2. COUNT
STUDY PROJECT
If we do not want to believe that ideas are innate
or Godgiven, but the result of subjective thinkers'
conceptual activity, we have to devise a model of
how elementary mathematical ideas could be
constructed – and such a model will be plausible
only if the raw material it uses is itself not
mathematical.
(Glasersfeld, 64)
3. COUNT
STUDY PROJECT
“mathematics deals with ideas. Not pencil marks or
chalk marks, not physical tirangles or physical sets,
but ideas (which may be represented or suggested
by physical objects” (Hersh, in Glasersfeld, 64)
Ideen gründen auf Erfahrung/Erleben
→ Wie entstehen Ideen aus Erfahrung ?
11. COUNT
STUDY PROJECT
Diese (erste) Übersicht ist
einerseits chronologisch motiviert, was
●
aber der Diskussion bedarf
● Stichwort “Horizontale Decalage”
● was entsteht nacheinander / oder auch zeitgleich?
nicht erschöpfend
●
12. COUNT
STUDY PROJECT
Leitfragen
●
● Wie wird Zahl kognitiv benutzt (über die ganze Spanne der
verschiedenen Stadien)?
● Welche Vorgänge legen Umgang mit und Erlernen von Zahlen nahe?
● Spielwelten / “echte” Welten
● Was wird gezählt ? (konkrete Objekte; Akte; Repräsentationen)
● Verbindung “early numerosity” und “ voll entwickelter Zahlbegriff”
● diese scheint nicht kontinuierlich zu sein
● Was ist “Verstehen” ?
● Kulturbedingte Unterschiede ?
13. COUNT
STUDY PROJECT
Ansatz
●
● Primat der Ordinalität
● (wie Brainerd, Dedekind)
● Kardinaliät folgt erst aus Ordinalität
● SchemaBegriff als Grundkonstrukt der Handlung und Repräsentation
15. “The first task, then, is the distinction of
individually discrete “things” in our
experiential field. To normal adult humans,
who are experienced managers of a more
or less familiar environment, it may seem
absurd to suggest that the segmentation
of their experiential world into discrete
things should not be an ontological given.”
(Glasersfeld p.64)
early arithmetic
Ausgangssituation
(relation/subit.)
Vielheit
subitzing
Objektidentität Objektpermanenz
Einheit SNWS
GLASERSFELD, STEFFE
prä Einheit
numerisch
16. “I believe that the first step in the setting of
a ‘real external world’ is the formation of the
concept of bodily objects and of bodily
objects of various kinds. Out of the multitude
of our sense experiences we take, mentally
and arbitrarily, certain repeatedly occurring
complexes of sense impressions (partly in
conjunction with sense impressions which
are interpreted as signs for sense experiences
of others), and we correlate to them a
concept – the concept of the bodily object.
Considered logically this concept is not identical
with the totality of sense impressions
referred to; but it is a free creation of the
early arithmetic
human (or animal) mind.”
(relation/subit.) (A. Einstein in Glasersfeld, p64)
Vielheit
subitzing
Objektidentität Objektpermanenz
Objektidentität
Einheit SNWS
PIAGET, GLASERSFELD
prä
numerisch
17. “I believe that the first step in the setting of
a ‘real external world’ is the formation of the
concept of bodily objects and of bodily
“The first task, then, is the distinction of objects of various kinds. Out of the multitude
individually discrete “things” in our of our sense experiences we take, mentally
experiential field. To normal adult humans, and arbitrarily, certain repeatedly occurring
who are experienced managers of a more complexes of sense impressions (partly in
or less familiar environment, it may seem conjunction with sense impressions which
absurd to suggest that the segmentation are interpreted as signs for sense experiences
of their experiential world into discrete of others), and we correlate to them a
things should not bean ontological given.” concept – the concept of the bodily object.
Considered logically this concept is not identical
with the totality of sense impressions “Husserl
referred to; but it is a free creation of the
human (or animal) mind.” proposed that the mental operation that
unites different sense impressions into the
concept of a “thing” is similar to the operation
that unites abstract units into the concept of a
number.”
(Glasersfeld, p.65)
early arithmetic
(relation/subit.)
Vielheit
Vielheit
subitzing
Objektidentität Objektpermanenz
Einheit SNWS
GLASERSFELD, STEFFE
prä
numerisch
18. “I believe that the first step in the setting of
a ‘real external world’ is the formation of the
concept of bodily objects and of bodily
“The first task, then, is the distinction of
objects of various kinds. Out of the multitude
individually discrete “things” in our
of our sense experiences we take, mentally
experiential field. To normal adult humans,
and arbitrarily, certain repeatedly occurring
who are experienced managers of a more Husserl
complexes of sense impressions (partly in
or less familiar environment, it may seem proposed that the mental operation that
conjunction with sense impressions which
absurd to suggest that the segmentation unites different sense impressions into the
are interpreted as signs for sense experiences
of their experiential world into discrete concept of a “thing” is similar to the operation
of others), and we correlate to them a
things should not bean ontological given.” that unites abstract units into the concept of a
concept – the concept of the bodily object.
number
Considered logically this concept is not identical
with the totality of sense impressions Brouwer (1949) proposed that the
referred to; but it is a free creation of the
human (or animal) mind.” perceiving subject’s selfdirected
attention “performs identifications
of different sensations and of different
complexes of sensations, and in this way,
in a dawning atmosphere of forethought,
creates iterative complexes of sensations”
(in Glasersfeld, p. 65)
early arithmetic
(relation/subit.)
Vielheit
subitzing
Objektidentität Objektpermanenz
Objektpermanenz
Einheit SNWS
PIAGET, GLASERSFELD
prä
numerisch
19. “I believe that the first step in the setting of
a ‘real external world’ is the formation of the
concept of bodily objects and of bodily Brouwer (1949) proposed that the
“The first task, then, is the distinction of
objects of various kinds. Out of the multitude perceiving subject’s selfdirected
individually discrete “things” in our
of our sense experiences we take, mentally attention “performs identifications
experiential field. To normal adult humans,
and arbitrarily, certain repeatedly occurring of different sensations and of different
who are experienced managers of a more Husserl
complexes of sense impressions (partly in complexes of sensations, and in this way,
or less familiar environment, it may seemproposed that the mental operation that
conjunction with sense impressions which in a dawning atmosphere of forethought,
absurd to suggest that the segmentation unites different sense impressions into the
creates
are interpreted as signs for sense experiences
of their experiential world into discrete concept of a “thing” is similar to the operation
iterative complexes of sensations”
of others), and we correlate to them a
things should not bean ontological given.” that unites abstract units into the concept of a
concept – the concept of the bodily object.
number
Considered logically this concept is not identical
with the totality of sense impressions
referred to; but it is a free creation of the
human (or animal) mind.”
early arithmetic
(relation/subit.)
Vielheit
subitzing
subitzing
Objektidentität Objektpermanenz
ULLER, WYNN, MARAMSSE, STARKEY,
prä Einheit SNWS ANSARI, GEARY, STRAUSS/CURTIS
numerisch
20. “I believe that the first step in the setting of
a ‘real external world’ is the formation of the
concept of bodily objects and of bodily Brouwer (1949) proposed that the
“The first task, then, is the distinction of
objects of various kinds. Out of the multitude perceiving subject’s selfdirected
individually discrete “things” in our
of our sense experiences we take, mentally attention “performs identifications
experiential field. To normal adult humans,
and arbitrarily, certain repeatedly occurring of different sensations and of different
who are experienced managers of a more Husserl
complexes of sensations, and in this way,
complexes of sense impressions (partly in
or less familiar environment, it may seemproposed that the mental operation that
conjunction with sense impressions which in a dawning atmosphere of forethought,
absurd to suggest that the segmentation unites different sense impressions into the
creates
are interpreted as signs for sense experiences
of their experiential world into discrete concept of a “thing” is similar to the operation
iterative complexes of sensations”
of others), and we correlate to them a
things should not bean ontological given.” that unites abstract units into the concept of a
concept – the concept of the bodily object.
number
Considered logically this concept is not identical
with the totality of sense impressions
referred to; but it is a free creation of the
human (or animal) mind.”
early arithmetic
early arithmetic
(relation/subit.)
(relation/subit.)
Vielheit
subitzing
Objektidentität Objektpermanenz
ULLER, WYNN, MARAMSSE, STARKEY,
prä Einheit SNWS ANSARI, GEARY, STRAUSS/CURTIS
numerisch
21. “I believe that the first step in the setting of
a ‘real external world’ is the formation of the
concept of bodily objects and of bodily Brouwer (1949) proposed that the
“The first task, then, is the distinction of
objects of various kinds. Out of the multitude perceiving subject’s selfdirected
individually discrete “things” in our
of our sense experiences we take, mentally attention “performs identifications
experiential field. To normal adult humans,
and arbitrarily, certain repeatedly occurring of different sensations and of different
who are experienced managers of a more Husserl
complexes of sensations, and in this way,
complexes of sense impressions (partly in
or less familiar environment, it may seemproposed that the mental operation that
conjunction with sense impressions which in a dawning atmosphere of forethought,
absurd to suggest that the segmentation unites different sense impressions into the
creates
are interpreted as signs for sense experiences
of their experiential world into discrete concept of a “thing” is similar to the operation
iterative complexes of sensations”
of others), and we correlate to them a
things should not bean ontological given.” that unites abstract units into the concept of a
concept – the concept of the bodily object.
number
Considered logically this concept is not identical
with the totality of sense impressions
referred to; but it is a free creation of the
human (or animal) mind.”
early arithmetic
(relation/subit.)
Vielheit
subitzing
Objektidentität Objektpermanenz
Einheit SNWS
STEFFE/GLASERSFELD, MARMASSE
prä SNWS
numerisch
23. Grounding (Verwurzelung)
Die Entstehung erster Schemata
Erhaltung
distributing
oneone
correspondence
turntaking
tagging
alignment
24. Grounding (Verwurzelung)
Die Entstehung erster Schemata sensomotorSchemata (zB. Kelly Mix)
werden in direkter Interaktion mit der
Umwelt gebildet
können nicht eigenständig von dieser
ablaufen
Erhaltung sind das Grundmaterial für Abstraktionen
distributing
oneone distributing
correspondence
turntaking
turntaking
tagging
tagging
alignment
alignment
MIX
„The construction of number concepts“
25. Grounding (Verwurzelung)
Die Entstehung erster Schemata
"From action to abstraction"
● Was ist die Erfahrungsgrundlage mathematischer
Konzepte?
● nicht Dinge
● sondern Handlungen
Erhaltung
● Warum?
oneone ● Dinge entsprechen mathematischen Definitionen
correspondence
nicht.
● Punkt und Linie haben keine Ausdehnung
● Eine Linie wird aufs Papier gezeichnet. Was ist die
Linie?
● der Strich auf dem Papier ?
● das Ziehen der Hand ?
GLASERSFELD
26. Grounding (Verwurzelung)
Die Entstehung erster Schemata
"From action to abstraction"
● "I propose to think of "point" as the very center of the
area in the focus of attention."(66)
● "What you have to focus on, of course, is not the
Erhaltung wire, nor the space it leaves, but the movements,
beause in movement we feel direction but no lateral
oneone extension."(66)
correspondence
● "To my mind, both these approaches are more
adequate than merely saying that a point has no
extension. They come closer to describing what one
can do to arrive at the concept that has no sensory
instantionation."(67)
● "The line, then, is a reflective abstraction from a
GLASERSFELD uniform movement we make."(67)
27. Grounding (Verwurzelung)
Die Entstehung erster Schemata
Erhaltung
distributing
oneone
correspondence
turntaking
tagging
alignment
GLASERSFELD
28. Grounding (Verwurzelung)
Die Entstehung erster Schemata
Erhaltung
distributing
oneone
oneone
correspondence
correspondence
turntaking
tagging
Objekte gegenüberstellen
alignment
z.B.
SNWS
Objekte mit der SNWS SNWS
verbinden
29. Grounding (Verwurzelung)
Die Entstehung erster Schemata Erhaltung verschiedster Art
Masse, Volumen, Anzahl
Ausgangspunkt für Multiplikation ?
Operationale
Operationale
Verknüpfung
Verknüpfung
Erhaltung
Erhaltung
distributing
oneone
correspondence
Park/Nunes untersuchen die
Frage, ob Multiplikation auf der
turntaking
Grundlage
● wiederholter Addition
tagging oder
● operationaler Verknüpfung
entsteht
alignment
PIAGET
PARK / NUNES
30. Grounding (Verwurzelung)
Die Entstehung erster Schemata
a second dimension ?
a plane requires movements
in two directions
Operationale
Operationale
Verknüpfung
Verknüpfung what's the experiential basis
for the concept “plane”
Erhaltung
Erhaltung
distributing
oneone
correspondence
turntaking
tagging
alignment
32. Operationen (abstrahierende)
psychologische und mathematische Ideen
Operationale Linking
Verknüpfung metaphors
Conceptual
Symbolizing
Blending
Reflektierende
Reflektierende
Abtraktion
Abtraktion
In short, I submit that the three
elementary concepts of arithmetic
– unit, set, and number – are abstractions,
not from physical
objects or other sensory material,
but from mental operations that
thinking subjects
must carry out themselves. DUBINSKY, ARBIB, PIAGET
33. Operationale Linking
Verknüpfung metaphors
Removal from the sensorymotor level Conceptual
Symbolizing
requires Blending
what Piaget has called "reflective abstraction,"
that is, Reflektierende
Reflektierende
Abtraktion
in our terms, the focusing of attention not Abtraktion
on sensorymotor signals but on the results or
products of prior attentional operations.
Something that has been constructed by means
of an attentional pattern is now reprocessed
and used as raw material
for a new sequence of focused and
unfocused pulses.
DUBINSKY, ARBIB, PIAGET
34. Operationale Linking
Verknüpfung metaphors
In the case of the unitary items, this creates
Removal from the sensorymotor level
an abstract, or arithmetic unit, that, Conceptual
Symbolizing
requires
in our view, represents Piaget's Blending
what Piaget has called "reflective abstraction,"
"element stripped of its qualities"
that is,
(cf. the above quotation from Piaget, 1970).
Reflektierende
Reflektierende
Abtraktion
in our terms, the focusing of attention not
The reprocessing of a unitary item does two things: Abtraktion
on sensorymotor signals but on the results or
It separates the attentional pattern
products of prior attentional operations.
(that created the unity)
Something that has been constructed by means
from whatever sensorymotor material it
of an attentional pattern is now reprocessed
contained and focuses an attentional pulse on it.
and used as raw material
In doing so, it creates a
for a new sequence of focused and
new unit that is again bounded by
unfocused pulses.
unfocused pulses.
DUBINSKY, ARBIB, PIAGET
35. Operationale Linking
Verknüpfung metaphors
Conceptual
Symbolizing
Symbolizing Blending
Reflektierende
Abtraktion
The fact is that number
words have become symbols for
us, and as such they symbolize
the counting procedure that leads up
to them, without our having
to carry out
that procedure or even having
to think of it. GLASERSFELD
36. However, once
patterns of mental operations have
been abstracted, they become
mathematical concepts
through association with symbols that
can “point” Operationale Linking
to them without invoking their Verknüpfung metaphors
actual execution.
Conceptual
Symbolizing
Symbolizing Blending
Reflektierende
Abtraktion
GLASERSFELD
41. Anschauung
O. WIENER Zahlkonzept
Verfahren
Verfahren
Produktivität
Systematitzität
Ich habe etwas verstanden,
Kompositionalität
wenn ich weiß wie es sich verhält,
und wie ich damit umgehen kann.
O.WIENER: kognitive Repräsentation (der Zahl)
besteht aus einer Sammlung von Verfahren, mit dieser
umzugehen, und einer Parametrisierung (zB einer
Lautfolge)
Zahlkonzept / Repräsentation
“völlig losgelöst”
42. Anschauung
GLASERSFELD Zahlkonzept
Verfahren
Verfahren
Produktivität
Systematitzität
“mental operations abstracted from experience” (62)
Kompositionalität
Zählen mit Stellenwertsystem
intensionaler Zahlbegriff
Wortfolge wird nun systematisch generiert
statt endlicher, auswendig gelernter Folge (SNWS)
Zahlkonzept / Repräsentation
“völlig losgelöst”
43. Anschauung
FODOR/PYLYSHYN Zahlkonzept
Verfahren
Produktivität
Produktivität
Systematitzität
Produktivität: Systematitzität
Es können unbeschränkt Repräsentationen generiert Kompositionalität
Kompositionalität
werden
→ Zählen ist nicht mehr auf eine endliche Menge von
einzelnen “ZahlIndividuen” beschränkt.
Zahlkonzept / Repräsentation
“völlig losgelöst”
44. Anschauung
FODOR/PYLYSHYN Zahlkonzept
Verfahren
Produktivität
Produktivität
Systematizität:
Systematitzität
Repräsentationen sind nicht atomar Systematitzität
“representations need an articulated internal
structure” (Fodor/Pylyshyn 1988, 24) Kompositionalität
Kompositionalität
bei Zahlen ist dies der Vorgang des Zählens, und
viele weitere Eigenschaften / Strukturen, die sich
daraus ergeben
“3 kommt vor 4”
“4 kommt nach 3”
“30 kommt vor 40”
“40 kommt nach 30”
Zahlkonzept / Repräsentation
“völlig losgelöst”
45. Anschauung
FODOR/PYLYSHYN Zahlkonzept
Verfahren
Produktivität
Produktivität
Kompositionalität:
Systematitzität
Systematitzität
zwei Zahlen zusammengenommen ergeben wieder
eine Zahl
Kompositionalität
setzt Systematizität voraus Kompositionalität
wenn man es weitertreibt, so kann man etwa Z
verstehen als N(positiv) und N(negativ)
→ es lassen sich unterschiedlichste Konstrukte mit
dem Zählen anfertigen
(operationale Verknüpfung?
“Fläche” Glasersfelds ?) Zahlkonzept / Repräsentation
“völlig losgelöst”
46. Anschauung
Anschauung Zahlkonzept
Verfahren
Produktivität
aber wie ist eine Zahl im Denken vorhanden?
Systematitzität
Glasersfeld (1981):
“symbolisiert” (siehe Abstraktion) ? Kompositionalität
→ als Zeichen für einen Prozess, der
im Moment nicht ausgeführt wird, aber bei Bedarf
ausgeführt werden kann
→ es muss Zugriff auf in einem bestimmten
Kontext relevante Verfahren
ermöglichen
Zahlkonzept / Repräsentation
“völlig losgelöst”
47. Anschauung
Anschauung Zahlkonzept
Verfahren
Produktivität
Systematitzität
Vorstellen usw.
Kompositionalität
Wilhelm Wundt bemerkte in der „Logik“, Band 1:
“Wer sich von den Eigenschaften des Dreiecks im
Allgemeinen Rechenschaft geben will, denkt sich
ein bestimmtes Dreieck." Es war dies ein Argument im
philosophischen Streit darum, ob alles Denken
bildhafter Natur ist – also sich als
Bilder verstandener Vorstellungen bedient.
Zahlkonzept / Repräsentation
“völlig losgelöst”