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Typisierung des Verständnisses mentaler Modelle mittels empirischer Datenerhebung am Beispiel der Quantenphysik
Abstract and Figures
Verständnisprozesse sind zentraler Teil des Lernens und der Bildung. Beim Erlernen von Quantenphysik sind diese Prozesse jedoch oft durch klassische Vorstellungen blockiert. Zur genaueren Erörterung dieser Problematik wird eine Studie vorgestellt, die klassische und quantenphysikalische mentale Modelle von Lernenden erhebt und zu dem allgemeinen Modellverständnis in Beziehung setzt.
Die Datenerhebung erfolgte per Onlinefragebogen und deckt eine breite Probandengruppe ab, zu der neben Lernenden verschiedener Schul- und Hochschulformen auch Lehrerinnen und Lehrer sowie viele andere Berufsgruppen gehören. Die empirischen Daten weisen darauf hin, dass die jeweilige Gestalt und Funktionalität der mentalen Modelle unabhängig voneinander in Bezug auf ihre Realitätstreue interpretiert werden. Aus dieser Beschreibung werden vier Verständnistypen mentaler Modelle abgeleitet und in die derzeitige
naturwissenschaftsdidaktische Erkenntnislage eingeordnet.
Keywords: mental model, conceptual development, understanding, physics education, chemistry education, biology education, educational science, neurology
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... One of the more recent endeavors to describe how learners understand models in science, refers to what we summarize as the Gestalt-Function-hypothesis according to which (quantum) models are perceived depending on the degrees of 'Fidelity of Gestalt' and 'Functional Fidelity' in students' thinking: The Gestalt-Function-hypothesis was first in principle proposed by Ubben and Heusler [12,17] and was based on an exploratory factor analysis of students' statements about physics models in general and about statements of various models for the atomic shell in particular. The results indicated a latent two-factor structure describing the ideas about models held by the 3108 participants surveyed. ...
... However, the results of this first study were only a description of how models of a very minor part of quantum physics (namely atoms) were perceived by learners, and it was thus by no means clear that the two-factor structure reported was generalisable to other areas of quantum physics, or even physics and sciences in general. Still, there were reasonings that implied a more general applicability of the proposed two-factor structure of learners' model perceptions: it was argued in [17] that in neurological research, visuo-structural and functional-conceptual processes were roughly mediated by different brain regions and that the structures might be thus transferable to model perceptions in general [18,19]. Therefore, several follow-up (exploratory) studies have been conducted that examined whether the Gestalt-Function-hypothesis held true for describing learners' perceptions of models in other areas of quantum physics, in astrophysics and even in mathematics, which we will shortly elaborate upon in the following. ...
... Interestingly, not only visual but also olfactory statements and statements about touch were documented regarding molecules: [36] not only reported that molecules were seen as colored balls, but also that they smell like the substances they constitute and also feel the same (e.g., hot or cold). At this point, it has to be emphasized that the term gestalt is not only limited to visual cues but auditory, haptic or olfactory cues as well [11,17]. Still, in this article we focus on gestalts of visual modality. ...
In previous research, it has been argued that many of the student (mis-)conceptions of quantum concepts described in the literature as widespread among learners can be traced back to poorly developed (quantum) model perceptions that hinder the learning of quantum physics. In particular, it has been shown that the degrees of two cognitive dimensions, namely Functional Fidelity and Fidelity of Gestalt, in students’ thinking account for a substantial amount of the variance in students’ model perceptions in quantum physics and may therefore be useful for describing and understanding the (development of) students’ conceptions of quantum physics topics. So far, however, the cognitive dimensions Functional Fidelity and Fidelity of Gestalt have only been investigated in exploratory studies. In this article, we report the results of a confirmatory factor analysis of data collected from N = 179 secondary school students using an instrument adapted from the literature to assess learners’ perceptions of the photon model. The results of our study provide empirical evidence that the two-factor model of learners’ model perceptions in the quantum context is indeed a good fit to the data. Together with literature from science education research on students’ conceptual development, and taking into account earlier findings on Fidelity of Function and Gestalt Fidelity we derive a plausible description of students’ conceptual development in the context of quantum physics – leading to what we call the Fidelities-Model of Conceptual Development. We discuss this framework in the light of previous research and argue for its potential generalisability beyond the teaching and learning of quantum physics topics. The implications of our findings for both science education research and practice are presented.
... physics and their effects on learning (e.g., see Michelini et al., 2000;Weissman et al., 2022;Aehle et al., 2022;Pospiech, 2021;Seskir et al., 2022) have been in the focus of quantum physics education research. Recently, research has indicated that students' understanding of models can facilitate the development of a deeper students' conceptual understanding of quantum physics aspects (Ubben, 2020;Ubben andBitzenbauer, 2022, 2023). Considering that developing an understanding of models holds significance not only in the realm of quantum physics education but also for acquiring knowledge of science in general, adopting a learning approach that considers this perspective has the potential to not only enhance our understanding of teaching and learning quantum physics but also contribute to the broader field of science education research. ...
... physics and their effects on learning (e.g., see Michelini et al., 2000;Weissman et al., 2022;Aehle et al., 2022;Pospiech, 2021;Seskir et al., 2022) have been in the focus of quantum physics education research. Recently, research has indicated that students' understanding of models can facilitate the development of a deeper students' conceptual understanding of quantum physics aspects (Ubben, 2020;Ubben andBitzenbauer, 2022, 2023). Considering that developing an understanding of models holds significance not only in the realm of quantum physics education but also for acquiring knowledge of science in general, adopting a learning approach that considers this perspective has the potential to not only enhance our understanding of teaching and learning quantum physics but also contribute to the broader field of science education research. ...
... In general, older high school students or university students are found to be of the latter two types (Ubben and Heusler, 2021). The difference between them is the amount of Fidelity of Gestalt ascribed to the model, with Functional Fidelity being relatively high in the respective age group (Ubben, 2020;Ubben and Bitzenbauer, 2022). Therefore, this aspect was decided to be the focus of talking about models in the course as well: One approach to enable learners to abstract quantum physical concepts from given models is therefore to make them explicitly aware of the low Fidelity of Gestalt of models. ...
Fostering students' understanding of models is a challenge. However, in particular for learning quantum physics an elaborate understanding of models is required. We investigated activities to foster students' functional thinking about (quantum) models in a synchronous online course. The results of an evaluation study (N = 59) showed that the participants improved in their quantum physical thinking about photons and had slightly improved their understanding of physics models in general. A correlation analysis indicates that there are no significant correlations between the students' general understanding of models in physics and their functional understanding of quantum models. Implications of our findings for both teaching and future research with regard to quantum physics education are discussed.
... Whether in formal or informal settings, it is essential for students learning quantum physics to work with and understand models: One of the key challenges in learning quantum physics is that many of the phenomena that are observed at the quantum level cannot be directly observed or intuitively understood in terms of classical physics or thinking. Hence, models are needed to bridge this gap by providing conceptual frameworks and visual representations that allow to reason about and predict the behavior of quantum systems and are thus tools for describing and understanding quantum phenomena and principles (Ubben, 2020). Therefore, learning about quantum physics, at its core, is about learning about models as has been pointed out by Stefani and Tsaparlis (2009). ...
... This finding is consistent with previous studies in the fields of (a) electrons in the atomic hull Ubben and Heusler (2021) and (b) photons Bitzenbauer and Meyn (2021b). A possible explanation previously given for this observation (Ubben, 2020;Ubben and Bitzenbauer, 2022) is indicated by neurological research, where hemispheric specialisations regarding function (left hemisphere) and appearance (right hemisphere) were found to work independently of each other (Levy and Trevarthen, 1976). For a more extensive review of this field of research, see Gazzaniga (2005). ...
... Our findings may inform teaching practice of quantum sciences in general and quantum optics in particular: As high-achieving students tended to have a low score in Fidelity of Gestalt, designing educational content with the goal of reducing the Gestalt-oriented thinking among students is likely to improve the understanding of learners (Dutt, 2011). However, as pointed out in Ubben (2020) as well, this step in cognitive development is the step of abstraction, namely, of letting go of the context and making knowledge transferable to other contexts. This step is often described as being the last step in conceptual development by various theoretical frameworks (see Section 7.2). ...
Learning quantum physics is essential for understanding the physical world. However, learning about quantum phenomena and principles poses a challenge as many of the phenomena that are observed at the quantum level cannot be directly observed or intuitively understood in terms of classical physics or thinking. Models play an important role in learning quantum physics by providing conceptual frameworks and visual representations that allow reasoning about and predicting the behavior of quantum systems. Therefore, understanding models is an essential part of learning quantum physics. In this article, we report the results of an exploratory survey study (N = 116) investigating the relationship between secondary school students' conceptual understanding and model thinking in quantum optics with a particular focus on photons. The findings suggest a strong positive correlation between students' functional understanding of the photon model and their conceptual understanding of quantum optics. This study contributes to our understanding of how students learn and make sense of quantum concepts through the use of models and may inform the development of instructional strategies for quantum physics education and outreach.
... Eine umfassende Darstellung über den Stand der Schülervorstellungsforschung in der Quantenphysik gibt Ubben in seiner Arbeit, auf die insbesondere für die detaillierte Beschreibung der einzelnen Studien verwiesen sei [207]. An Lernendenvorstellungen zur Natur des Lichts: Untersuchungen zur Natur des Lichts beziehen sich in aller Regel auf den Dualismus von Wellen und Teilchen. ...
... Das Fallen lassen deterministischer Vorstellungen ist für Lernende eine große Hürde, wie Müller bei [175, S. 214f] schreibt. In der bereits oben zitieren Studie von Bethge, die in den Arbeiten[207,143] genauer erläutert ist, konnten drei Antwortkategorien Lernender zur Wahrscheinlichkeitsdeutung gefunden werden (vgl.Sprechen über klassische Wellen und Teilchen könnte die Ausprägung berichteter Hybridvorstellung eindämmen. Stattdessen mag die Betonung zentraler Begriffe der Quantenphysik, wie die Präparation, geeignet sein die Quantenphysik -nun losgelöst von der Mechanik -zu entwickeln und somit verbreitete Lernschwierigkeiten zu vermeiden. ...
... Mehr als die Anfänge sind nun gemacht: ein anerkanntes begriffliches Rahmenwerk zum Lernen von und Sprechen über Quantenphysik ist längst etabliert[143,110]. Auch über die Struktur der Vorstellungen Lernender liegen immer genauere Erkenntnisse vor[207]. Außerdem wurde in verschiedenen Arbeiten in der jüngeren Vergangenheit auf die Notwendigkeit einer Modernisierung des Quantenphysikunterrichts ganz konkret hingewirkt[26,214]. ...
Die Quantenphysik bildet schon heute das Fundament zahlreicher aktueller Technologien. Zukünftige Quantentechnologien, wie Quantencomputer, werden sowohl in der Industrie als auch für die Gesellschaft an Bedeutung gewinnen.
In vielen nationalen und internationalen Schulcurricula ist die Quantenphysik als Thema für den Physikunterricht mittlerweile fest verankert. Aber trotz des enormen Bedeutungszuwachses von Quantentechnologien ist der Unterricht zur Quantenphysik an Schulen nach wie vor von semi-klassischen Modellen und quasi-historischen Zugängen geprägt, während moderne Begriffe der Quantenphysik häufig unberücksichtigt bleiben. Die Folge sind oft klassisch-mechanistisch geprägte Vorstellungen Lernender zur Quantenphysik.
Hier setzt diese Arbeit an: mit dem Erlanger Unterrichtskonzept zur Quantenoptik wird ein Konzept vorgestellt, mit dem Lernende der gymnasialen Oberstufe Quanteneffekte anhand quantenoptischer Experimente kennen lernen. Konzepte der Quantenoptik, wie die Präparation von Quantenzuständen, die Antikorrelation am Strahlteiler und die Einzelphotoneninterferenz verhelfen Lernenden zu einem modernen Bild über Quantenphysik. Im Rahmen einer summativen Evaluation im Mixed-Methods-Design mit 171 Schülerinnen und Schülern zeigte sich, dass Lernende mit dem Erlanger Unterrichtskonzept zu quantenphysikalisch dominierten Vorstellungen gelangen und verbreitete Lernschwierigkeiten vermieden werden können.
... Eine große Problematik hierbei ist allerdings, dass viele Ideen, Prozesse und Phänomene der Quantenphysik aus physikalischen Gründen nicht einfach im dreidimensionalen Raum visualisiert werden können. Es ist daher eine große Herausforderung, Lernenden Repräsentationen an die Hand zu geben, von denen ausgehend sie die komplexen quantenphysikalischen Phänomene verstehen können und die möglichst wenige inadäquaten Vorstellungen fördern (Ubben, 2020). Eine der größten Problematiken während Lernprozessen ist, dass Repräsentationen -wie etwa Visualisierungen -häufig von Lernenden als "gestalttreu" empfunden werden. ...
... Eine der größten Problematiken während Lernprozessen ist, dass Repräsentationen -wie etwa Visualisierungen -häufig von Lernenden als "gestalttreu" empfunden werden. In solchen Situationen wird den Repräsentationen zu viel Realitätsgehalt zugeschreiben, sodass Lernende durch das Festhalten an den Bilden daran gehindert werden, die zugrunde liegenden Ideen zu abstrahieren (Ubben & Bitzenbauer, 2022;Ubben, 2020). Aufgrund dieser Problematik ist ein Ansatz, Abstraktion durch Repräsentationen, die gut an die mathematischen Beschreibungen der Quantenphysik anknüpfbar sind, zu verwenden (Schecker et al., 2019). ...
Full text: https://ojs.dpg-physik.de/index.php/phydid-b/article/view/1362
Translation on arXiv: https://arxiv.org/abs/2312.06269
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Quantenphysikalische Modellierungen sind fachlich komplex und oft unanschaulich. Dieser Artikel stellt einige Ansätze vor, wie quantenphysikalische Ideen dennoch mittels haptischer Modelle dargestellt werden können. Dazu werden Modelle aus dem 3D-Drucker, Modelle aus Papierstreifen und Modelle aus Stoff gegenübergestellt. Neu ist insbesondere die Verwendung von Reißverschlüssen statt Papierstreifen, sodass ein einfaches „Aufschneiden“ und „Zusammenkleben“ möglich wird. Die Modelle sind dabei vor allem mit dem Ziel entwickelt worden, unter Verwendung von wenig mathematischem Grundwissen topologische Ideen zu transportieren und zu visualisieren.
... Physics Education Research (PER), amongst other things, uncovers how learners understand physics topics and concepts and how to support them in their learning processes [19][20][21]. Particularly, research on learners' conceptions of topics in quantum physics, has become a topic of interest in PER in recent years [22][23][24][25][26][27]. It is evident that quantum concepts like quantum entanglement are frequently linked with mysticism and science fiction [28], and students tend to be unaware of applications of quantum physics concepts in their everyday lives [29]. ...
Quantum entanglement is a challenging concept within the field of physics education, often eluding a full grasp by both educators and learners alike. In this paper, we report findings from a two-phase empirical study into the views of entanglement held by pre-service physics teachers and physics students from various universities. In the first phase, we utilized a questionnaire consisting of open-ended questions which was completed by 31 pre-service physics teachers. The study participants’ ideas were explored using qualitative content analysis which led to the creation of rating scale items used in study phase 2. These items were administered to a broader cohort including 73 physics university students in order to capture the learners’ agreement or disagreement with the questionnaire statements, and hence, helped to validate and substantiate the in-depth insights from study phase 1. Key findings revealed widespread accurate notions, like the need to consider the entire system when examining entangled states. However, less elaborated views were also identified, including ideas such as that measurements of entangled states always show perfect (anti-)correlation. Another striking observation was the confusion between quantum entanglement and superposition. In the case of quantum teleportation, many participants seemed to have a basic grasp of the concept, although a number of misconceptions were apparent, notably the idea that quantum entanglement enables faster-than-light communication. Practically, the findings can assist educators in anticipating and addressing widespread (mis-)conceptions, paving the way for more effective instruction in quantum mechanics and its real-world applications, such as quantum cryptography and computing.
... Modelle sind in der Physikdidaktik u. a. dadurch charakterisiert, dass sie bestimmte funktionale Eigenschaften darstellen, aber nicht die eindeutige und vollständige Realität der Objekte abbilden (Kircher 2015). Allerdings verstehen Lernende Modelle oft genau so, also als Abbildung der vollständigen Realität (Ubben, 2020). Daher ist es essentiell, besonders beim Einsatz der hier vorgestellten sehr bildlichen Modelle eine Diskussion mit den Lernenden zu führen und Modellkritik zu üben. ...
Sektormodelle sind Anschauungsmodelle für nicht-euklidische Geometrie. Sie wurden entwickelt, um die Grundkonzepte der allgemeinen Relativitätstheorie in zwei bzw. drei Dimensionen auf anschauliche Weise darzustellen (Zahn & Kraus, 2014; Zahn & Kraus, 2019). Für eine Einführung in die Allgemeine Relativitätstheorie kann man mit solchen Modellen beispielsweise den Raum in der Nähe eines Schwarzen Lochs oder das Innere eines Neutronensterns darstellen.
Der Einsatz von Sektormodellen ist nicht nur im Physikunterricht möglich. Auch im Mathematikunterricht können sie genutzt werden, um sphärische und hyperbolische Geometrien darzustellen. Sektormodelle erlauben diverse Aktivitäten der Lernenden: beispielsweise können Lernende anhand der Modelle mit einfachen Mitteln Krümmungen bestimmen und den Verlauf von Geodäten untersuchen. Die einfacheren ebenen Sektormodelle können Lernende ausgehend von der Metrik auch selbst berechnen und konstruieren (Kraus & Zahn, 2016).
In diesem Beitrag geben wir eine kurze Einführung in Sektormodelle und stellen Umsetzungen mit dem 3D-Drucker, sowie ihr Potential für den Einsatz im Physikunterricht vor.
... Im Rahmen einer Fragebogenstudie (N = 3108) extrahierten sie faktorenanalytisch zwei unabhängige Faktoren zur Beschreibung mentaler Modelle der Teilnehmenden an der Studie: die Gestalt und die Funktionalität. Unter der Gestalttreue verstehen die Autoren der Arbeit dabei, "inwieweit physikalische Modelle als gestalttreue Abbilder der Realität gesehen werden" (Ubben, 2020 ...
Jüngste Ergebnisse der Beforschung gängiger Modellvorstellungen zu Elektronen in der Atomhülle von Ubben und Heusler legen zwei unabhängige Faktoren des Modellverständnisses Lernender nahe: die Gestalttreue und die Funktionalitätstreue. Mit diesen beiden Faktoren wird die Beschreibung von Typen des physikalischen Modellverständnisses ermöglicht. In diesem Beitrag zeigen wir anhand der Ergebnisse eines Vorstellungsfragebogens zur Quantenphysik (N = 118) faktorenanalytisch, dass sich die Faktoren der Gestalttreue und der Funktionalitätstreue auch aspektspezifisch für andere Inhaltsbereiche der Quantenphysik zeigen. Am Beispiel der Eigenschaft Ort in der Quantenphysik wird anhand von Ergebnissen einer Interviewstudie (N = 25) begründet, dass mit dem Erlanger Unterrichtskonzept zur Quantenoptik der Übergang zu einem funktionalen Modellverständnis in der Quantenphysik bei Lernenden der gymnasialen Oberstufe gefördert werden kann.
Im naturwissenschaftlichen Unterricht setzt hypothesengeleitetes Experimentieren bei den Lernenden eine Grundvorstellung über die untersuchten Zusammenhänge voraus. Die Arbeit mit einem dynamischen Modell soll im didaktischen Konzept der physikalischen Modellbildung die Lücke zwischen einer naturwissenschaftlichen Fragestellung und einer im Experiment überprüfbaren Hypothese schließen.
Die Anwendung des aus erkenntnistheoretischen Überlegungen abgeleiteten Konzepts wird in einer explorativen Studie im Mixed-Methods-Design analysiert, in der sich N= 41 Lehramtsstudierende mit GeoGebra-Modellen auf ein Experiment im Praktikum der Optik vorbereiten. Anhand von Videoaufnahmen werden Faktoren und Verhaltensmuster der Arbeit mit einem dynamischen Modell identifiziert. Diese Merkmale werden in
Abhängigkeit individueller Lernvoraussetzungen auf Zusammenhänge mit der Qualität und Quantität der formulierten Hypothesen sowie den Handlungen im Experiment untersucht.
Aus den Daten wird ersichtlich, dass die Formulierung komplexer Hypothesen und ihre Überprüfung in einem Experiment erst bei gründlicher Arbeit mit dem dynamischen Modell erfolgt. Als Ergebnis werden Gestaltungsempfehlungen für dynamische Modelle abgeleitet, die eine zielgerichtete Untersuchung des modellierten Systems ermöglichen. Sie werden in einem Unterrichtskonzept für die Optik umgesetzt, das sich
durch die Modellierung optischer Phänomene und die Überprüfung der Modellaussagen in Experimenten auszeichnet.
We conducted a quasiexperimental study in order to investigate the effect of a teaching concept on quantum physics based on coincidence and correlation experiments with heralded photons on preuniversity students' conceptions of quantum physics (experimental group, N = 150). We compare the results with the traditional curriculum's effect (control group, N = 130) at German secondary schools using a questionnaire to assess students' conceptions of quantum physics adapted from the literature. The results show that students introduced to quantum physics using the quantum optics concept acquire conceptions of quantum physics that are significantly less influenced by classical mechanistic and deterministic conceptions than those of the control group. In more detail, correlation and principal component analysis results indicate that the conceptions acquired by experimental group students are more consistent than those of the control group students.
The Heegaard splitting of SU(2) is a particularly useful representation for quantum phases of spin j-representation arising in the mapping S1→S3, which can be related to (2j,2) torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions (I,J,K), which are crucial to understand the Hopf mapping S3→S2. We discuss the relationship between observables on the Bloch sphere S2, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a haptic model of entanglement and gauge symmetries is proposed, which may also be valid for physics education.
In teaching sciences, models are often used to introduce, elaborate or simplify real-world phenomena or concepts. It is, however, often the case that misconceptions arise from or are facilitated by these teaching models during their transition to mental models of the individual learners. For instance, models are often seen as direct replicas of something real—scaled versions of reality. Even though for architectural models, this approach is sufficient, in physics, other model types must also be taken into account. In particular, in quantum physics, the ability for abstract model building is essential. In our exploratory study with 3108 participants, the dispositions towards models in physics in general and models of the atomic hull in particular were analysed. Based on this quantitative data, two independent dimensions of the participants’ mental models were extracted: (i) Functional Fidelity and (ii) Fidelity of Gestalt. Based on these empirical findings, four main types of mental models are proposed.
This study focused on determining the elements of mental models of atomic structure and views on visual representations of models of atomic structure in two sub-cohorts of student teachers studying at a university in Turkey. In total, 141 student teachers participated in this study. In the first cohort, the focus was on 73 freshman science student teachers’ drawings of mental models of atomic structure. The analysis showed a wide variety of individual aspects in the students’ minds when asked to sketch the structure of atoms. The majority of students preferred to draw two-dimensional structures, neglecting the atom’s space-filling character. Concerning the details of atomic structure, the majority of students emphasized only the most essential components of atoms, namely protons, neutrons, and electrons. It was quickly recognizable that these elements were arranged according to different analogies or representations of historical models, particularly related to Bohr’s atomic theory and different representations thereof. Overall, the different visual representations of atomic models the students see in school, almost exclusively serve as the basis for their ideas about atomic structure. Current atomic theory, like quantum mechanical models, are generally not used when students are asked for a “contemporary” model of atoms. Rather it seems that concreteness and functionality are the primary factors leading to the selection of an atomic model when requested. This study is supplemented by data collected from the second cohort of 68 prospective teachers consisting of a diverse group of students ranging from freshman to senior level. The students in this cohort were asked for their preferred illustrations of atoms in textbooks. Open-ended questions about atoms led to further insights. The analysis of the prospective teachers’ drawings indicated that a more careful approach to teaching is necessary to clarify the relationships between different models of atomic structure and to allow students to understand what an appropriate and contemporary understanding of atomic structure should encompass.
A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the SU(2)-representation of spin-jstates in C2j+1 before Hopf-mapping and the number p=2j of nodes in the stellar representation in CP1 after Hopf mapping.
Previous research has documented significantly larger income-related gaps in children’s early cognitive development in the United States than in the United Kingdom, Canada, and Australia. In this study, we investigate the extent to which this is a result of a more unequal income distribution in the United States. We show that although incomes are more unequal in the United States than elsewhere, a given difference in real income is associated with larger gaps in child test scores there than in the three other countries. In particular, high-income families in the United States appear to translate the same amount of financial resources into greater cognitive advantages relative to the middle-income group than those in the other countries studied. We compare inequalities in other kinds of family characteristics and show that higher income levels are disproportionately concentrated among families with advantageous demographic characteristics in the United States. Our results underline the fact that the same degree of income inequality can translate into different disparities in child development, depending on the distribution of other family resources.
Electronic supplementary material
The online version of this article (10.1007/s13524-018-0738-8) contains supplementary material, which is available to authorized users.
We represent the world in a variety of ways: through percepts, concepts, propositional attitudes, words, numerals, recordings, musical scores, photographs, diagrams, mimetic paintings, etc. Some of these representations are mental. It is customary for philosophers to distinguish two main kinds of mental representations: perceptual representation (e.g., vision, auditory, tactile) and conceptual representation. This essay presupposes a version of this dichotomy and explores the way in which a further kind of representation – procedural representation – represents. It is argued that, in some important respects, procedural representations represent differently from both purely conceptual representations and purely perceptual representations. Although procedural representations, just like conceptual and perceptual representations, involve modes of presentation, their modes of presentation are distinctively practical, in a sense which I will clarify. It is argued that an understanding of this sort of practical representation has important consequences for the debate on the nature of know-how.
