| Von Glasersfeld's answers - January IV 2006  |


Dear Professor von Glasersfeld,


Greetings. I have had occasion, recently, to re-read your
paper, Learning as a Constructive Activity, in Claude Janvier's 1987
edited collection. (For me,this is memory lane, as I heard you
present this paper in Montreal at PME-NA in 1983--one of the first
professional meeting I attended in my mathematics education career.)
I am drawn back to this paper in my effort to better understand the
role of metacognition in conceptual development. In the section titled
"The Construction of Viable Knowledge," you begin by distinguishing
your position from the non-mentalist position of the behaviorists. You
approach this by highlighting the importance of reflection in
conceptual development, where reflection is defined in metacognitive
terms: "I am using 'reflection' in the sense in which it was
originally introduced by Locke, i.e., for the ability of the mind to
observe its own operations" (p. 11). You then go on to argue
for establishing this sort of metacognitive reflection as the "primary
goal" of instruction: "the primary goal of mathematics
instruction has to be the student's conscious understanding what he or
she is doing and why it is being done" (von Glasersfeld, 1987, p. 12).
I find myself struggling to reconcile the centrality of this
metacognitive sort of reflection with what I understand
(unfortunately, only dimly) of Piaget's notion of "reflective
abstraction." As I understand it, Piaget used the notion of reflective
abstraction to focus our attention on the "schemes" that arise out of
the child's experience of his or her repeatedly performed actions. It
is these schemes that are the foundation for the conceptual structures
we might seek to facilitate in our mathematics classes. As I
understand it, the metacognitive sort of abstraction (does Piaget call
it "reflected abstraction?") also can evolve in this process of
development, but this is a secondary sort of effect--not what we should
primarily focus on as educators. My educational concern is
that reform pedagogies in mathematics education have tended to focus
too strongly on promoting STUDENTS' metacognitive self-awareness, at
the expense of highlighting the need for the TEACHER to have strong
models of the students' cognitive structures--a pedagogical focus that
may have evolved, in part, out of how our community has construed your
perspectives. I would appreciate any direction you can offer to me of
writings--your own, Piaget's, or other's--that might help me better
understand these issues. Thank you for enriching our educational
lives with your exploration of constructivist frameworks over many
David Kirshner




Dear Mr. Kirshner,
I take it as a great compliment that you remembered my paper after twenty
years. It's that sort of thing that seems to justify one' efforts! You will find my view
of "metacognition" in the answer to the preceding question. Yout question is deeper.
I do believe that Lock's definition is still valid, but it defines reflection in general and
Paget's "reflective" or "reflecting abstraction" concerns a very particular application. If
your library has the following book:
L.P.Steffe (Ed.), Epistemological foundations of mathematical experience.
New York: >Springer, 45-67. 1991, you can read my chapter on Abstraction,
Re-Presentation, and Reflection; if it doesn't, here is an excerpt that
might answer your question (I have added the mentioned references in
[...]; From empirical abstractions, whose raw material is sens-
ory-motor experience, Piaget, as I said earlier, distin- guished three
types of reflective abstraction. Unfortunately, the French labels
Piaget chose for them are such that they are inevitably confused by
literal translation into English.  The first "reflective" type
derives from a process Piaget calls reflechissement, a word that is used
in optics when something is being reflected, as for instance the sun's
rays on the face of the moon. In his theory of cognition, this term is
used to indicate that an activity or mental operation (not a static
combination of sensory elements) developed on one level is abstracted
from that level of operating and applied to a higher one, where Piaget
then considers it to be a reflechissement. (Moessinger & Poulin-Dubois,
1981 [Moessinger, P. & Poulin-Dubois, D. Piaget on abstraction, Human
Development, 1981, 24, 347-353.], I have translated this as "projection",
which captures something of the original sense.) But Piaget stresses
that a second characteristic is required: Reflective abstraction
always involves two inseparable features: a "reflechissement" in
the sense of the projection of something borrowed from a
preceding level onto a higher one, and a "reflexion" in the sense
of a (more or less conscious) cognitive reconstruction or
reorganization of what has been transferred. (Piaget,1975; p.41)
[Piaget, J. L'equilibration des structures cognitives. Paris:
Presses Universitaires de France, 1975.] At the beginning of the
first of his two volumes on reflective abstraction (Piaget et al. 1977)
[Piaget, J. & collaborators, Recherches sur l'abstraction
reflechissante, Vol.I & II. Paris:Presses Universitaires de France,
1977.], the two features are again mentioned: Reflective
abstraction, with its two components of "reflechissement" and
"reflexion", can be observed at all stages: from the sensory-motor
levels on, the infant is able, in order to solve a new problem,
to borrow certain coordinations from already constructed
structures and to reorganize them in function of new givens. We do
not know, in these cases whether the subject becomes aware of any
part of this. (Piaget et al., 1977, Vol.I; p.6). In the same passage he
immediately goes on to describe the second type of reflective abstraction:
In contrast, at the later stages, when reflection is the work of thought,
one must also distinguish thought as a process of construction and thought as a process
of retroactive thematization. The latter becomes a reflecting
on reflection; and in this case we shall speak of "abstraction
reflechie" (reflected abstraction) or pensee reflexive (reflective thought).
Since the present participle of the verb reflechir, from which both the
nouns reflechissement and reflexion are formed, is reflechissante,
Piaget used "abstraction reflechissante" as a generic term for both types.
It is therefore not surprising that in most English translations the distinction
was lost when the expression "reflective abstraction" was introduced as the standard term.
The situation is further confounded by the fact that Piaget distinguished a
third type of reflective abstraction which he called "pseudo-empirical".
When children are able to represent certain things to themselves but are
not yet fully on the level of concrete operations, it happens
that the subjects, by leaning constantly on their perceivable
results, can carry out certain constructions which, later on,
become purely deductive (e.g. using an abacus or the like for the
first numerical operations). In this case we shall speak of
"pseudo-empirical abstraction" because, in spite of  the fact
that these results are read off material objects as though they
were empirical abstractions, the perceived properties are actually
introduced into these objects by the subject's activities.
(Piaget, et al., 1977; Vol.I, p.6). To recapitulate, Piaget
distinguishes four kinds of abstraction. One is called "empirical"
because it abstracts sensory-motor properties from experiential
situations. The first of the three "reflective" ones, projects and
reorganizes on another level a coordination or pattern of the sub-
ject's own activities or operations. The next is similar in that it also
involves patterns of activities or operations, but it includes the
subject's awareness of what has been abstracted and is therefore called
"reflected abstraction". The last is called "pseudo-empirical" because,
like empirical abstractions, it can take place only if suitable
sensory-motor material is available. (I cannot give you the page
numbers because three years ago a fire destroyed my library. Fortunately
I had most of my papers stored elsewhere!) If this doesn't answer your
question, don't hesitate to ask again


Best wishes

Ernst von Glasersfeld




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