space in the class. The study shows results in continuity with others emerged in the previous ICMEs and some new ones, and offers an analysis of the novelty of the software in introducing new ways to support learning communities in the construction of mathematical meanings. The study is conducted in a semiotic-cultural framework that considers the introduction and the evolution of signs, such as words, gestures and interaction with technologies, to understand how students construct mathematical meanings, working as a community of practice. The novelty of the results consists in the presence of two technologies for students: the “private” graphic calculators and the “public” screen of the connectivity software. Signs for the construction of knowledge are mediated by both of them, but the second does it in a social way, strongly supporting the work of the learning community.

Proportional Reasoning in Nursing Practice

We investigate how expert nurses undertake the calculation of drug dosages on the ward. This calculation is error-critical in nursing practice and maps onto the concepts of ratio and proportion. Using episodes of actual drug administration gleaned from ethnographic study, we provide evidence that experienced nurses use a range of correct proportional-reasoning strategies based on the invariant of drug concentration to calculate dosage on the ward instead of the single taught method they describe outside of the practice. These strategies are tied to individual drugs, specific quantities and volumes of drugs, the way drugs are packaged, and the organization of clinical work. The centrality of mathematics to school curricula has been justified partly in terms of its importance for the world of work, and many researchers have attempted to articulate the relationships between school-taught mathematics and the mathematics found in occupational and everyday lives. We believe that the study of mathematics in work provides a particularly fruitful setting for illuminating fundamental issues concerning the construction of mathematical meanings more generally. In essence, it offers an opportunity to focus attention on the ways in which professional discourses shape mathematical meanings and, reciprocally, on how the use of mathematics-however defined-structures the discourse of work. One might hope that such insight will afford leverage on didactical strategies within and beyond workbased settings. Until the middle of the 1980s, research in this field was undertaken largely within a framework in which the researchers assumed that mathematics was unproblematically visible in workplace settings and that it consisted mainly of calculation (see, for example, Fitzgerald, 1981). Later, researchers, for example Wolf (1984) and Harris (1991), adopted methodologies that were considerably more sensitive to the complexity of the practice/mathematics relationship. Yet all using such approaches encountered a fundamental obstacle: Most employees did not describe their activities in mathematical terms, and they often declared that they used very little mathematics in their work. The fundamental difficulty, it seems, is that the math

The Construction of Mathematical Meanings: Connecting the Visual with the Symbolic

In this paper, we explore the relationship between learners' actions, visualisations and the means by which these are articulated. We describe a microworld, Mathsticks, designed to help students construct mathematical meanings by forging links between the rhythms of their actions and the visual and corresponding symbolic representations they developed. Through a case study of two students interacting with Mathsticks, we illustrate a view of mathematics learning which places at its core the medium of expression, and the building of connections between different mathematisations rather than ascending to hierarchies of decontextualisation.

Change in Teaching Mathematics: A Case Study

The purpose of this case study was to examine teacher’s learning in the setting of the classroom. In an ongoing mathematics research project based on constructivist views of learning and set in a second-grade classroom, the teacher changed in her beliefs about learning and teaching. These alterations occurred as she resolved conflicts and dilemmas that arose between her previously established form of practice and the emphasis of the project on children’s construction of mathematical meaning. The changes that occurred as the teacher reorganized her practice were analyzed and interpreted by using selected daily video recordings of mathematics lessons along with field notes, open-ended interviews, and notes from project meetings. The analyses indicated that changes occurred in her beliefs about the nature of (a) mathematics from rules and procedures to meaningful activity, (b) learning from passivity to interacting and communicating, and (c) teaching from transmitting information to initiating and guiding students’ development of knowledge.