Definitions


 * List of Terms** (last updated 2/22/2010-NF added dictionary definitions of //cognition//, //knowledge//, //learning//; it would likely be interesting to compare some of these dictionary definition terms to the ways in which various authors have defined them.)

Cognition Cognitive psychology Conceptual relativism Constructivism Constructivist teaching Dialectic Distributed intelligence Epistemic individual Epistemology Experimental psychology Knowledge Learning Metaphysics Ontology Orthography Positivist Epistemology Practical thinking Pragmatic realism Reality with a capital "R" Reflective abstraction Reification Sense (i.e.: "making sense") Sociocultural theory Symbolic interactionism Veridicality Vygotskian approach

Piagetian terms (see bottom)

Definitions: **- Cognition** // “the mental action or process of acquiring knowledge and understanding though thought, experience, and the senses. A result of this; a perception, sensation, notion, or intuition. From //cognoscere 'get to know.'" New Oxford American Dictionary.

//**- Cognitive psychology**// “involves accounting for teachers’ and students’ inferred interpretations and understandings in terms of internal cognitive structures and processes” (Cobb, 2007, p. 4). Cobb (2007) further distinguishes between theories of this type that are concerned with the process of learning versus those that address the development of reasoning in specific mathematical domains (p. 19). The unit in this perspective is the //epistemic individual//. “Cognitive psychology is premised on the assumption that students’ development involves qualitative changes in their mathematical reasoning” (Cobb, 2007, p. 21). Cobb (2007) also claims that often the purpose of research in this domain is to create frameworks that cab be used to inform instructional decisions, yet has found that frameworks themselves are rarely sufficient (p. 21). See Gravemeijer (1994b) for more on how cognitive psychology does not support heuristics for instructional design. In cognitive psychology we can characterize “mathematical learning as a constructive process in the course of which students successively reorganize their sensory-motor and conceptual activity” (Cobb, 2007, p. 22). “The origins of cognitive psychology can be traced to Piaget’s interest in problems of genetic epistemology” (Cobb, 2007, p. 28).

- // **Conceptual relativism** // (as defined by Putnam) (Cobb, 2007, p. 11)

- // **Constructivism** // (Cobb, 2007, p. 10) see also radical constructivism

- // **Constructivist teaching** // is a pedagogy that “claims to translate the theoretical contention that learning is a constructive activity directly into instructional recommendations…is closely associated with the dubious assertion that ‘telling is bad’ because it deprives students of the opportunity to construct knowledge for themselves” (Cobb, 2007, p. 5). Other researchers include Noddings (1990), Ball and Chazan (1994), and J. P. Smith (1996).

- **Dialectic**: the art of investigating or discussing the truth of opinions

- // **Distributed intelligence** // Pea (1985; 1987; 1993) identifies the unit to include the individual, the tools, and the social contexts relative to the individual. This theory developed from cognitive science—not cognitive psychology. The difference is that cognitive science is concerned with the observer’s viewpoint, not the actor’s viewpoint. Cobb (2007) identifies John Seeley Brown (1989), Alan Collins (1992), and James Greeno (1997) as having made contributions to cognitive science prior to their work in distributed intelligence theory (p. 25). As cited in Cobb (2007), research by Dorfler, 1993 and Pea (1993) characterize distributed cognition in that “artifacts do not merely serve to amplify cognitive processes but instead reorganize them” (p. 26). The unit of analysis in this domain includes the classroom norms, discourse, and the ways in which tools are used (Cobb, 2007, p. 27).


 * -** // **Epistemic individual** // refers to what Thompson and Saldanha (2000) call the “idealized student” where particular students are not the main subject of inquiry. This conceptualization can be considered as part of the cognitive theory (Cobb, 2007, p. 20). “Researchers working in this cognitive tradition account for variations in specific students’ reasoning by using the constructs that comprise their framework to develop explanatory accounts of each student’s mathematical activity. This approach enables the researchers to both compare and contrast the quality of specific students’ reasoning and to consider the possibilities for their mathematical development” (Cobb, 2007, p. 20).

- // **Epistemology** // “is concerned with inquiries into both the nature of knowledge and the process by which we come to know” (Cobb, 2007, p. 10).


 * -** // **Experimental psychology** // “refers to the psychological research tradition in which the primary methods employed involve experimental and quasi-experimental designs, preferably with the random assignment of subjects” (Cobb, 2007, p. 4); “the psychological tradition whose primary contributions to mathematics education have involved the development of assessment instruments, particularly norm-referenced tests, and the findings of studies that have assessed the relative effectiveness of alternative curricular and instructional approaches” (Cobb, 2007, p. 15). I think a good example of this would be Senk & Thompson (2003). Proponents of this tradition tend to argue their methods are scientific while others are not (Cobb, 2007, p. 17). Danzinger (1990) contends that experimental psychology is primarily concerned with satisfying administrators (see Cobb, 2007, p. 18).

 **-** // **Knowledge** // “**1** facts, information, and skills acquired by a person through experience or eduction; the theoretical or practical understanding of a subject. • what is known in a particular field or in total; facts and information. • Philosophy true, justified belief; certain understanding, as opposed to opinion. **2** Awareness or familiarity gained by experience of a fact or situation." //New Oxford American Dictionary.//

- //Learning//** “the acquisition of knowledge or skills through experience, practice, or study, or by being taught. • konwledge acquired in this way" //New Oxford American Dictionary.// -** // **Metaphysics** // is defined from Aristotle’s work to be the science of things transcending what is physical or natural.


 * -** // **Ontology** // is defined to be the branch of metaphysics dealing with the nature of being.

- **//Orthograph//y** is defined to be the conventional spelling system of a language; the study of spelling and how letters combine to represent sounds and form words. Cited in Scribner (1984) and Saxe (1988) as referring to the way digits combine via the place-value system to "spell" the numeral for a given number.

- // **Positivist epistemology** // refers to “practical reasoning is accounted for in terms of the application of abstract theoretical principles to specific cases” (Cobb, 2007, p. 4). Cobb cites work by Schon (1983, 1986, 1987) including the book entitled //The Reflective Practitioner//.

- //**Practical thinking**// "refers to all thinking that is embedded in larger activities and that functions to carry out the goals of those activities" (Scribner, 1984, p. 9).

- // **Pragmatic realism** // according to Putnam involves “questions concerning the existence of abstract mathematical and scientific entities should be addressed not by making claims about reality … but by examining disciplinary practices” (Cobb, 2007, p. 11).


 * -** // **Radical Constructivism

- Reality with a capital “R”** // is “an imaginary realm t hat has been the center of philosophical debate since the time of Plat. Putnam (1987) refers to it as Reality … to distinguish it from the world in which our lives take on significance and meaning” (Cobb, 2007, p. 10).


 * //-Reflective abstraction

-Reification//**


 * //- Sense//** Originally defined by Leontiev, referenced in Lave, Murtaugh, and de la Rocha page 73. "... sense designates personal intent, as opposed to meaning, which is public, explicit, and literal. Sense derives form the relations of actions and goals to motivated (higher order) activities of which they are a particular realization." To paraphrase, sense is the personalization of public knowledge. One can know something without it making sense. The degree to which knowledge influences an individual to organise actions into motivated - or high order - activities, is the degree to which that knowledge makes sense to them.


 * -** // **Sociocultural theory** // has emerged from work by Vygotsky (1962; 1978; 1981) and deals with the individual in the context of society and culture (Cobb, 2007, p. 15). This theory is concerned with “the process by which people develop particular forms of reasoning as they participate in established cultural practices” (Cobb, 2007, p. 22). Scribner (1984) describes the goal of this theory as: "to understand the formative role of culture in cognitive development" (p. 10). Theorists in this tradition investigate the unit as the “individual-in-cultural-practice” (Cobb, 2007, p. 22). Leont’ev (1978; 1981) was also a prominent theorist in this domain (Cobb, 2007, p. 23). A relevant body of research within the sociocultural practice includes the notion of “a community of practice” (Cobb, 2007, p. 24). “Distributed theorists usually conduct detailed analysis of reasoning processes that are stretched over people and the aspects of their immediate environment that they use as cognitive resources” (Cobb, 2007, p. 27). Examples of studies we've read include: Lave, Murtaugh & de la Rocha (1984); Scribner (1984); and Saxe (1988).

- // **Symbolic interactionism** // involves “negotiation of meaning, students should be encouraged to continually discuss their differing interpretations” (Cobb, 2007, p. 5).


 * -Veridicality**: coinciding with reality

- // **Vygotskian approac****h** // involves that “students’ use of symbols and other tools profoundly influences both the process of their mathematical development and its products” (Cobb, 2007, p. 13).


 * __Piagetian Terms__**
 * Accommodation ** : changes in a cognitive structure to include new information
 * Adaptation ** : adjustment to new information about the environment
 * Assimilation ** : incorporation of new information into an existing cognitive structure
 * Centration ** : tendency of preoperational children to focus on one aspect of a situation and neglect others
 * Circular reactions ** : processes by which an infant learns to reproduce desired occurrences originally discovered by chance
 * Class inclusion ** : understanding of the relationship between a whole and its parts
 * Conservation ** : awareness that two objects that are equal according to a certain measure remain equal in the face of perceptual alteration so long as nothing has been added to or taken away from either object
 * Decenter ** : to think simultaneously about several aspects of a situation
 * Deferred imitation ** : reproduction of an observed behavior after the passage of time
 * Egocentrism ** : inability to consider another person’s point of view
 * Equilibration ** : the tendency to seek a stable balance among cognitive elements
 * Genetic epistemology ** : the study of the origins of knowledge
 * Horizontal decalage ** : inability to transfer learning about one type of conservation to other types, which causes a child to master different types of conservation tasks at different ages
 * Hypothetical-deductive reasoning ** : ability to accompany the state of formal operations, to develop, consider, and test hypotheses
 * Invisible imitation ** : imitation with parts of one’s body that one cannot see
 * Irreversibility ** : a preoperational child’s failure to understand that an operation can go in two ore more directions
 * Object permanence ** : the understanding that a person or object still exists when out of sight
 * Organization ** : integration of knowledge into systems
 * Representational ability ** : capacity to store mental images or symbols of objects and events
 * Schemes ** : organized patterns of behavior used in particular situations
 * Seriation ** : ability to order items along a dimension
 * Symbolic function ** : ability to use mental representations to which a child has attached meaning
 * Theory of mind ** : awareness and understanding of mental processes
 * Transduction ** : a preoperational child’s tendency to mentally link particular experiences, whether or not there is logically a causal relationship
 * Transitive inference ** : understanding of the relationship between two objects by knowing the relationship of each to a third object
 * Visible imitation ** : imitation with parts of one’s body that one can see