Synthesis


 * Monday, Feb. 22**: Sketching out and supporting big ideas.


 * Constructivism** (cognitive theory) including:


 * Radical Constructivism** (Piaget, von Glasersfeld)
 * Everything you know is constructed.
 * Corollary: Behaviors are not enough to tell if a student "knows" in the same way you do
 * Corollary: "Transfer" of knowledge (in the sense of a computer transfering data) is impossible because the students' constructions are unpredictable
 * Corollary: Constructions are often domain-dependent
 * Learning during instruction involves folding back (Pirie & Kieren, 1994)
 * Learning is an active process
 * A learner constructs his or her own mathematical knowledge according to the goals he or she has in mind when constructing this knowledge
 * The role of culture/society, communities of practice, and tools/resources outside of the individual is not a point of focus (it is part of the environment/experience and is not given preferential treatment). The focus in constructivist theories is on the individual.


 * Social constructivism** (Bauersfeld, Ernest, Confrey?)
 * An attempt to place culture/society, communities of practice, and tools/resources outside of the individual on an equal footing with the individual.


 * Sociocultural theory** (Vygotsky)
 * Learning is active
 * Learning takes place in social milieu
 * At a minimum, a strong relationship was noted between sociocultural and constructivist theories. Some felt Vygotsky to be a constructivist
 * The role of culture/society, communities of practice, and tools/resources outside of the individual is central to or focused on by these theories
 * Situated theories of learning?


 * Learning is an active and inductive process**


 * Learners transform experiences into knowledge in different ways. Enactive (manipulating objects), Iconic (governed by perceptual organization), Symbolic (by means of language, reasoning, and other systems of meaning). Bruner (date?)
 * Concrete knowledge to image having to (Pirie and Kieren, 1994)
 * Knowledge is built inductively?
 * Additional support
 * Sfard and Linchevski (1994) - the notion of reification. Object/Process duality
 * Dubinsky (1991) - Reflective abstraction and Action-Process-Object-Schema (APOS) theory
 * Piaget (1978) - Reflective abstraction (?)
 * Simon, Tzur, Heinz, and Kinzel (2004) - "Key to the identification of an activity sequence is that it is available to students at their //current// conceptual level." (p. 322, emphasis in original)


 * Monday, March 8.** Struggling to understand the difference between "reflection" and "reflective abstraction"


 * Both are essential to the learning of mathematics**
 * Confrey (1990), p. 109
 * Dubinsky (1991)
 * Piaget (1964)
 * Simon, Tzur, Heinz, and Kinzel (2004)
 * Sfard and Linchevski (1994)
 * Steffe and D'Ambrosio (1995) include a section (pp. 155 - 156) on "Reflection in Teaching Mathematics"
 * von Glasersfeld (1995)


 * Reflection defined**: Steffe and D'Ambrosio (1995). Quoting von Glasersfeld: "the capability that allows us to step out of the stream of direct experience, to re-present a chunk of it, and to look at it //as though// it were direct experience" (p. 155, emphasis in original).
 * A social or individual action
 * A construct that should be defined in one's writing
 * "Appropriately encouraged after one or more individuals have made what Piaget called a reflective abstraction" (Steffe and D'Ambrosio, 1995, p. 155).
 * Simon, Tzur, Heinz, and Kinzel (2004) elaborates on the role of reflection in learning. Reflection plays a role in resolving the learning paradox. Reflection on activity-effect relationships leads to learning.


 * Reflective abstraction defined**: See von Glasersfeld (1995), "Whatever results reflection upon these mental processes produces, are then called 'reflective abstraction.'" (p. 69). So reflective abstraction is the result of reflection upon mental processes.
 * An individual, mental action
 * A Piagetian construct (does not beg a definition in writing)
 * A type of abstraction formed through reflection
 * Reflective abstraction is "the big picture" that reification, encapsulation, interiorization, objects, and processes make up
 * It is both the product and the process ... ? I had trouble with this (I will reflect on it)
 * So, an object //is// an encapsulation (reification). Reifications, encapsulations, interiorizations ... these are all reflective abstractions
 * Reflective abstraction is also an action (e.g. the reification of a process into an object)
 * The product and process aspects of reflective abstraction are inexorably linked. One cannot be extracted from the other
 * Reification (Sfard and Linchevski, 1994) is one type of reflective abstraction
 * Encapsulation and interiorization (Dubinsky, 1991) are types of constructions that result from reflective abstraction
 * Objects and processes (Dubinsky, 1991; Sfard and Linchevski, 1994) are reflective abstraction

For next time, what characterizes situatedness? In what ways does situatedness account for the social aspects of learning that "active intellectual engagement" may neglect?


 * Monday, March 15.** Sitatuated cognition.

Situated cognition is a kind of reaction to the first wave of the cognitive revolution. The first wave was a reaction to the stimulus-response paradigm of behaviorism and is represented by "mainstream cognitive psychology." In some ways, however, mainstream cognitive theories are still "nonhuman," for example information processing theory, and are still essentially stimulus-response.

The second wave of the cognitive revolution evolved as a reaction to the limited emphasis on affect, context, culture, and history. (Cobb and Bowers, 1999, p. 5).

From the situated perspective, knowing is an activity that is situated with regard to an individual's position in the world of social affairs (Cobb and Bower, 1999, p. 5).

Situated cognition and sociocultural theories both derive from the work of Vygotsky and so it is difficult to clearly delineate them
 * Both theories account for context in the construction of knowledge.
 * Theories of situated cognition are typically focused on one community, while sociocultural theory may take a broader perspective.
 * Change in identity and change in forms of participation are more in line with a sociocultural perspective than with a situated perspective
 * In sociocultural theory, practices in the community are somewhat more established than in a situated perspective and a novice changes their forms of participation as they become a member of the community. In the situated perspective, the community itself may evolve.
 * For example the practices of mathematics education are currently established and we are being enculturated into the practices of the community. This is a sociocultural perspective.
 * As a local, classroom community, we negotiate shared understandings of the readings. Our practices and understandings evolve. This is a situated perspective.
 * To some extent, time scale seems to be a distinguishing characteristic. Of course, from a sociocultural perspective, culture evolves. But what is more important is the way that an individual is enculturated into a (relatively) fixed/ currently established set of cultural practices.
 * Another distinguishing characteristic is the breadth. In a situated perspective, the "local microworld" is of interest. In sociocultural theory, the culture into which individuals are enculturated is a larger, more global entity.
 * Still, these perspectives have much in common. The notions of "novice" and "apprenticeship," for example, are shared.

Situated cognition is not oriented toward a particular epistemology (whereas constructivism and main stream cognitive psychology are). The teacher is a "semiotician" (van Oers, 2000). She is a representative of the community, an influencing participant of the community. This doesn't mean that individual students are unimportant. On the contrary, they are significant participants. Epistemology simply does not come up in situated cognition. What is important are the established and emerging practices of communities.

Next week: more on situated cognition and sociocultural theory.


 * Wednesday, March 24, 2010 (And continued Monday, March 29, 2010)**

What are the strengths/affordances and weaknesses/limitations to each theory? What metaphor (Sfard, 1998) does each align itself with?

__Behaviorism__ (“Old school” “hard science”), experimental studies?

Learning outcomes, not processes Acquisition metaphor Logical positivism Transfer? Benny – instructional failure Back to basics (learning objectives)

// Strengths // Facts/memorization Tangible effects of instruction Makes assessment (of factual/procedural knowledge) straightforward/objective Change sequence of stimuli to optimize intended behavior, reinforce desired response

// Weaknesses // Cannot assess reasoning

Question: can you have quantitative (for instance a pre-post test) studies from a non-behaviorist research perspective?

__Constructivisim__

von Glasersfeld, Piaget

Post-epistemological

Meaning: making as part of learning (Van Engen) Kant, Dewey (see Phillips, 1995) Piaget

The focus is on an individual’s conception (the answer, how they got there, and the justification of the answer), ie, unit of analysis = the individual

Focuses on measuring reasoning

Focuses on processes of thinking of students (individual and qualitative differences) that can inform instructional design

Affordances: Well-developed. A variety of theories offer flexibility Multiple perspectives on "how knowledge is constructed" Can inform instructional design (e.g., Simon, 1995; Confrey, 1990; Dubinsky, 1991; Pirie & Kieren, 1994; Sfard & Linchevski, 1994?)

Limitations: Does not focus on social aspects of learning Implications for instruction can be incomplete/problematic (not a "full resource") (see Cobb & Yackel, 1996; Cobb & Bowers, 1999; Cobb, 1998)

__Sociocultural__

Strongly present in the 1990s Ethnography Evolved from AM to PM

Vygotsky, individual mind in culture (AM, focused on individual cognition) Leont'ev (PM?) Wertsch

Knowledge is embedded in culture. Culture is a lens Local/Global culture Participation with established cultural practices

Unit of analysis: AM - focus on individual cognition PM - activity in context of culture

Meaning:

Affordances: Can switch between local/global focus

Limitations: PM may cause loss of focus on learning?


 * Monday, April 5. "Learning is active" from a sociocultural perspective (the ideas seemed to be forming, so the notes are not so strong)**

Vygotsky (1978) supports notion that learning is active from a social perspective. ZPD. Lave and others consider this a somewhat passive point of view. But still, the learner must be active to interact with the capable other.

Wenger (p. 53) "living meaningfully implies ... an active process of producing meaning that is both dynamic and historical." In Wenger, construction of meaning is knowing. Meaning is constantly renegotiated. Constanty transforming.

See also Alexei Leont'ev and James Wertsch in Confrey.

Brown, Collins, and Duguid, cognition is a product of activity. On the very front, p. 32.

Sfard's PM. If learning is participation...

Wenger. Participation may be peripheral.

In Vygotsky, you initially require a capable other. Their help is removed bit by bit. Now you are able to function independently. "Self-regulated mental processes." This is, by nature, an active process (from this perspective, learning is an active process).

See also Stein and Brown. The pie charts.

With respect to Wenger's negotiation of enterprise, shared goals, mutual engagement. If an individual does not share the goals or the values of a communitiy, the individual may "drop out" of the community. In Wenger, participation is more than simply "being there."

You cannot "make" someone be active. Can only provide opportunities/incentives for activity.

In the participation metaphor, there is a greater emphasis on process than there was in the acquisition metaphor. The emphasis is not solely on the process. Product, in Wenger, may be tied up with the shared goals of the community. Or the shared enterprise of the community.

**Wednesday, April 7, 2010**

 * ~ Modes of Belonging ||
 * Engagement || Imagination || Alignment ||
 * Participating in the group

Needs both participation and reification

“Involvement with mutual negotiation” p. 173

Participating - may not be negotiating meaning - (p. 4) involvement in practices of community and construction of identities - One can participate without active engagement - Broader than engagement (e.g., p. 57, 131) || Requires material to work with reification

Identity in a broader contexts

Transcending time and space

Needs both participation and reification

Recognizing experiences with others

Ability to take an outsider’s perspective

“Risk taking” || Also places identity in a broader context

Needs both participation/ reification

Local-global link “in a sense”

Focusing on finding commonalities ||

“constellations of practice”- communities of practice nested within another community (albeit larger) (p. 136): The flower is participation; the computer is reification

Identity formation is a dual process: identification and negotiability

(see p. 199, 2nd paragraph, 3rd paragraph; p. 200-201). This is almost like a meter stick to determine the value of meaning; ownership of meaning does not imply there is a single meaning; there is A position based on the valued meaning(s) among the plurality. ||
 * Identification || Negotiability ||
 * || Economies of meaning: a value-laden collection of meanings may exist within one community and it may span communities

When one positions himself in a community whose meaning is not quite valued, then the ownership of meaning is not quite high