Überblick über Einsatzgebiete, Techniken und Nutzen von Virtual Reality und Augmented Reality in der Logistik, erstellt vom Virtual Dimension Center (VDC) in Fellbach.
The document outlines a study on how people understand positional number systems. It discusses:
1) Using qualitative interviews and an online study to present subjects with "problems" in made-up number systems to see how they make sense of them.
2) Early results showing subjects extracting known operations like repetition and enlarging sequences to continue patterns. They also coordinate operations sequentially to test solutions.
3) The proposed quantitative study would measure subjects' ability to extract regularities, coordinate operations to solve problems, and mentally enact the system through training, consolidation, and testing phases. This quantifies the cognitive mechanisms involved in learning number systems.
The document summarizes a study project on the development of mathematical ideas. A team of 4 researchers - Sven Spöde, Stefan Schneider, Benjamin Angerer, and Alexander Blum - are investigating how elementary mathematical ideas could be constructed from non-mathematical raw materials. The study will examine mechanisms of abstraction in psychological and mathematical ideas, such as temporal order and number concepts. The methodology involves reviewing theoretical literature, integrating with developmental psychology and education studies, and qualitative studies on problem solving. The intended outcome is a mechanism for conceptual development and an empirically-motivated framework bridging psychology and AI.
This document discusses Imre Lakatos' view of mathematical discovery as involving proofs and refutations rather than purely deductive proofs. It summarizes Lakatos' analysis of Euler's formula for polyhedra, where counterexamples led to improvements in the conjecture and proof through methods like monster-barring, exception-barring, and lemma-incorporation. Lakatos saw this process as more reflective of human mathematical reasoning compared to the formalist view of deductive proofs alone determining validity. The document outlines Lakatos' view that science involves a similar logic of conjectures and refutations.
Imre Lakatos nació en Hungría en 1922 y desarrolló la teoría de los Programas de Investigación Científica para explicar el crecimiento del conocimiento científico. Según Lakatos, la ciencia avanza a través de programas de investigación que proponen teorías sucesivas para resolver anomalías, protegiendo un núcleo firme central. Los programas son progresivos si generan nuevas predicciones y degenerativos si no lo hacen. Lakatos criticó el falsacionismo de Popper y propuso en su lugar un "fals
Este documento resume la vida y obra de Imre Lakatos, un filósofo húngaro de la ciencia. Describe su metodología de los programas científicos de investigación, la cual propone como una mejora al falsacionismo de Popper. También resalta las críticas que Lakatos hizo a Popper, como su noción ingenua de falsación y falta de contextualización histórica. Finalmente, enumera algunas de las publicaciones más importantes de Lakatos.
Überblick über Einsatzgebiete, Techniken und Nutzen von Virtual Reality und Augmented Reality in der Logistik, erstellt vom Virtual Dimension Center (VDC) in Fellbach.
The document outlines a study on how people understand positional number systems. It discusses:
1) Using qualitative interviews and an online study to present subjects with "problems" in made-up number systems to see how they make sense of them.
2) Early results showing subjects extracting known operations like repetition and enlarging sequences to continue patterns. They also coordinate operations sequentially to test solutions.
3) The proposed quantitative study would measure subjects' ability to extract regularities, coordinate operations to solve problems, and mentally enact the system through training, consolidation, and testing phases. This quantifies the cognitive mechanisms involved in learning number systems.
The document summarizes a study project on the development of mathematical ideas. A team of 4 researchers - Sven Spöde, Stefan Schneider, Benjamin Angerer, and Alexander Blum - are investigating how elementary mathematical ideas could be constructed from non-mathematical raw materials. The study will examine mechanisms of abstraction in psychological and mathematical ideas, such as temporal order and number concepts. The methodology involves reviewing theoretical literature, integrating with developmental psychology and education studies, and qualitative studies on problem solving. The intended outcome is a mechanism for conceptual development and an empirically-motivated framework bridging psychology and AI.
This document discusses Imre Lakatos' view of mathematical discovery as involving proofs and refutations rather than purely deductive proofs. It summarizes Lakatos' analysis of Euler's formula for polyhedra, where counterexamples led to improvements in the conjecture and proof through methods like monster-barring, exception-barring, and lemma-incorporation. Lakatos saw this process as more reflective of human mathematical reasoning compared to the formalist view of deductive proofs alone determining validity. The document outlines Lakatos' view that science involves a similar logic of conjectures and refutations.
Imre Lakatos nació en Hungría en 1922 y desarrolló la teoría de los Programas de Investigación Científica para explicar el crecimiento del conocimiento científico. Según Lakatos, la ciencia avanza a través de programas de investigación que proponen teorías sucesivas para resolver anomalías, protegiendo un núcleo firme central. Los programas son progresivos si generan nuevas predicciones y degenerativos si no lo hacen. Lakatos criticó el falsacionismo de Popper y propuso en su lugar un "fals
Este documento resume la vida y obra de Imre Lakatos, un filósofo húngaro de la ciencia. Describe su metodología de los programas científicos de investigación, la cual propone como una mejora al falsacionismo de Popper. También resalta las críticas que Lakatos hizo a Popper, como su noción ingenua de falsación y falta de contextualización histórica. Finalmente, enumera algunas de las publicaciones más importantes de Lakatos.
This document discusses approaches to distinguishing science from pseudoscience, including logical positivism and critical rationalism. Logical positivism held that sense experience and positive verification are the only ways to obtain authentic knowledge, while critical rationalism proposed falsifiability as the demarcation criterion. Specifically, a theory is scientific if it is logically consistent and falsifiable without ad hoc adjustments. For example, observing a single black swan would falsify the conjecture that all swans are white. While an improvement, Popper noted that confirmation is still not a valid criterion for demarcation, as theories shape what is observed.
The document summarizes the key ideas from the book "Where Mathematics Comes From" by George Lakoff and Rafael Núñez. The authors argue that mathematics is not a purely abstract, logical system but is grounded in human cognition and experience. They claim that mathematical concepts are built up over time through image schemas, metaphors, and conceptual blends that originate from basic embodied experiences like containment and motion. According to Lakoff and Núñez, abstract mathematical ideas are understood via more concrete embodied concepts, and mathematical inferences are inherited from the spatial logics of these source domains through conceptual metaphor.
The document outlines Thomas Kuhn's model of scientific progress, which includes five stages:
1) Normal science operates within an established paradigm and focuses on puzzle-solving.
2) Anomalies that do not fit the paradigm begin to emerge.
3) As anomalies accumulate, they lead to a period of crisis as the paradigm is challenged.
4) A scientific revolution overthrows the existing paradigm and replaces it with a new one that can account for the anomalies.
5) The new paradigm becomes established as normal science, and the cycle repeats.
Imre Lakatos
Imre Lakatos, nacido Imre Lipschitz (Debrecen, Hungría, 1922 - Londres, 1974), fue un matemático y filósofo de la ciencia húngaro de origen judío que logró salvarse de la persecución nazi cambiando su apellido. En 1956 huyó a Viena escapándose de las autoridades rusas luego de la fallida revolución húngara abortada por los soviéticos y posteriormente se estableció en Londres, donde colaboró en la London School of Economics.
En sus comienzos se adscribió a la escuela de Karl Popper. Lakatos, en lo que él denomina el falsacionismo sofisticado reformula el falsacionismo para poder resolver el problema de la base empírica y el de escape a la falsación que no resolvían las dos clases anteriores de falsacionismo que él llama falsacionismo dogmático y falsacionismo ingenuo. Lakatos recoge ciertos aspectos de la teoría de Thomas Kuhn, entre ellos la importancia de la historia de la ciencia para la filosofía de la ciencia. Lakatos cuestiona a Popper, pues la historia de la ciencia muestra que los científicos no utilizan la falsación como criterio para descartar teorías enteras, como Popper defendía, sino para hacer que éstas se desarrollen y perfeccionen. Y, por otra parte, la confirmación de los supuestos científicos también es necesaria, según Lakatos, pues nos permite mantenerlos vigentes.
Thomas Kuhn & Paradigms (By Kris Haamer)Kris Haamer
Thomas Kuhn was a physicist and philosopher known for his work "The Structure of Scientific Revolutions" which introduced the concepts of paradigms and paradigm shifts. A paradigm is a universally accepted scientific theory that provides models and solutions for a community of scientists. According to Kuhn, normal science operates within an existing paradigm until anomalies emerge that cannot be explained, creating a crisis and leading to a new paradigm that better explains the facts. This process of paradigm shifts advances scientific understanding as new theories provide more accurate ways of viewing reality.
The poem describes Ulysses, an aging king who feels restless and desires further adventure and knowledge. He addresses his crew, urging them to set sail once more to explore beyond the known world. Though aged and weaker than in his youth, Ulysses maintains a determined spirit to continue seeking and learning until his death.
The document discusses research methods and the scientific method. It provides an overview of key figures in the development of science like Galileo, Popper, Kuhn, and Lakatos. It describes Galileo's experiment dropping objects from the Leaning Tower of Pisa to test hypotheses. It also summarizes Popper's concept of falsifiability, Kuhn's idea of paradigms, and Lakatos' attempt to find common ground between Popper and Kuhn.
The document discusses Karl Popper's theory of falsification and its evolution over time. It explains that Popper argued scientific theories are never truly verified, but can be falsified by a single contradictory observation. Theories should aim to be falsifiable to be considered scientific. Later, Popper acknowledged natural selection as testable despite initial doubts. The document also examines criticisms of falsification, such as that theories may not be falsified even when observations contradict them, depending on how the theory is modified in response.
The document discusses research into how numerical concepts and representations develop. It proposes that new concepts are formed through extracting operations from existing concepts, coordinating those operations, and applying and evaluating them through a process of trial and error. An exploratory study found that participants used known operations like counting and arithmetic to develop a system of representations for a fictional base-4 number system. The document concludes that conceptual understanding is grounded through explicitly constructing new operations from older ones.
The document discusses different ways of representing numbers, from the traditional Hindu-Arabic numeral system to more abstract set-theoretic definitions. It also describes experiments conducted to study how people learn and make sense of non-standard base notation systems like a quaternary (base-4) numbering system. The qualitative studies found people struggled with issues like missing symbols, order of variation in multi-digit sequences, and interpreting the value of the symbol "A". The quantitative study aimed to corroborate these findings and investigate specific problems people encountered. Preliminary results found performance was generally good, but that people commonly struggled interpreting the symbol "A" and its value or position in the numbering system.
This document discusses psychologically informed aspects of a general mechanism of intelligence. It explores how developmental psychology, theoretical analysis, and problem-solving tasks can provide insights into these mechanisms. Key concepts discussed include abstraction mechanisms like extraction, coordination, encapsulation and generalization. Experimental investigations observing people solve problems can provide insights into how these abstraction principles intertwine and develop schemas. Points of interest include how subjects generate successors with production rules, justify patterns through analogy to base 10, and operate with numbers by transferring them into base 10 first before operating.
The document is a case study discussing the counting of RSVQ viruses. It mentions counting efforts over time, describes legends for buchstaben for counting, and notes that counting should continue for a longer time on the next slide.
The document is a series of repetitive strings that do not provide any meaningful information. It consists of the repeated phrase "04bs count, case study" with some additional random strings interspersed. The document does not convey any essential facts or details that could be summarized.
This document discusses approaches to distinguishing science from pseudoscience, including logical positivism and critical rationalism. Logical positivism held that sense experience and positive verification are the only ways to obtain authentic knowledge, while critical rationalism proposed falsifiability as the demarcation criterion. Specifically, a theory is scientific if it is logically consistent and falsifiable without ad hoc adjustments. For example, observing a single black swan would falsify the conjecture that all swans are white. While an improvement, Popper noted that confirmation is still not a valid criterion for demarcation, as theories shape what is observed.
The document summarizes the key ideas from the book "Where Mathematics Comes From" by George Lakoff and Rafael Núñez. The authors argue that mathematics is not a purely abstract, logical system but is grounded in human cognition and experience. They claim that mathematical concepts are built up over time through image schemas, metaphors, and conceptual blends that originate from basic embodied experiences like containment and motion. According to Lakoff and Núñez, abstract mathematical ideas are understood via more concrete embodied concepts, and mathematical inferences are inherited from the spatial logics of these source domains through conceptual metaphor.
The document outlines Thomas Kuhn's model of scientific progress, which includes five stages:
1) Normal science operates within an established paradigm and focuses on puzzle-solving.
2) Anomalies that do not fit the paradigm begin to emerge.
3) As anomalies accumulate, they lead to a period of crisis as the paradigm is challenged.
4) A scientific revolution overthrows the existing paradigm and replaces it with a new one that can account for the anomalies.
5) The new paradigm becomes established as normal science, and the cycle repeats.
Imre Lakatos
Imre Lakatos, nacido Imre Lipschitz (Debrecen, Hungría, 1922 - Londres, 1974), fue un matemático y filósofo de la ciencia húngaro de origen judío que logró salvarse de la persecución nazi cambiando su apellido. En 1956 huyó a Viena escapándose de las autoridades rusas luego de la fallida revolución húngara abortada por los soviéticos y posteriormente se estableció en Londres, donde colaboró en la London School of Economics.
En sus comienzos se adscribió a la escuela de Karl Popper. Lakatos, en lo que él denomina el falsacionismo sofisticado reformula el falsacionismo para poder resolver el problema de la base empírica y el de escape a la falsación que no resolvían las dos clases anteriores de falsacionismo que él llama falsacionismo dogmático y falsacionismo ingenuo. Lakatos recoge ciertos aspectos de la teoría de Thomas Kuhn, entre ellos la importancia de la historia de la ciencia para la filosofía de la ciencia. Lakatos cuestiona a Popper, pues la historia de la ciencia muestra que los científicos no utilizan la falsación como criterio para descartar teorías enteras, como Popper defendía, sino para hacer que éstas se desarrollen y perfeccionen. Y, por otra parte, la confirmación de los supuestos científicos también es necesaria, según Lakatos, pues nos permite mantenerlos vigentes.
Thomas Kuhn & Paradigms (By Kris Haamer)Kris Haamer
Thomas Kuhn was a physicist and philosopher known for his work "The Structure of Scientific Revolutions" which introduced the concepts of paradigms and paradigm shifts. A paradigm is a universally accepted scientific theory that provides models and solutions for a community of scientists. According to Kuhn, normal science operates within an existing paradigm until anomalies emerge that cannot be explained, creating a crisis and leading to a new paradigm that better explains the facts. This process of paradigm shifts advances scientific understanding as new theories provide more accurate ways of viewing reality.
The poem describes Ulysses, an aging king who feels restless and desires further adventure and knowledge. He addresses his crew, urging them to set sail once more to explore beyond the known world. Though aged and weaker than in his youth, Ulysses maintains a determined spirit to continue seeking and learning until his death.
The document discusses research methods and the scientific method. It provides an overview of key figures in the development of science like Galileo, Popper, Kuhn, and Lakatos. It describes Galileo's experiment dropping objects from the Leaning Tower of Pisa to test hypotheses. It also summarizes Popper's concept of falsifiability, Kuhn's idea of paradigms, and Lakatos' attempt to find common ground between Popper and Kuhn.
The document discusses Karl Popper's theory of falsification and its evolution over time. It explains that Popper argued scientific theories are never truly verified, but can be falsified by a single contradictory observation. Theories should aim to be falsifiable to be considered scientific. Later, Popper acknowledged natural selection as testable despite initial doubts. The document also examines criticisms of falsification, such as that theories may not be falsified even when observations contradict them, depending on how the theory is modified in response.
The document discusses research into how numerical concepts and representations develop. It proposes that new concepts are formed through extracting operations from existing concepts, coordinating those operations, and applying and evaluating them through a process of trial and error. An exploratory study found that participants used known operations like counting and arithmetic to develop a system of representations for a fictional base-4 number system. The document concludes that conceptual understanding is grounded through explicitly constructing new operations from older ones.
The document discusses different ways of representing numbers, from the traditional Hindu-Arabic numeral system to more abstract set-theoretic definitions. It also describes experiments conducted to study how people learn and make sense of non-standard base notation systems like a quaternary (base-4) numbering system. The qualitative studies found people struggled with issues like missing symbols, order of variation in multi-digit sequences, and interpreting the value of the symbol "A". The quantitative study aimed to corroborate these findings and investigate specific problems people encountered. Preliminary results found performance was generally good, but that people commonly struggled interpreting the symbol "A" and its value or position in the numbering system.
This document discusses psychologically informed aspects of a general mechanism of intelligence. It explores how developmental psychology, theoretical analysis, and problem-solving tasks can provide insights into these mechanisms. Key concepts discussed include abstraction mechanisms like extraction, coordination, encapsulation and generalization. Experimental investigations observing people solve problems can provide insights into how these abstraction principles intertwine and develop schemas. Points of interest include how subjects generate successors with production rules, justify patterns through analogy to base 10, and operate with numbers by transferring them into base 10 first before operating.
The document is a case study discussing the counting of RSVQ viruses. It mentions counting efforts over time, describes legends for buchstaben for counting, and notes that counting should continue for a longer time on the next slide.
The document is a series of repetitive strings that do not provide any meaningful information. It consists of the repeated phrase "04bs count, case study" with some additional random strings interspersed. The document does not convey any essential facts or details that could be summarized.
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”