Modeling, Symbolizing, and Tool Use in Statistical Data Analysis

Abstract

P. 1 Kaput (1991) and Thompson (1992) argue in constructivist terms that the tools and symbols students use profoundly influence both the course of their mathematical learning and its results, the increasingly sophisticated mathematical ways of reasoning that they develop. P. 1 Sfard (2000) and Dörfler (2000) have both formulated discourse perspectives in which they contend that mathematical reality is symbolized into being. p. 1 Meira (1998) and van Oers (1996) [who draw on sociocultural theory] explicitly frame mathematics as an activity that is mediated by the use of tools and symbols. p. 1 focus on symbolizing as an activity; a concern for the way that students actually use tools and symbols. p. 1 Nemirovsky (1994) distinguishes between symbol systems and symbol use p. 2 in all three theoretical perspectives, the ways that symbols are used and the meanings that they come to have are seen to be mutually constitutive. p. 2 (cognitive science) A concern for symbol use entails the dissolution of a distinction central to mainstream cognitive science, that between external representations or symbols on the one hand and internal representations or concepts on the other. p. 2 analyses that focus on people’s activity with symbols treat them as an integral aspect of their mathematical reasoning rather than as external aids to it (cf. Cobb, 2000; Meira, 1995). p. 2 the process of learning to use symbols in general, and conventional mathematical symbols in particular, is cast in terms of participation. p. 2 Symbol use is then seen not so much as something to be mastered, but as a constituent part of the mathematical practices in which students come to participate. p. 2 (research perspective) the perspective of a classroom researcher who is motivated by an interest in instructional design. p. 2 (unit of analysis) account for the mathematical learning of the classroom community taken as a unit of analysis in its own right P. 2 The theoretical construct I use to account for this collective learning is that of a classroom mathematical practice. p. 2 the taken-as-shared ways of symbolizing negotiated by the teacher and students constitute one of the core aspects of a mathematical practice. p. 2 focus specifically on the taken-as-shared ways of symbolizing that were constituted as part of this process of collective mathematical learning p. 2 (Chain of signification): offers an account of the classroom community’s mathematical learning that is cast in semiotic terms. p. 3 (classroom design experiments) of up to a year in duration. In the course of these experiments, we develop sequence of instructional activities for students and analyze students’ mathematical learning as it occurs in the social situation of the classroom. Work of this type falls under the general heading of design research in that it involves both instructional design and classroom-based research. P. 3 develop conjectures about both 1) possible learning routes or trajectories, and 2) the specific means that might be used to support and organize this learning. p. 3 These conjectures are then tested and modified on a daily basis once the design experiment begins as informed by ongoing analyses of classroom events. p. 3 (differences in mathematical reasoning) these conjectures cannot be about the anticipated learning of each and every student in a class given that there are significant qualitative differences in their mathematical reasoning at any point in time. p. 3 Descriptions of planned instructional approaches written so as to imply that all students will reorganize their reasoning in particular ways at particular points in an instructional sequence are, at best, questionable idealizations. P. 3 (collective mathematical development) It is feasible to view a hypothetical learning trajectory as consisting of conjectures about the collective mathematical development of the classroom community. p. 3 (mathematical practice) comprises three interrelated mathematical norms: 1) a taken-as-shared purpose, 2) taken-as-shared ways of reasoning with and talking about tools and symbols, and 3) taken-as-shared forms of mathematical augmentation. p. 3 (learning trajectory) an envisioned learning trajectory can be viewed as consisting of an anticipated sequence of classroom mathematical practices together with conjectures about the means of supporting their evolution from prior practices. p. 3 (symbols and meaning) the claim that the ways that symbols are used and the meanings they come to have are mutually constitutive receives support from several different theoretical perspectives. This claim implies that instructional design should support the evolution of taken-as-shared ways of symbolizing as part of the process of supporting the development of mathematical meaning. P. 3 (RME) in RME a concern for symbolizing finds expression in the heuristic of emergent models p. 3 (RME) The overall intent of the RME approach is to support a so-called reinvention process in which students initially mathematize their informal reasoning in problem situations that are experientially real to them. p. 4 (hypothetical learning trajectory in RME) involves the conjecture that models of informal reasoning will gradually take on a ‘life of their own’ and be transformed through use into models for more general mathematical reasoning that are independent of situation-specific imagery. p. 4 (Evolution in RME) Gravemeijer et al. (2000) clarify this distinction between models-of and models-for by identifying four types of mathematical activity: (1) activity in the task setting, in which interpretations and solutions involve knowing how to act in the setting (often out of school settings). (2) referential activity, in which models-of signify activity in the setting described in instructional activities (posed mostly in school). (3) general activity, in which models-for make possible a focus on interpretations and solutions independently of situation-specific imagery. (4) reasoning with conventional symbolizations, in which the support of models-for mathematical activity is no longer needed. symbolizing is viewed as a primary means of supporting collective mathematical learning in approaches consistent with RME. P. 4 RME does not adopt what is termed a modeling point of view. p. 4 (modeling point of view): a model is considered to capture mathematical structures and relationships implicit in starting point situations. p. 4 (a model as characterized in RME) originates from the taken-as-shared ways of acting and reasoning about the settings that constitute the starting points of an instructional sequence. As a consequence, the increasingly sophisticated mathematical ways of knowing of both the classroom community and of the participating students are viewed as originating from tool-mediated activity in those settings rather than from the settings per se. p. 4 (learning trajectory and symbolizing): the focus when formulating learning trajectories is on the progressive reorganization of taken-as-shared ways of reasoning rather than on mathematical relationships that might be abstracted directly from starting point situations. P. 4 (changes in symbolizing and changes in meaning) my purpose is to exemplify the interplay between changes in ways of symbolizing and changes in meaning as an aspect of the evolution of classroom mathematical practices. p. 4 (suitable domain) Statistical data analysis therefore constitutes a suitable domain in which to explore the interplay between the development of ways of symbolizing and the development of mathematical meaning. p. 4 (Distribution) this focus on distribution allowed us to frame our instructional intent as that of supporting the gradual emergence of a single, multi-faceted mathematical notion rather than a collection of, at best, loosely-related concepts and inscriptions. P. 6 (remediating) our immediate goal was not one of remediating certain competencies and skills. Instead, the challenge was to influence the students’ beliefs about what it means to do statistics in school. In doing so, it would be essential that they actually begin to analyze data in order to address a significant question rather than simply manipulate numbers and draw specific types of graphs. p. 7 Describes the tool: This tool was designed to provide the students with a means of each individual data point is inscribed as a horizontal bar The students could select the color of each bar to be either pink or green, thus enabling them to enter and compare two data sets. p. 7 the teacher was able to initiate a shift in the discourse during this session such that the students began to speak about the bars as attributes of individual cases that had been measured. p. 7 Assisted by the teacher, another student challenged Casey’s argument p. 9 (mathematical practice) The mathematical practice that emerged as the students used the first minitool can therefore be described as that of exploring qualitative characteristics of collections of data points. p. 9 Options provided by the second minitool P. 9 Our intention in designing this minitool was to build on the ways of reasoning about data that the students had developed as they used the first minitool. P. 9 the students were able to use the second minitool to analyze data almost immediately and it was apparent that the axis plot inscription signified a set of data values rather than merely a collection of dots spaced along a line. p. 10 (public classroom discourse) This was the first occasion in public classroom discourse in which a student had described a data set in global, qualitative terms by alluding to its shape. P. 10 (recast/revoice) The teacher then recast Karen’s analysis as a way of characterizing the global shift of which Janice had spoken. As a consequence of such revoicing, it gradually became taken-as-shared P 10 illustrative episode p. 12 (discussion) During the remainder of the discussion, the teacher attempte