Math II (562417), страница 16
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The medium is a questionnaire listing fourteen different mathsemantic problems. The questionnaire names each problem, gives an example, states why it is a problem, and then asks the two questions, "Did you study this in class; if so in what class?" and "Do you think such instruction should be given?" (with "English," "Math," and "Other" listed as choices, and room given to write in the others).
The questionnaire makes no attempt to resolve the problems. Its purpose is to collect information, not to teach. However, to avoid a single bias toward math or English, the questionnaire has two versions, one slanted toward each subject. Thus, where the math version speaks of "multiples," "adding and multiplying," "downward comparisons," and "countable and uncountable things," the English version refers to "plurals," "combining," "marked adjectives," and "mass versus count nouns." Both versions, however, provide examples in exactly the same wording, ask the same two questions, and request the same classifying information (age, sex, educational status and attainment, field of study or work, geographic location, etc.).
By now, the initial surveys will have been run at FSU, the primary site, and (for comparative purposes) at a private school near Philadelphia. An interact version will have collected responses from other places, which, with the assistance of Dr. John Gough of Deakin University, should include a batch from Australian math educators.
Preliminary results of the survey have probably also been announced by now in a symposium, "Making Mathematics Meaningful," scheduled for May 4, 2001, at FSU, the 30th in the mathematics series held there annually.
The initial returns (fewer than 100) already show some interesting results. In total, the "should be taught" responses run about double the "have studied" responses, although this varies by problem. Both English and math figure to some degree in the answers to every problem, but respondents assign some (like "approximations") more, but never exclusively, to math, and others (like "plurals versus collectives") more to English. Some interesting problems, like "adding (combining) unlike things," have many respondents picking both math and English, a choice made possible but nowhere mentioned in the questionnaires.
If you have access to the internet, you can see the English questionnaire at http://www.mathsemantics.comlQNRE.shtml. For the math version, just change QNRE to QNRM.
The survey is by invitation only, one questionnaire per person. Therefore, if you'd like to send in a response, please pick just one or the other, whichever you like. Then, in the box for group invitation code, enter TMM-24 (which means "The Mathsemantic Monitor, article #24," the number of this one). If you want to receive e-mail information on the survey's results, there's a box for that also, near the end.
If you wish to participate in any other way in the research, you can mention that in a final "comments" box. Please note that, to the extent possible, we hope all on-campus distributions of the questionnaires will involve two-person teams, one member to represent the more math-related fields and the other to represent the more English- or humanities-related fields.
Dr. Lemmert, assisted by Dr. Baugher and me, is planning to give a short mathsemantics seminar this summer or fall. This will permit collection of after-seminar responses, to show the extent to which the seminar raises mathsemantics awareness and how this alters views regarding where corrective instruction should be given.
Drs. Lemmert, Baugher, and I, as a team, intend to write reports for our respective audiences; and Dr. Gough, for his. If all goes well, you'll be receiving reports in future articles in this journal series. Look for them. Meanwhile, you could check at http://www.mathsemantics.com/MS-Research.html for news.
A persona of aviation consultant, demalogician, etc., Edward MacNeal, a regular contributor to these pages. Viking published his Mathsemantics: Making Numbers Talk Sense in 1994. The International Society for General Semantics published his MacNeal's Master Atlas of Decision Making in 1997. Copyright (C) 2001 Edward MacNeal.
ABSTRACTION IN CONTEXT: EPISTEMIC ACTIONS
Source: Journal for Research in Mathematics Education, Mar2001, Vol. 32 Issue 2, p195, 28p, 1 chart, 1 graph
Author(s): Hershkowitz, Rina; Schwarz, Baruch B.; Dreyfus, Tommy
We propose an approach to the theoretical and empirical identification processes of abstraction in context. Although our outlook is theoretical, our thinking about abstraction emerges from the analysis of interview data. We consider abstraction an activity of vertically reorganizing previously constructed mathematics into a new mathematical structure. We use the term activity to emphasize that abstraction is a process with a history; it may capitalize on tools and other artifacts, and it occurs in a particular social setting. We present the core of a model for the genesis of abstraction. The principal components of the model are three dynamically nested epistemic actions: constructing, recognizing, and building-with. To study abstraction is to identify these epistemic actions of students participating in an activity of abstraction.
Key Words: Abstraction; Activity; Construction of knowledge; Context
Abstraction has been the focus of extensive interest in several domains, including mathematics education. Many researchers have taken a predominantly theoretical stance and have described abstraction as some type of decontextualization. In this article, we propose a different view of abstraction and show how our view leads to a fresh approach to research on abstraction.
We are practitioners who are informed about recent theoretical research, but we are also deeply involved in a curriculum design, development, and implementation project. This curriculum has been built around extended problem situations. We have been considering not only what abstraction could mean in the framework of this curriculum project but also how processes of abstraction manifest themselves empirically in project classrooms. Thus, although the outlook of this article is theoretical, our thinking about abstraction has emerged from the analysis of experimental data.
Our empirical approach led us to focus primarily on process aspects of abstraction rather than on outcomes. We see abstraction as a process in which students vertically reorganize previously constructed mathematics into a new mathematical structure. We also pay careful attention to the multifaceted context in which processes of abstraction occur: A process of abstraction is influenced by the task(s) on which students work; it may capitalize on tools and other artifacts; it depends on the personal histories of students and teachers; and it takes place in a particular social and physical setting. We thus take a sociocultural point of view, as opposed to a purely cognitive or a purely situationist one.
While investigating processes of abstraction in a number of case studies, we identified three observable epistemic actions that are characteristic for abstraction: constructing, recognizing, and building-with. We present one case study, a teaching interview of a Grade 9 student. We designed a task to encourage this student to capitalize on outcomes of previous abstractions to construct new knowledge. We show how the three epistemic actions emerged during the teaching interview. Finally, we propose a model of abstraction that integrates the three epistemic actions in a dynamically nested manner; the nesting takes into account how the student makes use of previous abstractions. Moreover, the context in which the process of abstraction occurs is vital in the characterization of the components of the model. The model constitutes the main result of this article. It is operational in the sense that it provides means to empirically study abstraction.
THEORETICAL BACKGROUND
Abstraction has been an object of intense inquiry in philosophy. Not only did Plato and his followers see in abstraction a way to reach "eternal truths," but modern philosophers such as Russell (1926) characterized abstraction as one of the highest human achievements. Rather than provide a review of the vast domain of research on abstraction, we will refer only to work that is immediately relevant to this article.
Cognitivist Approaches to Abstraction
Classical cognitive psychologists considered (a) the extraction of commonalties from a set of concrete exemplars and (b) the corresponding categorization as the main features of abstraction (e.g., Rosch & Mervis, 1975). To them, abstraction is the transition from concrete to abstract, that is, to the set of commonalties. Piaget's (1970) idea of reflective abstraction led to a remarkable extension of this classical approach. It allowed him to deal with the categorization of mental operations and thus with abstraction of mental objects. The outcomes of reflective abstraction, the schemes, are the building blocks of knowledge at every level of development. Reflective abstraction extracts schemes from a pattern of related actions. This process leads to constructive theoretical models that are logically consistent.
Following Piaget, several mathematics educators have proposed descriptions of the process or mechanism by which students shift their foci from the concrete to the abstract (see Dreyfus, 1991, for a brief review). For most of these educators, abstraction proceeds from a set of mathematical objects (or processes) and consists of focusing on some distinguishing properties and relationships of these objects rather than on the objects themselves. The product of abstraction consists of the class of all objects that have the distinguishing properties and enter into the distinguishing relationships. This process of abstraction is thus a process of decontextualization--of ignoring both the objects and some of their features and relations, often those linked to a particular realization or representation. The process is linear, proceeding from the objects to the class or the structure, which may then be considered an object on a higher level. In the classical approach, abstract is considered an intrinsic property of this new object; this property is, however, not directly accessible.
In spite of the animated theoretical debate that has taken place on the nature of abstraction, little experimental research is available. For example, Stevenson (1998) has recently stated, "Although there is little or no empirical support for [von Glasersfeld's] specific assertions about the progressive abstraction of concepts, he does provide a clear account of how they might develop" (p. 94). We surmise that the lack of experimental evidence is due to the difficulty of observing the processes of abstraction (as opposed to the products, for which there is more evidence). A notable exception is a study conducted by Goodson-Espy (1998), who observed abstraction during problem solving. Although firmly anchoring her study in the framework proposed by Sfard (1991), Goodson-Espy used the notion of the levels of abstraction theoretically proposed by Cifarelli (1988/1989). For example, the lowest level, namely the ability to recognize characteristics of a previously solved problem in a new situation, is called recognition, a notion we will discuss; in other words, abstraction depends on the personal history of the solver.
This view of dependence on the personal history of the solver conforms to the views of theorists who recognize the importance of context in processes of abstraction. Not only personal history but also the use of tools and social interactions are contextual factors that may influence abstraction processes. The contradiction between decontextualization and the dependence of the abstraction process on context is only apparent. Two separate notions of context are involved: the context of mathematical objects and a set of external factors. The person who abstracts gradually ignores the context of the various mathematical objects. However, the set of external factors may influence this process of abstraction. In the cognitivist approach, the context that may influence the process of abstraction is thus considered as a set of external factors. In the next subsection, we take a different position with respect to context.
Recently, several authors have criticized the classical approach and proposed other approaches. For example, Ohlsson and Lehtinen (1997) stated that to identify an object as an instance of an abstraction, the knower must already possess that abstraction in some way. The cognitive mechanism of abstraction is the assembly of existing ideas into more complex ideas. Thus, the process does not lead unidirectionally from concrete to abstract. Concrete and abstract are not separate entities but are linked rather than detached during the process of abstraction.
Even more fundamentally, Confrey and Costa (1996) criticized the primacy given in the classical approach to the very notion of mathematical object. They claimed that this primacy may reinforce a narrow perspective of the mathematics community because it separates mathematical thinking from its origins in social contexts and neglects the development and use of mathematical tools. Others have expressed criticism of the fact that abstraction is considered as a mental activity of a solipsistic character in which the role of the environment (social interactions, tools) is disregarded (e.g., Greeno, 1997). Noss and Hoyles (1996) situated abstraction in relation to the conceptual resources students have at their disposal: When students progress through a succession of activities (in a social context, in the presence of tools), they learn to attune practices from previous contexts to new ones. Therefore, according to Noss and Hoyles, students do not detach from concrete referents at all. On the contrary, there is a process of webbing, in which students connect to previous similar activities and capitalize on the tools they have at their disposal to construct new mathematical knowledge. However, Noss and Hoyles did not clearly articulate the link between webbing and the construction of new knowledge and thus did not provide a framework within which to investigate the process of abstraction.
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