Week+3

__**From: Pirie & Kieren (1994)**__
 * How does the Pirie & Kieren model for growth in understanding relate to the Dubinsky discussion of interiorization, coordination, encapsulation generalization, and reversal? Could these be added to their model in some way? (DOS)
 * Figure 1 (p. 167) shows a visual representation of the eight levels of growth in understanding that fit with their theory. Do the authors present enough warrant to distinguish these levels from one another? Should the model make use of fewer (or more) levels? (JLK)
 * Figures 11, 12, and 13 represent mappings of students' thinking and iterative transitions between different levels of understanding. What implications do these maps, when combined with the authors' main points, have for the practice of mathematics instruction? (More specifically, what about lecturing?) (JLK)
 * Three features of their theory include 'don't need' boundaries, folding back, and complementaries of acting and expressing. Each of these notions seems to suggest qualitatively different aspects for describing the process for students growing in their mathematical understanding. How does the theory describe or take into account a variety of different occurrences for each of these aspects? Is the theory "defined enough" to clear questions related to these aspects? (AJ)
 * In the maps of specific students, is there some typical expected pattern? Is there a competing theory that handles the folding differently? (JMG)

__**From: Dubinsky (1991)**__


 * Of the five kinds of construction of Piaget, I am the least clear about "encapsulation". Are there other mathematical examples to help clear this one up? (DOS)
 * Dubinski presents a discussion of the inadequacies of traditional teaching practices (p 120) with respect to the construct of reflective abstraction. Do his arguments seem to hold? If so, what are the implications for mathematics instruction? (JLK)
 * Dubinski briefly discusses (p 123) the implications for the use of computers in supporting the growth of reflective abstraction. What are some examples of the use of mathematics software (or graphing calculators) that could be used in future research efforts to examine and observe students' reflective abstraction? (JLK) Since the article is 20 years old, did anyone 'run with' Dubinsky's claim and produce anything in the realm of technology related to this specific work? (JMG)

__**From: Sfard & Linchevski (1994)**__


 * Sfard and Linchevski (1994, p.89) claim a strong connection between their discussion of //reification// to Dubkinski's (1991) discussion of //reflective abstraction//: What are some of the differences between these two concepts? (JLK, JMG, AJ)

__**In-Class Discussion**__


 * Commonalities**
 * Iterative nature of learning
 * Folding back (Pirie)
 * Reification itself (Sfard)
 * Schemas and their construction (Dubinsky)
 * Most often not linear (doesn't follow strict steps)
 * Often 'gaps' (Sfard, p.195)
 * Implication that instruction needs to change to conform to how students learn
 * Dualities
 * Process-Object duality (Sfard & Dubinsky)
 * Acting-expressing relationship (Pirie)
 * Operational/structural (Sfard)