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

decades.

 

David Kirshner

ANSWER:

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