Math II (562417), страница 17
Текст из файла (страница 17)
In the perspective on abstraction we take in this article, we may similarly be seen as critical of the cognitivist approach. We will discuss two essential differences between our approach and the cognitivist approach. One difference is rooted in how context is conceived, and the other is whether processes of abstraction are linear or dialectic.
Context From a Sociocultural Perspective
Regarding context, the most salient difference between cognitivist and sociocultural approaches concerns the unit of analysis of human behavior. In his analysis of human development, Vygotsky (1934/1986) pointed out that the study of individual actions is doomed to failure. Rather, one must identify the meaningful cultural activities in the course of which individual actions occur. For example, when defining the zone of proximal development, Vygotsky made clear that the learner does not imitate isolated actions modeled by a more capable peer but participates in activities that are meaningful for him. In activity theory, Leont'ev (1981) articulated Vygotsky's implicit view on context. According to this theory, context can be defined as the interconnected collection of factors that frame the structure and meaning of human actions. The activity rather than the individual human action is the unit of analysis because it is "the minimal meaningful context for understanding individual actions" (Kuutti, 1996, p. 28). Activities are chains of actions related by a common content and carried out cooperatively or individually; the common content is designated by the term object. The object can be material, but it can also be intangible (e.g., a problem to be solved or a common idea), as long as the participants in the activity can share it for manipulation or transformation. The activity is driven by the participants' overall goals, which are termed motives. Participants are aware of their motives. Although the motives of participants may vary, the motives of all participants in an activity must be compatible. An activity always includes various artifacts (e.g., instruments, ideas, signs, procedures) through which actions are mediated. Artifacts may be created, manipulated, and transformed during an activity. An outcome of an activity can be an artifact to be used again in later activities (Bodker, 1997). In particular, ideas, strategies, or conceptions may be such outcomes.
The context of an activity is not only an external, objective description of the material conditions of the activity but also includes subjective components such as a participant's personal history, conceptions, and social relationships. As a consequence, context becomes an inseparable component of the activity because participants choose to carry out actions that seem relevant to them in the given context. This inseparability between context and activity is in contrast to the role of conditions that facilitate or alter (mental) actions, a role assigned by cognitivist researchers to contextual factors.
Van Oers (1998) used the activity-theory view of context for conceptualizing abstraction: "Starting from an assumption that conceives of context as constitutive of meaning, it becomes clear that the notion of 'decontextualization' is a poor concept that provides little explanation for the developmental process toward meaningful abstract thinking" (p. 135). One way to describe abstraction without reference to decontextualization has been proposed by Davydov (1972/1990). Davydov's theory leads us to the second major difference between the cognitivist and the sociocultural views of abstraction.
The Dialectic Nature of Processes of Abstraction
Davydov (1972/1990) developed an epistemological theory to account for a dialectical connection between abstract and concrete. Some of the tenets of his theory are similar to principles of activity theory: Practical activity serves as a basis for human thought; during such activity, people are aware of a motive and use tools. They take into consideration not only the properties of artifacts but also their potentialities, and the corresponding thought processes are different in nature. Thought processes that relate to the properties of artifacts may concern similarities and differences and lead to categorization; in contrast, thought processes that relate to potentialities inherent in artifacts include hypothetical thinking and the construction of justifications.
According to Davydov, cognition thus functions at two levels, the level of empirical thought and the level of theoretical thought. One's goal in empirical thought is to interconnect features of reality (for example, by observing similarities of and differences between things), whereas one's goal in theoretical thought is to reproduce reality. As posited by Davydov, everyday conceptions are generally attained through empirical thought. In contrast, scientific concepts are often attained through theoretical thought.
Davydov (1972/1990) described theoretical thought as "an idealization of the basic aspect of practical activity involving objects[1] and of the reproduction in that activity of the universal forms of things, their measures, and their laws" (p. 298). This activity gradually turns into a cognitive experimentation characterized by the fact that one (a) mentally transforms objects during the activity and (b) forms a system of connections between these objects. Theoretical thought consists then of the expression of the symbol-mediated being of objects, of their universality, or (as worded by Davydov) a "theoretical reproduction of reality" (p. 302).
For scientific concepts, empirical thought does not lead to the attainment of abstract knowledge, because this knowledge consists of connections throughout a whole system. To proceed to the construction of abstract knowledge, one needs dialectical logic. The learner needs to consider the links of the new theoretical knowledge with other components within a comprehensive whole to consider possible contradictions and integration. Davydov, in his method of ascent, proposed a description of the genesis of abstraction.
Abstraction starts from an initial, simple, undeveloped first form, which need not be internally and externally consistent. The development of abstraction proceeds from analysis, at the initial stage of the abstraction, to synthesis. It ends with a consistent and elaborate final form. It does not lead from concrete to abstract but from an undeveloped to a developed form of the abstract in which new features of the concrete are emphasized.
Davydov's theory is incompatible with most cognitivist theories, because most cognitivist theorists consider abstraction as a move from the concrete to the abstract. An exception is the approach proposed by Ohlsson and Lehtinen (1997): Their view of abstraction as an organization of existing abstract entities into a more complex structure is compatible with Davydov's move from an undeveloped to an elaborate form of abstraction. There is, however, a difference. Davydov, on the one hand, starts from a single, undifferentiated abstract entity. His process of abstraction consists of establishing an internal structure with internal links and results in a differentiated and structured entity. Ohlsson and Lehtinen, on the other hand, start from two (or more) entities that have been previously constructed; these entities have internal structure (that may be ignored for the present purpose). The process of abstraction consists of one's establishing external connections between the two existing entities with the aim of integrating them into a single, more complex structure.
The dialectical approach described by Davydov is highly relevant to educational research and practice. Much school learning concerns scientific concepts that are not directly attainable through empirical thought. Designers and teachers need to elaborate ready-made sequences of tasks that are intended to mediate the construction of students' scientific knowledge. In this process, they could conceivably gain from relying on Davydov's theory. However, the theory is removed from practitioners, designers, and policymakers. Issues such as how to design sequences of activities leading students to abstraction or how to assess the abstractions they make cannot be addressed by a theory that is epistemological in essence and does not give tools for studying processes of abstraction. This shortcoming exists not because the theory is wrong but rather because practitioners, designers, and educational psychologists have not reworked this theory in the light of the lessons drawn in classrooms, from the development of materials (written, software, methods, etc.) and the description of learning episodes.
In the following sections, we will build on Davydov's theory from our practitioners' point of view and develop it into a functional definition of abstraction and a model to facilitate one's observation of processes of abstraction.
ABSTRACTION: A DEFINITION
As mentioned in the introduction, we come to this research from the point of view of practitioners of mathematics education, and this position confers particular characteristics upon the research. The research presented in this article is an integral part of a long-term research and development project; our main goal in this project is to design and create a learning environment in which students will be engaged in meaningful mathematics. We view our research as being within a comprehensive setting, which includes all aspects of curriculum development from design considerations to large-scale implementation.
A main component of the present cycle of the project (Hershkowitz et al., in press) is an introductory course on functions for Grade 9. The course consists of a sequence of tasks around problem situations with a contextual frame that is thought to facilitate the growth of the function concept; through these problem situations, the mathematical structure of the topic of function is transparent. Most of the inquiry is done in small groups with computational tools at the students' disposal, and students are asked to write group or individual reports in which they report, compare, critique, and reflect on their hypotheses and solution processes.
We were overwhelmed and surprised by what we observed in trial classrooms. What occurred there was different in nature from what we had observed in the previous two decades of development and research. The need to meaningfully describe the observed learning practices led us to adopt the perspective of activity theory. We attempt to analyze students' construction of knowledge when they are investigating problem situations in context. As such, our research is bottom-up research: We do not hypothesize a theory then collect data and check them as to their consistency with the theory. Rather, from an activity-theory perspective, we use naive observation and documentation of the students' actions while they are doing meaningful mathematics. During the analysis and interpretation, we then adopt theories that fit our overall approach to mathematics learning as well as additional theoretical ideas that emerge from the data and from the need to explain these data meaningfully. Therefore the purpose of the data is often to serve as basis or partial basis for the emergence of a theoretical idea. This is the perspective for the present study.
Our goal is to experimentally investigate abstraction; we aim to identify processes in which mathematical abstractions occur and situations that enable learners to appreciate and make efficient use of abstractions. The term abstraction is thus being used to refer both to a process and to an outcome; to distinguish between them, we will also use the terms process of abstraction or abstracting, on the one hand, and abstracted entity, on the other.
Our definition for abstraction will later serve as the theoretical guide to the establishment of an operational model for studying abstraction experimentally. This definition is a result of the dialectical bottom-up approach described above; by means of this approach, we reached successively refined descriptions of abstraction. Our definition is thus a product of our oscillating between our theoretical perspective on abstraction and experimental observations of students' actions, actions we judged to be evidence of abstracting. This oscillation between theoretical principles and experimental data is highly nonlinear. However, our presentation of the oscillation needs to be linear. Thus, the model and the ways in which it emerged from data will be described in the following section, whereas the definition itself and its theoretical roots will be discussed here. These roots, listed below, are based on the epistemological principles and the sociocultural background presented earlier.
• Abstraction is an activity (in the sense of activity theory), a chain of actions undertaken by an individual or a group and driven by a motive that is specific to a context.
• Context is a personal and social construct that includes the student's social and personal histories, conceptions, artifacts, and social interaction.
• Abstraction requires theoretical thought, in the sense of Davydov; it may also include elements of empirical thought.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
