Investigating Experimentally Problem Solving Strategies in ...

Angela Schwering, Ulf Krumnack, Kai-Uwe Kühnberger & Helmar Gust (eds.)

Investigating Experimentally Problem Solving Strategies in Geometric Proportional Analogies

PICS

Publications of the Institute of Cognitive Science

Volume 30-2010

ISSN: 1610-5389 Series title: PICS Publications of the Institute of Cognitive Science Volume: 30-2010 Place of publication: Osnabrück, Germany Date: September 2010 Editors: Kai-Uwe Kühnberger Peter König Sven Walter Cover design: Thorsten Hinrichs

© Institute of Cognitive Science

Investigating Experimentally Problem SolvingStrategies in Geometric Proportional Analogies

Editors:

Angela Schwering, Ulf Krumnack,Kai-Uwe Kuhnberger, Helmar Gust

Institute of Cognitive Science,University of Osnabruck

June 2008

Contents

1 Introduction 4

2 Background 52.1 Psychological and computational aspects of analogies . . . . . 5

2.1.1 What is an analogy? . . . . . . . . . . . . . . . . . . . 52.1.2 Stages in analogical reasoning . . . . . . . . . . . . . . 62.1.3 Types of analogies . . . . . . . . . . . . . . . . . . . . 122.1.4 Geometric proportional analogies: our focus . . . . . . 17

2.2 Gestalt psychology . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Gestalt theory . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Gestalt principles . . . . . . . . . . . . . . . . . . . . . 202.2.3 Application of Gestalt principles . . . . . . . . . . . . . 25

3 Experimental part 273.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Technical framework . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 General introduction to the Analogy Lab . . . . . . . . 283.2.2 Lab elements and basic functions . . . . . . . . . . . . 303.2.3 Experimental paradigms and programming issues . . . 323.2.4 Measure to prevent multiple participation . . . . . . . 403.2.5 External programs . . . . . . . . . . . . . . . . . . . . 42

3.3 Pretests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.1 First pretest . . . . . . . . . . . . . . . . . . . . . . . . 473.3.2 Second pretest . . . . . . . . . . . . . . . . . . . . . . . 523.3.3 Mini tests: HIT and Technologietag . . . . . . . . . . . 88

3.4 Final experiment . . . . . . . . . . . . . . . . . . . . . . . . . 973.4.1 Participants . . . . . . . . . . . . . . . . . . . . . . . . 973.4.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 983.4.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . 993.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.4.5 Comment classification . . . . . . . . . . . . . . . . . . 133

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4 Closing discussion 134

Bibliography 138

A First pretest 139A.1 List of analogies . . . . . . . . . . . . . . . . . . . . . . . . . . 139A.2 Analogy combinations . . . . . . . . . . . . . . . . . . . . . . 142A.3 Participant information . . . . . . . . . . . . . . . . . . . . . . 142A.4 Test environment . . . . . . . . . . . . . . . . . . . . . . . . . 145A.5 Results by analogy . . . . . . . . . . . . . . . . . . . . . . . . 146A.6 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A.7 Suggestions for improvement . . . . . . . . . . . . . . . . . . . 155

B Second pretest 157B.1 List of analogies . . . . . . . . . . . . . . . . . . . . . . . . . . 157B.2 Stimuli combinations . . . . . . . . . . . . . . . . . . . . . . . 160B.3 Example randomization . . . . . . . . . . . . . . . . . . . . . 160B.4 Randomization of Check/Radio options . . . . . . . . . . . . . 161B.5 Participant information . . . . . . . . . . . . . . . . . . . . . . 163

C Mini tests: HIT and Technologietag 165

D Final test 168D.1 Demographics . . . . . . . . . . . . . . . . . . . . . . . . . . . 168D.2 Stimuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169D.3 Solution classification . . . . . . . . . . . . . . . . . . . . . . . 176D.4 Solution analysis . . . . . . . . . . . . . . . . . . . . . . . . . 178

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Foreword

The work presented in this technical report was developed by the studyproject “Psychological and Computational Aspects of Analogy”. The Studyproject was divided in an experimental and a formal part. In the experimentalpart, different solutions for ambiguous geometric proportional analogies wereinvestigated. The web-based experiment reached 238 participants and gaveus valuable results for analyzing different perceptions and solution strategiesof participants. In the formal part, the study project developed a compu-tational approach to formalize different perceptions of geometric figures andsolve these ambiguous analogies. This part was inspired by the work fromMehdi Dastani and his colleagues, who developed a language of perceptionfor proportional analogies.

This technical report covers only the experimental results of the studyproject. It gives some details about the background from analogy research,describes the motivation from Gestalt Psychology and explains the experi-mental design. Two events in Osnabrueck - the “Hochschulinformationstag”and the “Technologietag” were used to carry out the experiment with a smallnumber of participants. It was followed by a comprehensive web-based ex-periment. In the end of this report, we discuss our findings with respect topossible explanations for different perceptions and different solution strate-gies.

The members of the study project were Clemens Bauer, Judith Degen,Irena Dorceva, Maxim Haddad, Martin Schmidt, Rolf Stollinski, Kae Sug-awara, Martesa Tantra and Radomir Zugic. It was held by Angela Schwer-ing, Ulf Krumnack, Kai-Uwe Kuhnberger and Helmar Gust at the Instituteof Cognitive Science, University of Osnabruck in the summer term 07 andwinter term 07/08.

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Chapter 1

Introduction

Clemens Bauer, Judith Degen, Irena Dorceva, Maxim Haddad,Martin Schmidt, Rolf Stollinski, Kae Sugwara, Martesa Tandra &Radomir ZugicInstitute of Cognitive Science, University of Osnabruck

Analogy-making in the scope of human perception underlies a high-levelcognitive process that transfers information from a particular entity (thesource) to another (the target) by drawing a comparison in order to showa similarity in some respect. In general, it can be inferred that if two enti-ties share some similarities, they probably also share many others as well.Analogies frequently occur in everyday life and thus play a central role inhuman cognition such as in reasoning and learning processes, or even in edu-cation. Furthermore, humans often use analogies to explain and understandnew phenomena by resorting to already known facts from a familiar domainbased on past experience.

While a multitude of analogy types are often built across different do-mains, (geometric) proportional analogies have the general form (A:B)::(C:D)(read ”A is to B as C is to D”) whereby A and B (the source domain) as wellas C and D (the target domain) comprise elements from the same domain. Inthe context of geometric figures, the main task for humans amounts to findinga suitable element for D such that the same structural relations between thesource elements A and B and the target elements C and D hold. However, theresulting solutions often differ significantly among humans due to the possi-bility of perceiving different conceptualizations of the same figures within agiven geometric analogy. It can therefore be claimed that analogy-making isa highly sophisticated cognitive process.

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Chapter 2

Background

Clemens Bauer, Irena Dorceva & Martesa TandraInstitute of Cognitive Science, University of Osnabruck

2.1 Psychological and computational aspects

of analogies

2.1.1 What is an analogy?

Analogies are partial similarities between different situations that hold fur-ther inferences (Bechtel and Graham [1999]). In other words, an analogy isa type of similarity in which the same arrangement of relations holds acrossdifferent objects. Analogies thus get the core of correspondences across differ-ent situations. Analogy-making is crucial for human cognition Kokinov andFrench [2003]. Many cognitive processes involve analogy-making: perceivinga square in a cubist painting as a human face, solving a mathematical prob-lem in a way similar to another problem previously solved, understandingmetaphors, communicating feelings or emotions, learning a foreign language,understanding poetry etc. (Gentner [2001]). All these cases require an ab-stract mapping to be established between two cases or domains based on theircommon structure or relations. This may require re-representation of one (orboth) of the domains in terms of the other one. The first domain is calledthe source, and the second is referred to as the target. Analogy-making is avery basic cognitive ability which probably starts with the simple ability of

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babies to repeat a behaviour that leads to a satisfactory outcome, e.g. cryingto get milk, then it slowly progresses to children’s being able to recognize ananalogy between a word or utterance and the corresponding real object andultimately culminates in the adult ability to make complex analogies betweenvarious situations. This seems to suggest that analogy-making serves as thebasis for numerous other kinds of human thinking and explains the impor-tance of developing computational models of analogy-making (Kokinov andFrench [2003]).

Analogy-making, or analogical reasoning, involves at least the followingsub-processes:

• Retrieval of a source for the analogy

• Mapping source onto the target

• Transfer, i.e., making inferences

• Evaluation of unmapped elements from the source to the target forapplicability

• Learning and consolidation based on the experience, which includesgeneralizing from specific cases and possibly developing general mentalschemas.

2.1.2 Stages in analogical reasoning

Five different stages in analogical reasoning are commonly distinguished: theretrieval, the mapping, the transfer, the evaluation, and the learning andconsolidation stage.

Retrieval

This has been extensively studied experimentally and it is now clear thatsuperficial similarity plays the major role in analogical retrieval, i.e. the re-trieval of a source for an analogy is easier if its objects, properties and generaltheme are similar to those of the target. Structural similarity, the familiarityof the domain from which the analogy is drawn, the richness of its repre-sentations and the presence of generalized schemas also facilitate retrieval(Kokinov and French [2003]). The process of analogical retrieval starts withthe comparison of two domains that are usually considered dissimilar. Inmost problems, our knowledge of the source domain is greater or more accu-rate than that of the target and the goal of the analogy is to say somethingabout the target from our knowledge of the source. The reasoning involved

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generally starts with a question about the target and the retrieval of a baserepresentation of it. Thus, the first step in forming an analogy is the retrievalof an analogue (Kokinov and French [2003]). While analogical reasoning isused everywhere in human cognition, it has been shown that the retrieval ofsuitable analogues is complicated. This can be exemplified in the experimentby Gick and Hoyloak (Gick and Holyoak [1983]) using Duncker’s radiationproblem:

Suppose you are a doctor faced with a patient who has an inoperable stomachtumor. You have at your disposal rays that can destroy human tissue when di-rected with sufficient intensity. How can you use these rays to destroy the tumorwithout destroying the surrounding healthy tissue?

Before participants were given this problem, they read various stories. Forsome participants, one of the stories involved a general who had to decide how toattack a fortress. The fortress was surrounded by mines, which would be set offif crossed by a large force. The general solved this problem by approaching thefortress with several small forces from multiple directions that would converge onthe fortress all at once. Using this problem as an analogue to the problem theywere supposed to solve, participants would know that they could use smaller dosesof radiation from multiple directions, which would arrive at the tumor at the sametime, thus allowing for the same large dose of radiation to be administered withouthaving to destroy tissue around a single location on the body.

The result of the experiment showed that participants who were not giventhe fortress story prior to reading the radiation problem solved it about 10%of the time. Participants who were given the fortress problem solved theradiation problem about 30% of the time. Although this was a significantincrease, it also meant that 70% of the time participants were not able toretrieve the relevant analogue. This shows that retrieval is a rather difficultstep and given that superficial similarity facilitates retrieval, as mentionedabove, one also has to take into account that people still have problems re-trieving suitable analogues. This may be due to the fact that people tens toretrieve items from memory based merely on surface similarity (i.e., basedon objects and attributes), rather than relational similarity (i.e., based oncommon relational structures) (Gentner et al. [1993]). Since analogies areall about relational similarity, this is a problem. The tendency to retrievesurface-similar analogues can also be an explanation for why experts arebetter at analogical reasoning in their domains of expertise than beginners(Novick [1988]). This last fact actually hints at the reasons for accepting

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domain-general theories of analogical reasoning. While expert reasoning mayappear to be using domain-specific processes, what may actually be happen-ing is that experts simply have more complex relational knowledge withinthat domain. More complex representations yield different types of resultsin analogical reasoning.

The next step after retrieving an analogue is to form an analogy betweenthe retrieved information and the target. This is the mapping step.

Mapping

This is unquestionably the core of analogy-making and, therefore, all com-puter models of analogy-making include mapping mechanisms, i.e., means ofdiscovering which elements of the source correspond to which elements of thetarget (Kokinov and French [2003]). In a typical case of analogical mappingthe most familiar situation is the source domain, which is then matched to aless familiar situation — the target domain. The familiar situation suggestsways of viewing the newer situation as well as further inferences about it.Analogical mapping requires aligning the two situations, that is, finding thecorrespondences between the two representations and projecting inferencesfrom the source domain to the target domain. This match is then evaluatedas are its implications for the solution. The problem with this is that onesituation can be mapped onto a second situation in many different ways.

The psychological processes that go on in analogical mapping were de-scribed by Gentner’s Structure-Mapping Theory (Gentner [1983]). Accordingto this theory, the comparison process involves finding an alignment betweenthe source and target representations that reveals a common relational struc-ture. On the basis of this alignment, further inferences are projected fromsource to target. Gentner [1983] says that people prefer to find an alignmentthat is structurally consistent, that is, there should be a one-to-one corre-spondence between elements in the source and elements in the target, andthe arguments of corresponding predicates must also correspond, thus havinga parallel connectivity. For example, in the analogy below, Peter in (a) couldbe put in correspondence with Peter in (b) (local entity match) or with Annein (b) (relational role match). People appear to consider both possibilitiesduring processing, but have to settle on one or the other by the end of theprocess.

(a) The dog rescued Peter.

(b) Peter rescued Anne.

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Another important theory for the mapping process was given by Chengand Holyoak [1985]. Their pragmatic mapping theory focused on the use ofanalogy in problem-solving and held that analogical mapping processes areoriented towards attaining goals (such as solutions to problems). Accordingto the pragmatic mapping theory, it is goal relevance that determines whatis selected in an analogy. Holyoak and Thagard [1989] later combined thispragmatic focus with structural factors in their multi-constraint approach toanalogy.

Analogical inference projection is a crucial part of the mapping process.Once an alignment is achieved, further inferences can be made by projectinginformation from the source domain onto the target domain. For example,in the above analogy, suppose we knew more about event (a), such as:

(a) The dog rescued Peter because he knows him. The dog has black hair.

(b) Peter rescued Anne.

In this case, the likely inference in (b) is that Peter rescued Anne becausehe knows Anne. This ability to request new inferences are central to the rolethat analogies play in reasoning. Importantly, analogical inference is ratherselective. For example, we are unlikely to make the inference (in the aboveexample) that Peter has black hair. This illustrates the selection problemin theories of analogical inference. If people projected everything knownabout the source onto the target, analogies would be useless in reasoning;it would even make things more complex (Holyoak and Thagard [1989]).However, this is not normally done and that is why the characterization ofthe selection process is a central aim of theories of analogy. For this at leastthree factors have been proposed. Holyoak and Thagard [1989] proposedgoal relevance: the inferences projected are those that fit with the reasoner’scurrent goals in problem-solving. A second factor, relational connectivity orsystematicity, was proposed in the theory underlying the Structure-MappingEngine (Gentner [1983]). Here, there is a higher preference for projectingmatching systems of relations connected by higher-order relations such ascause, rather than projecting local matches. In many cases, goal relevanceand systematicity will make the same predictions because problem-solvinggoals often involve a focus on causal systems (Gentner [1983]).

A third factor in selecting inferences, as proposed by Keane (Gentner[1983]), is adaptability. In this case the ease with which a possible infer-ence can be modified to fit the target is preferred. There is evidence for allthree of these criteria. Spellman and Holyoak [1996] showed that when twopossible mappings are available for a given analogy, people would select the

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mapping whose inferences are relevant to their goals. Evidence for system-aticity comes from the finding that when people read analogous passages andmake inferences from one to the other, they are more likely to transfer a factfrom the source to the target when it is causally connected to other matchingpredicates (Markman [1997]). Finally, Keane [1996] found evidence that thedegree of adaptability predicts which inferences are made from an analogy.

Experimental work has demonstrated that finding this type of structuralisomorphism between the source and target domains is crucial for mapping(Gentner [1983]). Object similarity also plays a role in mapping, althoughgenerally a secondary one. Another factor is the pragmatic importance ofvarious elements in the target, because people try to find mappings thatinvolve the most important elements in the target. Searching for the appro-priate correspondences between the source and target is a computationallycomplex task that can lead to combinatorial explosion if the limits are notconstrained (Kokinov and French [2003]).

Transfer

This is the process of introducing new knowledge into the target domainbased on the mapping (Kokinov and French [2003]). After the two domainsare mapped onto each other, information is transferred from the source do-main to the target domain. In Structure Mapping Theory, this involvesproducing candidate transfers or inferences. These are potential inferencesabout the target from the source that come from the entities that are part ofthe common relational structure but are present only in the source. Thoseentities are carried over to the target and placed in the corresponding posi-tion in the relational structure of its representation. If these inferences aregood fits, they are kept. If not, they are discarded (Kokinov and French[2003]).

For example, suppose a new type of public transport has been imple-mented in Osnabruck and you heard in the news that this is the same thatis used in New York. It is cheap, ecological and fast, but you also knowthat it is very dangerous at night. Transfer is when you wonder whetherthe analogous new public transport will also be unsafe for use at night inOsnabruck.

Transfer is present in one form or another in most models of analogy-making and is typically integrated with mapping. Transfer is considered bysome authors as an extension of the mapping already established, thus addingnew elements to the target in such a way that the mapping can be extended(Kokinov and French [2003]).

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Evaluation

This is the process of establishing the likelihood that the transferred knowl-edge will turn out to be applicable to the target domain. Once the commonalignment and the candidate inferences have been discovered, the analogy isevaluated. According to Gentner [1983] evaluating an analogy involves atleast three kinds of judgment:

1. structural soundness: whether the alignment and the projected infer-ences are structurally consistent;

2. factual correctness: whether the projected inferences are false, true orindeterminate in the target; and

3. relevance: whether the analogical inferences are relevant to the currentgoals.

In practice, the relative importance of these factors varies quite a bit, forexample, in domains where little is known or where there is disagreementabout the facts. In the example above, the evaluation process would have toassign our degree of confidence about whether the new public transport willalso be dangerous at night in Osnabruck. Evaluation is often implicit in themechanisms of mapping and transfer.

Learning and consolidation

Abstraction All the later processes involved in analogy-making lead toan effective learning of the individuals. This translates into a consolidatedknowledge that can be used to solve future analogies by retrieving priorknowledge. Thus in analogical abstraction, the common system that rep-resents the interpretation of an analogy is extracted and stored. This kindof schematic abstraction helps to promote transfer to new examples. Whenpeople are asked to compare two analogous passages, they are better ableto later retrieve and use their common structure (given a relationally similarprobe) than are people who were given only one of the stories (Gick andHolyoak [1983]). Further studies have shown that actively comparing twoanalogous passages leads to better subsequent retrieval than reading the twopassages separately. These findings are consistent with the claim that analog-ical alignment promotes the common structure and makes it more availablefor later use (Gick and Holyoak [1983]).

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Analogies in real-world reasoning Analogies are often used in common-sense reasoning to provide plausible inferences. It must be noted that analogy-making is not a deductive process (Bechtel and Graham [1999]). There is noguarantee that the inferences from a given analogy will be true in the targeteven if the analogy is carried out perfectly and all of the relevant statementsare true in the source. However, the set of implicit constraints describedabove make analogy-making a relatively ’tight’ form of inductive reasoning(Bechtel and Graham [1999]). There is a flip side to the lack of deductivecertainty in analogical reasoning. It means that analogies can suggest gen-uinely new hypotheses whose truth values could not be deduced from currentknowledge. Analogical comparison can lead to new learning in at least fourways: analogical abstraction, inference projection, difference detection, andre-representation.

Bechtel and Graham [1999] state that in analogical abstraction, the struc-ture common to the source and target is noticed and extracted. Sometimesthe common system is stored as a separate representation: this is referred toas schema abstraction (Gick and Holyoak [1983]). In inference projection, aproposition from the source is mapped to the target. If it is retained as partof the target structure, then learning has occurred. In difference detection,carrying out a comparison process leads people to notice certain differences,namely those connected to the common structure. In re-representation, twonon-identical predicates are aligned and decomposed (or abstracted) to findtheir commonalities, resulting in a re-representation that contains a commonpredicate: for example, comparing chase(dog, cat) and follow(detective, sus-pect) might result in pursue(entity1, entity2). A further kind of knowledgechange hypothesized to take place in scientific discovery is restructuring, inwhich the target undergoes a radical change in structure (Bechtel and Gra-ham [1999]).

2.1.3 Types of analogies

Analogies are known to play a key role in cognition, especially in creativeproblem-solving across areas, whether art or science. They are also part ofstandard intelligence tests. For researchers in cognitive science, analogiesopen a new door to get closer to understanding cognition and their rolein human cognition (Indurkhya [1989]). However, analogies come in a widerange of variations depending on the field of study and psychological content.There are significant variations in what is meant by the term “analogy” itself,but they can be reduced to at least three distinct usages. In the followingthese three basic senses of “analogy” are presented and the roles they play incognition (Indurkhya [1989]). Following an overview of the types of analogy,

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the particular type of analogy used in our study project is presented.

Analogy by rendition

This first use of the term “analogy” concerns seeing or perceiving an ob-ject or situation (target) as another object or situation (source) (Indurkhya[1989]). The main characteristics of this mode of analogy is that the similari-ties between the two objects or situations do not exist prior to viewing one asanother, but instead are created by the process. This can be made clear byan example from Schon [1963], where a development team, in trying to im-prove the performance of synthetic fibre paintbrushes, ended up viewing thepaintbrush as if it were a pump. In the example, it becomes clear that it wasnot that the researchers noted some similarities between the paintbrush andthe pump and then imported more features from the pump to the paintbrush,but rather the act of viewing the paintbrush as a pump created the similar-ities — similarities that were not there before. Put another way, wheneverwe notice that two objects or situations are similar, we can do so only withrespect to their existing ontologies and descriptions, their actual informationload. However, in analogy by rendition a completely new ontology is createdand a new level of description for the target object can emerge (Indurkhya[1989]). This type of analogy, according to Gordon [1961], tries to make theunfamiliar familiar or the familiar unfamiliar, depending on the situation.The research question is more along the line of trying to explain how newperspectives are created, which is a deep cognitive process that involves toomany aspects to be covered so easily. This mode of analogy is closely relatedto models and metaphors and all require elements of interpretation in theirenvironment to become meaningful (Indurkhya [1989]). However, this is onlyto illustrate how complex analogies can get when abstraction or change ofestablished environments come into play.

Proportional analogy

Another use of analogy is the so-called proportional analogy. This type ofanalogy is concerned with relations of the form “A is to B as C is to D”. As in“gills are to fish as lungs are to man”. The process referred to is usually thatof generating the fourth term (D) of a proportional analogy relation giventhe other three (Evans [1962]). At first sight, a proportional analogy seemsto be a syntactic process: that is, it appears that only the terms A, B, C, andD and their structures are important, but no meaning is involved. This couldbe more evident when abstract symbols are used such as geometric figures orletters. This is why the first computer models used such types of analogies

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and were moderately successful (Evans [1962]). However, these analogies alsohave an element of interpretation. For instance, to understand the analogy“gills are to fish as lungs are to man” requires one to interpret the role oflungs in the context of physiology of another species, in this case that of fish,since these are different. Hofstadter and Dennett [1981] point to the fact thatthis interpretation is not limited to proportional analogy relations involvingwords or other meaningful symbols. They discussed many cases where all ofthe four terms of the analogy relation were meaningless strings of characters,yet one needed to identify the roles of the various characters in a string andthen interpret these roles in the context of another string. Furthermore, whengeometric figures are used in such proportional analogies, there is also a needto interpret the figures. Coming up with an adequate interpretation stronglydepends on the figure or on the relations it has within or to another figureetc.

Figure 2.1 shows examples of abstract geometric figures. In these threeproportional relations shown, the first term is always the same. Dependingon the analogy in which it is embedded, it requires a different interpretationor a different way of “seeing” it. If we think back to analogy by rendition,this “seeing” strongly reminds us of the process involved there, so a strongseparation among the types of analogies is not possible. They all have someaspects that they share. In this case however, the interpretation is even morecomplex because in each step of the proportional analogy, the figures A, B,C, and D contribute to the context of the interpretation of the remainingfigures, while at the same time being interpreted in the context of the others(Indurkhya [1989]).

In the example given in Figure 2.1, figure A can be described as “twoinverted triangles that are put on top of each other in an off-centre manner”.In the second case, it is described as “a hexagon with an equilateral triangleon each side”. The symbols“hexagon”,“triangle”,“inversion”etc. that emergefrom the configuration of the analogy provide an ontology and structure toA. One could say that “high-level” concepts are obtained by grouping “low-level” representations of the other figures (Indurkhya [1989]). Thus, while Ais described using a vocabulary furnished by the other figures, this very act,at the same time, provides a vocabulary for describing other figures, whichin turn affects the vocabulary being used to describe A, and so on; they allinteract and one has to see the whole analogy as a singular interacting entity.

As with the previous type of analogy, the real challenge is to model theinteraction between the four terms of the analogy. Another important aspectof all proportional analogies is the symmetry property that they exhibit withrespect to interchanging the second (B) and the third term (C). That is, if“A is to B” as “C is to D” is a proportional relation, then so is “A is to C as

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Figure 2.1: Three proportional analogies that each require a different inter-pretation of the star in A

B is to D”. Any reasonable theory or model of proportional analogies oughtto have the same characteristic. Proportional analogies are interesting tostudying cognition because of the underlying process of shifting perspectivesand changing representations (Indurkhya [1989]). It is indeed a hallmark ofintelligence to be flexible and to be able to adjust mental models to solve theactual problem, and this in the best and most economic way. Therefore, bysuccessfully modelling these types of proportional analogies, we can get somemodest insight into the cognitive process of describing or seeing the samesituation from different points of view: a process that indeed lies at the veryheart of cognition (Indurkhya [1989]).

Predictive analogy

Predictive analogies help to explain a new domain by observing similaritieswith a known domain (source) and then transferring further information fromthe known domain to the new domain (target) (Gentner [2001]). Here themain difference to analogy by rendition is that the process involved in predic-tive analogies are less often applicable than that of analogy by rendition, sinceit requires that there be some existing similarities between the two domains.The target must have some ontology and this ontology must be triggered bysome existing similarities between the source and target (Indurkhya [1989]).In analogy by rendition, the ontology of the target is changed as a result ofthe process, and a new perspective (similarity) is created. The new similari-ties created between the source and target are from their existing ontologies.

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It is also important to notice that in the predictive analogies, the predictedsimilarities are made with respect to the existing ontologies and no new on-tologies are created. Predictive analogies are more powerful than analogyby rendition as they predict that there might be other similarities, whichanalogy by rendition cannot claim since one cannot know in advance howthe regrouping might affect the rest of the target. In contrast, predictiveanalogies do exactly that; possible similarities that have previously been en-countered in similar problems are sought and it is predicted that this will fitand help resolve the new problem.

This characteristic of predictive analogies, to solve the newly-encounteredproblem by trying to find a similar, previously-solved problem, is what makesthem so attractive for a heuristic process for problem-solving (Indurkhya[1989]). At the same time, it also makes it the most problematic one becausethe transfer or inference is not considered to take place in the logical sense ofthe word; this inference is not necessarily “true” but is considered to be “jus-tified”. This is to say that in some psychological sense a rational person findsit more reasonable than a random inference, but this degree of reasonable-ness, or justification, strongly depends on its association with an inferencefrom the analogy which increases with the amount of known similarity be-tween the source and target, and it can vary strongly amongst individuals(Indurkhya [1989]). Predictive analogy is also known as analogical reasoningand is applied in most cognitive computational models, such as StructureMapping Engine (Gentner [2001]).

A well-known example of a predictive analogy is the Rutherford analogy:

The atom is like the solar system.

The Rutherford analogy is given in Figure 2.2 with the source domain‘solar system’ on the left-hand side and the target domain ’atom’ on theright (Gentner [2001]).

A domain is represented as a structure (a graph) with objects (such assun, planet-i), attributes (such as yellow(sun)), and relations (such as at-tracts(sun, planet-i)). Besides the first order relations, which are definedover objects, there is a second-order relation, defined over relations: cause(attracts(sun, planet-i), revolves-around(planet-i, sun)). All these entitiesare represented as nodes in the graph. The arcs represent relations betweenthese entities, in other words, their roles (such as “subject” or “object”). Incontrast to proportional analogies, the representation (“description”) is fixed.Analogical reasoning is modelled as a structure preserving mapping of ob-jects from the source to the target. Following the principle of systematicity,mappings of larger structures are preferred. For the Rutherford example,

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Figure 2.2: Structure mapping for the Rutherford analogy, Gentner [2001]

the mapping of sun to nucleus and of planet-i to electron-i results in a largestructural congruence between both domains. Since each object can carryover nodes from the source to the target to which it is connected, the causalexplanation of why a planet revolves around the sun is transferred to thetarget domain, resulting in the inference that an electron revolves aroundthe nucleus because it is attracted by it (Schmid et al. [2003]).

Another aspect relevant to our study project is that proportional analo-gies can be viewed as special cases of predictive analogies. In the latter, theinferred relation becomes its cause relation, whereas in proportional analo-gies, the inference is typically not the complete pattern (or cause) but onlya sub-structure of the source that is mapped to the target (Schmid et al.[2003]). These substructures that can be influenced by Gestalt laws andrelations within the figures of geometric proportional analogies is what mo-tivates our study project.

2.1.4 Geometric proportional analogies: our focus

Geometric proportional analogies (GPAs) are a type of analogy formed be-tween two collections of geometric figures (Mullally and O’Donoghue [2005]).Geometric analogies then are graphical proportional analogies (Figure 2.3),where A, B and C are each identified by a geometric figure and they are alsoof the form A:B::C:D (“A to B as C to D”).

The source domain (A:B) identifies some transformation(s), which mustthen be applied to C, yielding the solution D. For example, the analogy inFigure 2.3 revolves around dividing the central polygon of part A to producepart B. This partitioning transformation must then be applied to C, whichallows us to generate D. There are two key points to note about GPAs. First,the change between the terms in the source domain (i.e. between A and B)is called the transformation. Second, parts A and C are used to identify the

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inter-domain mapping (Gentner [2001]), i.e. small square (A) maps to smallcircle (C) and central square (A) maps to central circle (C). The combinationof the transformation and mapping will yield D.

Figure 2.3: Simple geometric proportional analogy

Figure 2.3 shows proportional geometric analogies, as used in our studyproject and some solutions that might be interesting for the modelling ofa heuristic. As said above, the similarity between proportional analogiesand predictive analogies makes them interesting for our project because thetransfer or inference is not considered to be an inference in the logical sense ofthe word and the degree of reasonableness of the inference can vary stronglyamongst individuals (Indurkhya [1989]).

Figure 2.4: Examples of geometric analogies used in our study project

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Often different geometric figures exist which are all possible solutions for ananalogy; however, they might not be equally plausible and depend on theperception of the elements A and B (Schwering and et al. [2007]). When thehuman visual sensory system observes a geometric figure, it transforms theunstructured information into a structured representation of coherent shapesand patterns. Human perception tends to follow a set of Gestalt principlesfor organization: stimuli are experienced as a possibly good Gestalt, i.e.,as regular, simplistic, ordered and symmetrical as possible (Koffka [1935a];Kahler [1929]; Wertheimer [1929], Wertheimer [1954]). Gestalt psychologyidentified different principles according to which humans construct Gestalts(e.g. principles of proximity, similarity, symmetry and good continuation,see Gestalt chapter for details). Applying different Gestalt principles to thesame geometric figure might result in different perceptions: Although certainrules have been identified as to which Gestalt principles are cognitively pre-ferred (Wertheimer [1929]), there does not exist a fixed hierarchy of Gestaltprinciples. The perception of geometric figures may differ among humansand depending on context.

2.2 Gestalt psychology

2.2.1 Gestalt theory

Human perception is a complicated process with many steps involved beforean action is produced as an outcome of the perceptual process (Figure 2.5).There have been many theories originating from the study of human percep-tion that try to explain the process. Gestalt theory is one of the theorieswhich accounts for human perception. According to Gestalt theory, humanperception is a parsing process, by which a stimulus pattern (unstructuredinformation) is given a structured representation (Koffka [1935a]). Gestaltprinciples try to account for this parsing as a process that follows certainrules. Therefore, when the human visual system perceives a geometric figure,it transforms the unstructured information into a structured representation ofcoherent shapes and patterns (Dastani and Scha [2003]). The transformationof the unstructured visual information into less unstructured (or structured)information follows certain principles, namely the Gestalt principles (Kof-fka [1935a]). Intuitively, it can be said that the role of Gestalt principlesin explaining human perception is that of linking stimulus and perceptionvia a phenomenological method, i.e, when one describes what is perceived(reception-perception).

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Figure 2.5: Human perceptual process

2.2.2 Gestalt principles

Gestalt theory originated from the study of human perception as an out-come of concrete investigations in psychology, logic and epistemology. In1912, Wertheimer stated for the first time the principles of a Gestalt theory,with the main idea being that the whole is more than the sum of its parts(Wertheimer [1924]). The general idea of the Gestalt theory by Wertheimeris described further in the Gestalt principles, which are a series of laws ofhow human perception organizes small parts into the whole to make senseof perceived objects. Wertheimer [1924] identified several Gestalt principleswhich underlie human perception as follows:

1. Law of Proximity

According to the Law of Proximity as defined by Wertheimer [1912], elementsthat are closer together will be perceived as coherent objects. Therefore, thegroup of circles in Figure 2.6(a) forms columns of circles, rather than rowsof circles as in Figure 2.6(b). The organization into either rows or columnsof circles is based on the distance between the circles. The vertical distancebetween the circles in Figure 2.6(a) is closer than their horizontal distance,which is why they are perceived as columns of circles.

2. Law of Similarity

The Law of Similarity states that similar elements are grouped as an entity.The similarity itself depends on the relationship constructed based on color,size, etc. of the elements. An example of applying the Law of Similarityin object perception can be seen in Figure 2.7. There are two groups ofelements in Figure 2.7 with the grouping of the elements based on the shared

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(a) (b)

Figure 2.6: Law of Proximity

characteristic of the objects, their color. The circles are thus perceived asrows of white circles interspersed with rows of black circles.

Figure 2.7: Law of Similarity

3. Law of Common Fate

According to the Law of Common Fate, the grouping of several objects can bebased on direction, that is, if two or more objects move in the same direction,they are considered to be one unit. The application of the Law of CommonFate is illustrated in Figure 2.8.

The grouping of the black circles can be ascribed to the direction of the“movement” of alternating columns of circles, i.e. a group of circles that“move” upward and another group of circles that “move” downward.

4. Law of Objective Sets

The Law of Objective Sets states that humans have a certain tendency toperceive one aspect of an event as a figure or foreground and another as a

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(a) (b)

Figure 2.8: Law of Common Fate

ground or background. Moreover, a certain objectively ambiguous arrange-ment will be perfectly definite and unequivocal when given as a part of acomposition. This idea is illustrated in Figure 2.9. Depending on our per-ception of foreground/ background, we will see either a black profile of twohuman faces facing each other or a white vase.

Figure 2.9: Law of Objective Sets

5. Law of Direction

The Law of Direction states that lines are preferentially regarded as if theyfollowed the smoothest or the simplest path. When two lines intersect at acertain point, the lines are usually not assumed to deviate at the intersectionpoint. This is illustrated in Figure 2.10, for the case of intersecting lines.There are two lines in Figure 2.10. According to the Law of Direction, peoplewill perceive the first line as going from A to B and the second line from Cto D, rather than, for example, one going from A to C and the other from Cto D.

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Figure 2.10: Law of Direction

6. Law of Good Curve (Continuation)

The Law of Good Curve implies that a pattern is continued even after it stops,following certain regularities. Moreover, according to the Law of Good Curve,stimuli that seem to be a continuation of the previous stimuli are consideredto belong together. The Law of Good Curve is illustrated in Figure 2.11,where the dashed lines are perceived together as a continuous line.

Figure 2.11: Law of Good Curve

7. Law of Closure

The Law of Closure states that humans tend to perceive enclosed (geomet-rical) objects as a whole as opposed to an open structure. Therefore, spacesare enclosed by completing a contour and disregarding gaps in a figure. Ascan be seen in Figure 2.12, humans tend to perceive the central figure as onewhite triangle and three black circles, ignoring the gaps that make the circlesand the triangle incomplete.

8. Law of Symmetry

The Law of Symmetry states that symmetrical alignments are seen as be-longing together regardless of their distance, as illustrated in Figure 2.13.

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Figure 2.12: Law of Closure

Figure 2.13: Law of Symmetry

9. Law of Past Experience

Lastly, Gestalt theory also accounts for people’s habits and knowledge, whichare said to guide human perception. This is referred to as the Law of PastExperience. It is implied in the Gestalt Law of Past Experience that everyexperienced sensation is stored in long-term memory; current sensations arethen compared to already-stored information.

An example of this case is given in Figure 2.14. Although the written wordis not complete, we are able to make out that the word meant is “Gestalt”.According to the Law of Past Experience, this is because we have informationabout the word “Gestalt” somewhere in our long-term memory. Thus, whenwe get this new input, we are able to recognize that it is the word “Gestalt”.

Figure 2.14: Law of Past Experience

However, Koffka [1935b] argues that there is only one general principlethat guides human perception, that is, the Law of Pragnanz, which meanssimplicity or regularity. According to the Law of Pragnanz, perceptual or-

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ganization always follows the rule of simplicity. It could mean that whenthere is an interaction between various Gestalt laws, which result in differ-ent structures of a pattern then, following the law of Pragnanz, the simplestor the most regular of these structures is the one that is actually perceived(Metzeger [1928]).

2.2.3 Application of Gestalt principles

The fundamental formula of Gestalt theory is defined as follows that thebehaviour of a system is determined by the intrinsic nature of the whole, andthat Gestalt theory hope to determine the nature of such wholes (Wertheimer[1924]). Therefore, Gestalt theory is a way to investigate science and theproblem in science from its core, by assuming it as a whole part, and payingattention to the relationship holds between its parts, no longer investigate asystem as separation of parts (Wertheimer [1924]). In a relation with humanperception, stimulus recognition is a perceptual organization, which is doneby organizing a scene in the environment to form perceptually separate objectto facilitate object recognition and make perception have sense. The role ofGestalt principles is guiding the scene organization.

Inspired by Gestalt Principles which said were said to guide human per-ception, a general theory of pattern perception was developed, called Struc-tural Information Theory (SIT). The theory aims at explaining why a patternis perceived as having a certain structure, not a different one (Dastani andScha [2003]). The idea is the same as Gestalt principles that aim at explain-ing why such structure is perceived as having such and such representation,not a different one(s). SIT is therefore working based on the assumption thathuman perceptual system is sensitive to certain kinds of structural regular-ities of a pattern (Dastani and Scha [2003]). SIT accounts for complexitymeasure of a pattern. According to SIT, the chosen pattern would always bethe pattern with least information load (Dastani and Scha [2003]). This is,so to say, would be a formalization of Gestalt law of simplicity.

On our experiment of perception of geometric analogy, we choose a setof gestalt principles, with its working definition that Gestalt laws are theapparently innate mental laws that determines human perception of objects:

1. Law of Proximity

2. Law of Symmetry

3. Law of Similarity

4. Law of Good Continuation / Direction

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These laws are taken into consideration because of the possibility to for-malize them and their importance in cases of perceiving geometric figures,other than perception of others. Therefore, our set of analogy examplesthat were used as stimuli were created by taking either Law of Proximity,Symmetry or Good Continuation into consideration. Our assumption is thatdifferent application of Gestalt law in perceiving the same analogy wouldresult in different solution. Figure 2.15 and 2.16 provides illustration of howapplying different laws to the same analogy would result in many differentsolutions.

Figure 2.15: Law of Proximity in proportional geometric analogy

Figure D in Figure 2.15 is a result of applying the Law of Proximityto solve the proportional geometric analogy problem. Following the Law ofProximity, the circle and the square below are grouped together. The analogyis completed by moving the circle up toward the square, as what was donein the source domain (figure A and B).

However, the Law of Proximity is not the only Gestalt law that guides theproblem solving of this particular analogy. This analogy can also be solvedby applying the Law of Good Continuation (figure 2.16). According to theLaw of Good Continuation, the circles are grouped together, as well as thesquares. The analogy is completed by moving the circle down, against thesquare.

Figure 2.16: Law of Good Continuation in proportional geometric analogy

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Chapter 3

Experimental part

Clemens Bauer, Judith Degen, Irena Dorceva, Martin Schmidt,Rolf Stollinski & Kae SugwaraInstitute of Cognitive Science, University of Osnabruck

3.1 Motivation

Analogies play a central role in human cognition, for example in reason-ing and learning processes. They are often used to explain and understandnew phenomena by applying already known facts from a familiar domain.Analogies are used in education and science as well as in everyday life. TheCoUGAR study project is part of the AI group at the Institute for Cogni-tive Science, Osnabruck that has been studying analogies for the past fewyears and developed a formal framework for handling analogies called HDTP(Heuristic-Driven Theory Projection). The main drive of the project is toprovide empirical evidence for analogical cognitive processes that are involvedin the solving of these analogies.

The first underlying hypothesis is that human perception tends to followa set of Gestalt principles for organization and this also applies to geometricproportional analogies. Accordingly stimuli are experienced as a possiblygood Gestalt, i.e., as regular, simplistic, ordered and symmetrical as possible(Koffka [1935a], Kahler [1929], and Wertheimer [1954]). In other words, whenthe human visual sensory system observes a geometric figure, it transformsthe unstructured information into a structured representation of coherent

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shapes and patterns.The second hypothesis is that the solution to geometric proportional

analogies can best be studied by focusing more on recall memory rather thanon recognition memory. This based on the theory that recalling an item frommemory requires more information storage (i.e., memory strength) than justrecognizing an item. Thus, if assessed, it can yield more insight on the cog-nitive process involved in solving the analogies.

In order to address the former hypothesis, we will apply a set of differentGestalt principles to the same geometric figure in the analogy that mightresult in different perceptions: Although there have been some proposals fora hierarchy in the preference of Gestalt laws (Wertheimer [1924]), there doesnot exist a fixed hierarchy of Gestalt principles. Thus, we restrict our setof Gestalt principles (see Gestalt section). This will hopefully yield someinformation about the first-order perception and its influence on the solutionof analogies. For the latter hypothesis, our experiments are designed in away that the participants have to construct the solutions from a given set offigures, thereby shifting the analogical reasoning process to a recall memorycognitive task instead of just a recognition memory task. This conscious recallprocess induced by the setup of the experiment is further analyzed by imple-menting a comment option whereby the participant can enumerate the stepsin reasoning that produced the answer. The combination of disambiguatingthe Gestalt laws involved in the first-order perception, the induced recallmemory cognitive process to solve the analogy and the option to commenton the steps that produced a particular solution to an analogy is expectedto yield important information that can be used in HDTP to get better andmore accurate results in solving geometric proportional analogies.

3.2 Technical framework

3.2.1 General introduction to the Analogy Lab

During the course of the study project, in addition to the final experiment,two initial pretests and two supplementary experiments (TdT1; HIT2) werecarried out. Both were realized with the so-called Analogy Lab. This onlinelab, originally developed for the purpose of exploring analogical cognition, hasbeen reworked and refined in the course of time for the benefit and successof the study project. It was used as a programming environment to set upand conduct all of the above mentioned pretests and experiments.

1 TdT: Tag der Technologie2 HIT: Hochschulinformationstag

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When visiting the Analogy Lab3, guest users have the possibility of fa-miliarizing themselves with the general ideas, basic concepts, and the ter-minology of analogy-making. One can also look into pending experimentsdealing with analogies, especially into those that are concerned with geomet-ric proportional analogies. Although not limited to such analogies, the mainfocus of the Analogy Lab has been on the presentation of the general subjectmatter of analogies, including the research done by the ICS Analogy Groupas part of the Institute of Cognitive Science at the University of Osnabruck.Figure 3.1 illustrates a rough sketch of the lab’s architecture. The studyproject used this framework to conduct all experiments.

Lab Content

Lab Resources

WIKI & Graphics

User Management

Parameter Passing

User Data

Shared Data

Prolog

Lisp

Huskell

R

Octave

VSA…

B r o

w s

e r

B a

c k

–E

n d

Figure 3.1: The overall architecture of the Analogy Lab

By means of a typical web browser one can easily access all freely availableglobal contents and resources. However, in order to gain access to vital labareas and functions one has to be logged into the system as a registered userwith certain necessary user-rights granted by the lab’s super-administrator.In this way, it is possible to modify the lab by adding, editing or deletingdesired contents such as plain text, scripts, mathematical formulas, complexgraphics, or images. To facilitate work and to increase its usability in gen-eral, the Analogy Lab provides diverse back-end capabilities, i.e. external

3 Link to the Analogy Lab: http://mvc.ikw.uni-osnabrueck.de/labs/osw_en.php?&lab=analogy-lab&usr=#urls

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interface-to-server-side applications which can be called on demand. SGML(Standard Generalized Markup Language), common derivates such as HTMLand XML, JavaScript and even LaTeX code are likewise accepted by the sys-tem and can easily be implemented by the user.

In this context, it is worth mentioning that lab structures are organizedeach time as (web)pages in a tree-like fashion. The path to a page can becompared to a directory path on a data medium, and, indeed, local pagesare represented by directory paths on the lab server. Specifying a hithertonon-existing pathname would for example automatically create a new blankpage provided that certain naming rules are observed. In doing so, one cangenerate as many pages as desired and simultaneously arrange them in adesired way. As these lab pages can analogously be regarded as files on ahard-disk drive, they can effortlessly be edited as one desires using typicaldisk operations (creation, overwriting, copying, and deletion).

For the purpose of editing a page, numerous formatting possibilities areat one’s disposal, among others the creation of tables and lists, textual enu-merations and indentations, the integration of graphics and images, or thevisualization of formulas using LaTeX code. By this means, it was possibleto properly program all pretests and experiments that were conducted in thecourse of the study project.

3.2.2 Lab elements and basic functions

The upcoming section aims at exemplifying the basic working elements ofthe Analogy Lab by briefly explaining its functions and purposes. Figure 3.2displays the main window of the lab system in its guest user mode. The sep-arate column on the left-hand side contains all globally available pages thatare organized in a tree structure as explained before. A simple mouse-click onone of the listed item names is sufficient to jump to the desired page where-upon all existing subpages will be shown in addition (i.e. the page Final

entails the subpages Welcome, Description, Final Combo1, Feedback, andResults). The currently visible page (Welcome) is always highlighted in redletters whose particular content can be viewed within the embedded windowat the centre of the screen. The arrows as well as the plus and minus sym-bols in the upper left corner can alternatively be used to navigate throughthe pages and to increase or decrease available graphics, respectively. Thelogin fields are located in the upper right corner of the screen.This brings us to the lab’s administrative mode. In order to fully switchfrom the guest user to the administrator (admin) mode, one has to log intothe lab system. Here, one should be able to recognize that several morebuttons have appeared in the lower right corner of the screen (Figure 3.3).

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Figure 3.2: The Analogy Lab as seen in its guest user mode

By pressing the raw button, it is possible to hide all navigation elementsto bring up only the raw content of the currently active page. Pressing theslides button enforces a mode that shows entire pages as content slideswith slim navigation features only. The novel working window allows theuser to edit a page as desired. The main editor window just off the middle ofthe screen displays the underlying code of the currently called Welcome pagewhose actual content is concurrently shown next to the editor window on theleft-hand side.Towards the bottom of the lab window (Figure 3.3) one can find a separatesection which serves as a designated upload interface. In this way, it ispossible to upload files including images or other resources to the user spaceor other lab areas. The Durchsuchen button opens a single browser windowthat allows the user to search local drives and other data media for thedesired file to be uploaded to the server. As with pathnames, one equallyhas to comply here with certain naming rules when specifying directory pathswithin global lab areas or in the space of the current user. In this respect,the target path itself can be entered by using the provided field right to theupload button, which must be pressed in order to upload a file. The list

button lists the content of the current directory in a demarcated window

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Figure 3.3: The Analogy Lab in its administrative mode

that can be displayed on demand (not visible in Figure 3.3). Finally, theoff button in the rightmost lower corner disables the administrator modewithout logging out of the system.

3.2.3 Experimental paradigms and programming issues

The following paragraphs are intended to outline the various technical as-pects with reference to the pretests and experiments that were conductedin the course of the study project. Apart from general remarks on the ex-perimental paradigms, special emphasis is placed on diverse programmingissues comprising detailed comments on selected sample code as well as onimportant interaction elements that were implemented for the benefit of theproject and not least for the success of the experiments. In sum, the purposeof this section is solely to address crucial aspects, not to provide an overalloverview.

Both the two pretests and the subsequent experiments had in commonthat they were concerned with solving geometric proportional analogies ofthe form A is to B as C is to D. In all cases the participants were encouraged

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to complete D independently whereupon they were given two fundamentallydistinct ways of solving analogies. With the exception of the second pretestwhich followed a different experimental paradigm, the first way of solvinganalogies was to select one out of several given solutions (D1, D2, or D3) forD by clicking on the number of the solution that the participants found mostplausible.

A typical stimulus can be seen in Figure 3.4. Here, each object and tex-tual element was manually implemented with the aid of the lab environment.In order to give a comprehensible example, figure A as part of the analogy’ssource domain was basically composed by the following sample code:

1 @graph[0,0,800,250,white]

2 @rec_[38,-5|A]

3 @obj_[24,30,45,45|example/circle_black]

4 @obj_[24,116,45,45|example/square_white]

...

5 @graph/[]

To begin with, the @graph command is used to start a graphic area, @obj_signifies a graphical object such as an image, and @rec_ eventually designatesan empty rectangular field that can be filled with some content such as plaintext. The underscore characters in this context imply that the correspond-ing objects can only be dragged in admin mode. If they are left out, thepertaining objects can also be dragged in guest user mode. The @graph/[]

tag in line 5 finally closes a graphic area. The square brackets in each line[...] may contain various parameters that allow for an adjustment or amore detailed specification of the associated objects.Concerning the first line of the code, the embedded parameters yield a graphicarea whose size amounts to 800 pixels in the x-direction and 250 pixels inthe y-direction. This size is incidentally sufficient to cover the whole analogyshown in Figure 3.4. What is more, the graphic area is assigned a white

background colour whereas the numerical values 0,0 specify its starting po-sition within a two-dimensional hidden coordinate system. The second linecontains the code for the letter A at the relative position 38,-5. The HTMLtags and are used here to make the letter appear in bold type.Both the third and the fourth line specify the two geometric objects that canbe found in figure A. These objects are mere GIF format images that were pre-viously stored in the directory ”example” under the name of ”circle_black”and ”square_white”, respectively. From there they are now called, loadedand put on the screen. The values 24,30 and 24,116 designate their par-ticular positions in relation to the mentioned coordinate system whereas the

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Figure 3.4: One task consisted of selecting the most plausible solution for Din order to complete the given analogy

numbers 45,45 represent the current image size (width/height) which can ofcourse be changed as desired.

In this context, the following issue should not remain unmentioned withrespect to the preliminary tests and experiments. It was found that depend-ing on the browser and resolution used by the participant, most analogieswould be distorted due to shifting of the various objects included to set upthe whole scene (again, cf. Figure 3.4). In particular, it was taken into con-sideration that the position of the colons (: / ::) in relation to the objectswas an important factor in the perception of the symmetry within the anal-ogy. Therefore, in order to prevent any confusion or distraction that may becaused by changes in the positioning of the colons, it was ultimately decidedthat a static representation of the entire analogy would be used for the finalexperiment. The images were created using an external image processingapplication and saved in PNG format to obtain a high quality and to re-duce data size. The resulting PNG images were then inserted at appropriatepositions within the lab code on all relevant pages by using the HTML tag<img src=...>. By doing this, the amount of code could also be reducedwhich in turn facilitated work.The second way for the participants to solve geometric analogies entailedthe possibility of constructing a solution for D from the available geometricobjects within the so-called resources toolbar at the bottom of the screen(cf. Figure 3.5, p. 35). Essentially, this toolbar can be considered a specialgraphic area with certain useful characteristics. The purpose and function-ality can be briefly explained as follows: by clicking on the desired shapein the resources toolbar and simultaneously holding the mouse button, theparticipants could drag the selected object into the solution space for D.

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Figure 3.5: Another way of completing the given analogy was to constructan own solution by hand

Then, by releasing the mouse button again, they could drop their selectedobject into the desired position. Although it was only possible to drag anddrop one object at a time, the participants were allowed to use any objectfrom the toolbar as often as they wanted. Once they were satisfied with theirsolution for D, they clicked on the OK button to submit it and to proceed tothe next page of the experiment. A typical sample code for the toolbar wouldlook as follows:

6 @rec_[4,220|Resources toolbar:]

7 @graph/[+example|20:20:0:270:h|45]

Analogous to the previous sample code (lines 1-5, p. 33), line 6 providesall necessary commands and parameters to generate the text ”Resourcestoolbar:” in bold letters at the stated position (4,220). Line 7 is responsi-ble for the appearance of the actual resources toolbar on the screen. As onecan see (Figure 3.5), the toolbar itself contains numerous geometric shapeswhich first of all have to be stored as images (GIF, PNG, JPG, etc.) ina certain local directory on the lab server. From there all images in thespecified directory (here, it is again a folder called ”example”) will be loadedand displayed in the toolbar. Directory names marked with an initial ”+”cause the lab to display the associated toolbar, i.e., both in admin and guestuser mode. Otherwise, the toolbar appears only when a user is logged inas an administrator. The remaining parameters in line 7 relate to the widthand height (20:20), the modifiable position (0:270), and the orientation (h =horizontal alignment; v = vertical alignment) of the toolbar. The initial scaleof all object images that are inserted into the solution space is determinedby the last parameter (here: 45).

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The green trash can symbol in Figure 3.5 is also featured in the toolbar. Ifthe participants needed to remove any of the objects from the solution space,all they had to do was to click and drag the desired object onto the trash cansymbol and release it. In this way it was possible for them to subsequentlymodify a solution for D or to delete an unintentionally dragged object. Thetrash can is basically composed of a single line of code:

8 @obj_[585,183,45,45|objects/Trash]

As in the cases before, the parameters within the square brackets determinethe relative position (585,183) as well as the size of the image (45,45) thatrepresent the trash can. The higher the latter values, the larger the trash canimage and, as a consequence, the larger the area that functions as a trashcan. It is worth mentioning that if the source filename of any image containsthe string ”trash”, this object will function as a trash can so that the user’sattention is required to avoid unforeseen consequences.

Going back to Figures 3.4 and 3.5, one can see several button-like ele-ments which have not been addressed up to this point. However, they shouldnot remain unmentioned since these kinds of buttons were used as centralnavigation and dedicated interaction elements in order to guide the partici-pants through the lab pages, to let them submit their user data to the labserver, or to handle the stimuli in a pseudo randomized, fixed sequence (thelatter significantly changed with the development and programming of thefinal experiment).

As indicated, different types of functional buttons were applied to set upthe pretests and experiments. To start with, line 9 contains the sample codefor a local link button (@loc) with the label ”NEXT” written in bold letters.Functionally it has the purpose of skipping to the next branch (n) in the treestructure where all pages are organized (cf. Figure 3.6, p. 37).

9 @loc[n|NEXT]

10 @loc[n,n,d,u,p|Continue]

11 @loc[r,n,n,n,n,n|Random]

Instead of going to the next branch, local buttons can also cause the sys-tem to go to the previous branch (p), to the superordinate node (u), or tothe subordinate node (d) of the tree. It is likewise possible to combine any ofthese commands to reach a desired page with a single button press as shownin line 10 of the sample code. A special case is illustrated by line 11 wherethe system randomly (r) selects one of the next five (n,n,n,n,n) branches.In this way it is possible to access pages in an unpredetermined fashion.

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Figure 3.6: A typical local button (NEXT) as used in the pretests and ex-periments

The visible buttons in Figures 3.4 and 3.5 (labelled 1, 2, 3, and OK, re-spectively) do not only look different, they also pursue another functionalpurpose. Their related code is:

12 @sgs_[512,228|1|1:objects:+example|d]



15 @sgs_[515,190|OK|1:objects:+example|d]

The @sgs command, standing for ”send graphic state”, commonly storesthe current state of a graphic in a certain parameter and then jumps to thespecified target. In the cases at hand (lines 12-15), all graphics are storedin the parameter 1 and appended (+) to the file ”example” in the ”objects”folder. By doing so, it was possible to record both the participants’ choices(cf. Figure 3.4) and their individual creation for part D of the particularanalogy (cf. Figure 3.5) in a desired log file. As usual, the leading numericalvalues such as 770,228 define the buttons’ positions, the underscore charac-ters after the @sgs command prevent the buttons from being moved in guestuser mode, and, finally, the parameter d invisibly guides the participant tothe subordinate node once the associated button has been pressed. As it canbe quite cumbersome to only use the parameters d, n, u, or p in order tojump from the current to a required page, @sgs tags entail the useful optionto directly specify the pathname of an existing lab page.

Again, one method for the participants to solve the analogies that theywere confronted with (cf. pp. 34 ff.) entailed the construction of a solutionby resorting to the already discussed toolbar. After the participants hadsubmitted their solution, they briefly explained how they arrived at theirparticular solution in the previous step. For this purpose, a comment boxwas provided. Ultimately, the participants had to click on a button to sub-mit their comment and to proceed with the experiment (cf. Figure 3.7, p. 38).

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16 @form[comment]

17 @box[d2|50:4]

18 @form/[SEND|2:comment_U1-2:|objects:+RESULTS2|n]

Figure 3.7: Subjects were given the chance to comment on their createdsolution

The sample code above is almost self-explanatory: line 16 announces a form(@form) of the type comment whereas line 17 defines the overall size of thecomment box (4 lines, each time with 50 character spaces per line). Finally,line 18 contains the necessary closing tag for the form (@form/) as well asseveral parameters that are related to the submit button labelled with ”SEND”.These appended parameters cause the lab system to store the user commentin the log file ”RESULTS2” within the folder ”objects” at the very momentwhen the SEND button is pressed. Furthermore, the string ”comment_U1-2:” is automatically attached to the front of the corresponding comment forbetter identification purposes.

The last point of this subsection has special reference to the second pretest

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and the final experiment as they entail some technical aspects that have notbeen discussed yet. As stated in the introductory part (cf. pp. 28 ff.), theAnalogy Lab occupies substantial back-end capabilities and accepts, amongother things, HTML code or JavaScript.

In particular, the latter two possibilities were taken advantage of for thesecond pretest. Besides a comment box, it was decided that labelled checkboxes and radio buttons would be provided so that the participants couldconstruct their comments according to the module principle (cf. Figures3.8 and 3.9 below). To verify that at least one radio button or check boxwas selected, a short Java based script had to be written whose code isillustrated in Figure 3.10. This script simultaneously checked whether or notthe comment box was empty and displayed an adequate pop-up message ifthat was true.

For this purpose specifically, there are some built-in tags for general textformatting that one may conveniently use. The opening tag <OSW:SRC> andthe closing tag </OSW:SRC> have to be used for instance if one wants toembed raw HTML code and JavaScript (cf. Figure 3.10, p. 40). Such tagsare equally applied to access and call external programs from within the lab(cf. pp. 42 ff.). These programs usually run in parallel to the Analogy Labon a server and are going to be the topic of section 3.2.5. To specify, mostof the programs presented therein were written for the purpose of the twosupplementary experiments (TdT, Hit; cf. p. 28) and especially for the finalexperiment.

Figure 3.8: Check boxes as used in the second pretest

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Figure 3.9: Radio buttons as used in the second pretest

Figure 3.10: This Java script was used for the second pretest

3.2.4 Measure to prevent multiple participation

In order to prevent participants from taking part in the online test more thanonce, it was decided as a minimal measure to include a web browser cookiewhose presence would be checked before entering the online setup stored inthe Analogy Lab.

When first accessing the online test through the link on the CoUGAR home-page4, participants were given the choice between the two language versions,

4 Link: http://www.cogsci.uni-osnabrueck.de/~cougar/test/

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German and English. Clicking on either opened a link to the Analogy Labin a separate popup window (955 x 855 pixels with all toolbars removed) inthe raw viewing mode (v = 2), i.e. ,only the main frame (cf. Figure 3.11).

Figure 3.11: Sample code to generate a separate pop-up window in rawviewing mode

All potential participants were free to browse through the initial descriptivepages as many times as desired. Until the participant reached the Infor-

mation page (cf. Figure 3.12), no cookies were set. Upon entering his orher personal information on this page and pressing the SEND button, a Javascript was written that created and saved a browser cookie (e.g. under thename ”cougarf” in the case of the final experiment).

Figure 3.12: The information page of the final experiment

The intention was that once participants went past this Information page,they would complete the entire test and would therefore be excluded fromparticipating a second time. The expiry date for the cookie was tentativelyset for the end of the calendar year as it was seen as sufficient time until theonline experiment closed (cf. Figure 3.13).

Within the HTML code for the website test portal, a Java script was included(cf. Figure 3.14) that checked for the presence of the cookie from the AnalogyLab. The script worked such that it established the existence of the cookie. If

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Figure 3.13: The expiry date for the cookie was tentatively set for the endof the calendar year 2007

the cookie was already saved on the participant’s computer, it would not bepossible to go through the test a second time using the same web browser anda popup message was displayed accordingly. If no such cookie was present,i.e., this was the first time, the online test was accessed from the specificcomputer using the specific browser, the experiment continued as usual.

Figure 3.14: A Java script was written to check the presence of the cookiefrom the Analogy Lab

3.2.5 External programs

This final section aims at presenting several programs all of which were writ-ten in Python (a general-purpose, high-level programming language) in orderto support data analysis and to improve the usability of the conducted ex-periments, respectively. With the exception of the log parser that has to beexecuted locally, all other programs are stored on the lab server and calledfrom within the Analogy Lab by means of specifically placed tags. Special

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emphasis will be placed on their functionality and less on the underlyingcode.

Log parser

The log parser has proven itself a powerful and versatile tool to facilitate dataanalysis, especially with respect to the final experiment whose results werestored in a log file with thousands of entries. As it would have been a tedioustask to parse such a log file by hand, the log parser was created before thestart of the first pretest. Ever since, the program has been improved andadapted according to the current needs.

In order to function properly, one first of all has to make sure that any logfile is properly encoded in UTF-8, which then must be manually passed tothe log parser. After the read-in process, the log is converted to comply witha predefined, more suitable internal data structure. The parser then checksthis newly structured log file for inconsistencies and filters out incomplete andredundant task entries. Records of users with incomplete data sets will nolonger be incorporated in the final results. In the case of redundant entries,the chronologically first entry is chosen whereas all others are discarded.

Next, all solutions for D that were once created by the participants in thecourse of the construction condition (cf. pp. 34 ff.) are clustered. Clustersare generated by letting the geometric objects within a given solution snapto a virtual grid whose mesh size is always determined by the smallest objectwithin that particular solution. The object with the lowest, leftmost positionis finally taken as the grid’s origin. All existing objects are then snapped tothe nearest grid point. The signature of a separate object is dependent on itsrelative grid position, colour, shape, and size. The signatures of all availableobjects within a solution characterize the whole“scene”. Scenes with similarcharacteristics are regarded as identical and assigned to the same cluster. Inthis way, similarly created solutions for D are grouped together in a properfashion.

It is worth mentioning that the clustering process can be customized withrespect to the mesh size and origin of the grid. A manual adaptation mightbe necessary if the relative position of the objects in D (measured along thevertical axis) in comparison to the C-part of the analogy plays a decisive rolefor the evaluation. Moreover, clusters can be manually merged by name ifspecified in order to facilitate the analysis.

Once the clustering process is finished, the log parser generates a singleSVG image per cluster comprising all associated user IDs, their solutionsas well as their comments if available. As a last step, the parser splits theentire output file (among other things the participants’ reaction times and

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choices as well as all cluster names are included) by generating several CSVfiles so that the resulting data packages can be handled by programs such asMicrosoft Excel or Apple’s Numbers.

Feedback generator

The feedback generator was created in view of the final experiment to giveparticipants the possibility of comparing their task performance in relationto other participants. For this purpose, they saw one out of six predefinedsentences after the completion of the experiment. These sentences were pre-sented on a separate lab page and suitably selected by the feedback generatoraccording to the current participant’s average reaction time (fast; medium;slow) and the outcome of the clustering process (average; non-average). Thecorresponding tag read <OSW:feedback></OSW:feedback> and was embed-ded at a position within the lab code where the feedback sentence was in-tended to appear.

First of all, the six sentences for the English version of the final experimentwere:(Accordingly, there were six German equivalents)

1. Fast & non-average: You quickly perceive the relevant relations in theanalogies without getting distracted by the details. Your solutions areinnovative and demonstrate a high degree of creativity.

2. Medium & non-average: The time you invest in analyzing the analogiesand constructing their solutions is well balanced. The result is veryoriginal.

3. Slow & non-average: You are highly original in creating solutions to theanalogies. Further, your observations are very thorough and thoughtful.

4. Fast & average: You quickly perceive the relevant relations of the anal-ogy and don’t get distracted by details. The solutions you create arehighly preferred by the majority of participants.

5. Medium & average: You invest a sensible amount of time in the anal-ysis of the analogies. The solutions you create are highly preferred bythe majority of participants.

6. Slow & average: After thoughtful in-depth analysis of the analogiesyou extract the relevant relations and construct solutions that are verypopular among the majority of participants.

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As implied before, a single participant is classified by the feedback gen-erator as fast, medium, or rather slow as a function of the resulting averagereaction time. To this end, the time needed to complete the whole experimentis divided by the total number of tasks (note that e.g. the final experimentcomprised 30 different tasks in total). On the basis of the arithmetic meansof all previously available data sets it is now determined whether or not acertain participant is x standard deviations away (the value for x can bedefined by hand). People below this threshold of x standard deviations areconsidered to be fast. Otherwise, they are either regarded as average or evenslow.

Whether somebody can be rated as creative (non-average) or rather asnormal (average) basically depends on the cluster into which a single par-ticipant can fall per task during the construction condition. More precisely,the total number of people per task that“share” the same cluster as the cur-rent participant is summed up. This result is then divided by the number oftasks and people that have performed exactly those tasks. In this context, amodifiable threshold that can take a value between 0 and 1 comes into play.If this value is set for instance to 0.6, i.e., that people above these 60% willbe classified as creative (non-average). People equal to or below this 60%threshold will consequently be regarded as normal (average).

Stimuli randomizer

Prior to the final experiment, entire stimuli sets had consistently been pseudo-randomized based on a pre-defined sequence. As this situation was ratherunsatisfactory with respect to the validity of the final data analysis, a ded-icated program was written for the randomization of all available stimuli ofthe final experiment in a more sophisticated manner. The surrounding tag<OSW:random> [...button code...] </OSW:random> was incorporated into thecode of specific buttons to call the randomizer from within the lab. Thesebuttons had to be pressed unavoidably by the participants if they wanted toproceed with the experiment.

The stimuli randomizer keeps track of which stimuli pages exist, where onthe lab server they are located as well as the fact that the final experimentconsists of altogether 30 tasks that have to be completed by each participant.Moreover, it is also fed with the information that the final experiment isseparated into three discrete blocks of tasks (see Experiments for details).

Whenever a new participant requests his or her first stimulus page bypressing a specific button, the randomizer is called whereupon an individualfile is created to continuously store the progress of that participant. Thatis, after the tenth stimulus the randomizer forwards the current participant

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to the tutorial of the second task and to sequentially pick out ten differentstimuli to comply with the second block of the experiment. After the 20thstimulus, the randomizer consequently proceeds to assign another set of 10stimuli from the first block to the current participant in accordance with thethird block. Once the 30th problem has been completed, the randomizerforwards the participant to the experiment’s finishing pages.

It is worth mentioning that under all circumstances the randomizer en-sures a uniform distribution of all stimuli depending on the count of alreadyused stimuli. In this way, it cannot happen that a certain stimulus is chosensignificantly more often than the remaining stimuli.

Progress bar

The progress bar was introduced as a new feature during the development ofthe final experiment. It conveys the progress of the running experiment.

Simply put, the bar consists of 30 interchangeable PNG images, i.e., oneimage for each of the 30 tasks that the final experiment comprised. Theimages are stored on the lab server and are appropriately loaded from withinthe Analogy Lab with the aid of specifically placed tags (see below). For eachparticipant, the program in control of the progress bar keeps careful track ofthe number of already completed tasks and returns the link to the correctimage on the server to visualize the bar.

The embedded tag on each relevant lab page to display the correct progressbar image read: <OSW:progress></OSW:progress>

The background program in control of the progress bar always sourced nec-essary data from the stimuli randomizer. Thus, it was called by means of thesame surrounding tag as the randomizer:<OSW:random> [...button code...] </OSW:random>

This is the progress bar image referring to task 1 out of 30:

This is the progress bar image referring to task 30 out of 30:

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3.3 Pretests

Having decided to make use of the Analogy Lab environment and its fea-tures to conduct the online experiments, preliminary tests were carried outto ensure usability of the platform as well as the stimuli created by the Ex-perimental Group. Access to the all experiments was made available via alink on the CoUGAR homepage. Participation was limited to once per par-ticipant and only to one of the language versions since identical versions ofthe tests were created in German and English.

In the end, there were two rounds of pretesting for the purpose of checkinghow the test could best be set up in order to make it both accessible for theparticipants and at the same time convenient for subsequent analysis of thedata collected. In the first pretest, the two methods of choosing and creatingan own analogy solution were contrasted. Subsequently, in the second pretest,ways to collect more detailed information on the creation of solutions wereevaluated. As a result of the data and feedback accumulated in the twopretests, the features and structure for the final test could be finetuned andadditional modules were implemented to parse the results log, to cluster thesolutions constructed by the participants as well as to provide feedback tothe participants about their performance in relation to previous participants.

In addition, an abbreviated version of the final experiment was created forthe purpose of collecting data in a more controlled environment and makingfinal adjustments for the final online experiment. This version was run attwo university events which enabled collection of fair amounts of data fromthe visitors.

3.3.1 First pretest

The first online pretest was open for voluntary participation in the periodfrom 17 to 28 August 2007.

Participants

A total of 83 session IDs were logged, of which 27 were removed automaticallyusing the results log parser prior to statistical analysis because participantswere either unable to perform the experiment completely or there was somesort of discrepancy in their results log or IP address. In sum, 56 valid sets ofdata were collected and included in the evaluation of the pretest.

Participants were 56 selected acquaintances of the project members, rang-ing in age from 19 to 48 years. There were 30 males and 24 females withmore than half being students (60.7%) in various disciplines of study. Half

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of the participants were from Germany and others from all over the worldand 62.5% of the participants opted for the German version of the test. Themajority were right-handed and had never taken an IQ test before.

Procedure

Participants sat in front of a computer screen and completed the pretest on-line. They were instructed to solve geometric analogies of the form A:B::C:D(read ”A is to B as C is to D”). Their task was to complete D.

Before the actual test began, participants completed an information formto provide information about their gender, nationality, handedness, occu-pation, area of studies (if applicable) and when they had last taken an IQtest. These factors, though not our primary focus in the experiment, wereexamined for possible preference correlations.

The test consisted of two blocks, the “choice” (C) and the “make” (M)conditions. In the C condition the analogy solution was to be selected fromamong three alternatives (multiple choice). In the M condition participantscreated the solutions themselves by dragging and dropping shapes from atoolbar onto the solution space on the screen. Further, in the M conditionparticipants were asked to comment on how they had come up with theirsolution (i.e., what reasoning they had applied). The purpose of this wasto have an explanation (possibly already in terms of a rudimentary Gestaltinterpretation) for potential novel solutions that we had not thought of be-forehand.

Materials

Participants were randomly presented with one of four pseudo-random com-binations (listed in A.4) of 16 analogies (listed in A.1). Each combinationconsisted of a C condition followed by an M condition, each consisting of8 trials. Further, half of the analogies had unambiguous solutions (U); theother half comprised ambiguous analogies (A), the solutions to which hadpreviously been classified in terms of different Gestalt laws. The A analogieswere further divided into subgroups: one group of four analogies (A1-1, A1-4,A2-1, A2-4) consisted of variations of each other, differing in features suchas shape, color, dimension, and distance of objects to one another. Similarlywith a second group of three analogies (A1-2, A2-2, A2-3) and an additional,unrelated one. The motivation for varying features was to test whether a cer-tain solution preference was robust or whether it could be easily manipulatedby varying certain features.

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The multiple choice answers for the U analogies consisted of the “correct”answer plus two additional “incorrect” answers. For the A analogies, theanswers consisted of the ambiguous alternatives.

The U and A groups were further subdivided into U1, U2 and A1, A2,consisting of four analogies each. This was a purely technical decision: inhalf of the combinations U1 occurred in the C condition, in the other half itoccurred in the M condition. Thus, a pseudo-random distribution of analogieswas ensured over the test.

Results

Due to the Analogy Lab’s randomization function, the different combinationsoccurred a different number of times. The majority of participants (23) wereassigned to Combo1 (41.4%), followed by 12 each in Combo2 and Combo3(both 21.4%) and 9 in Combo4(16.1%). Thus, A1 and U1 occurred 21 times,A2 and U2 35 times in the C condition. In general, solution preferences areto be interpreted as tendencies, and not as statistically significant preferencedifferences.

Relative answer frequencies for the C condition are shown in Figures 3.15and 3.17, for the M condition in Figures 3.16 and 3.18. Comparison of the Cand M conditions by analogy are provided in Appendix A.5.

Unambiguous analogies

There were clear answer preferences for the unambiguous analogies, as ex-pected. (For specific solutions and distributions of the solutions to eachanalogy, see Appendix A.5.)

It should be noted, however, that U2-1 and U2-3 showed a second solutionpreference of 11% in the C condition. Since this is not repeated in the Mcondition, this second choice is attributed to lack of attention to detail in whatcould be considered the easier task of choosing from pre-given solutions.

In the M condition, participants created an additional interesting solutionfor U1-1 (25%) and U2-1 (15%). In both cases, solution 4 was created byintroducing an axis (horizontal in U1-1 and vertical in U2-1) along which thefigure in A is reflected.

From such cases, it can be seen that some of the analogies which hadpreviously been categorized as unambiguous and for which participants werepresented with pre-given solutions in the C condition needed further modi-fication and that interesting and creative results could mainly be obtainedfrom the task in the M condition.

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Figure 3.15: Relative frequencies of unambiguous analogies in the C condition

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Choice: unambiguous relative frequencies

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Figure 3.16: Relative frequencies of unambiguous analogies in the M condi-tion

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Ambiguous analogies

Compared to the unambiguous analogies, answers for the ambiguous analo-gies were more evenly distributed. Additionally, many more different solu-tions were created in the M condition. Although in general there was notenough data to determine whether seeming differences in answer preferencesare significant, these differences again are interpreted solely as tendencies.

The first group of ambiguous analogies (A1-1, A1-4, A2-1, A2-4) showeda general preference for color switching, especially in A1-1 and A2-4-M. Thesecond preferred solution was generated by rotating the entire figure by 180degrees. There was an increase in preference for this solution upon intro-ducing an additional shape to the analogy. The third solution, which wasexplained in terms of movement of the black object upwards, was highly

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Figure 3.17: Relative frequencies of ambiguous analogies in the C condition

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Answer 1 Answer 2 Answer 3

Make Choice

Make Choice

dispreferred in the C condition, except in A2-1. This may be explained bythe rectangular objects giving the impression of two parallel aligned lines,which in turn may facilitate perception of movement (i.e., following the Lawof Good Continuation). Comparing the C and M conditions revealed thatthe Good Continuation solution was more preferred in the M condition thanin the C condition for A1-1 and A1-4. This may be due to the fact thatparticipants perceive or represent the problem differently or reason about itdifferently when asked to create an own solution than when asked to choosefrom among given alternatives.

Figure 3.18: Relative frequencies of ambiguous analogies in the M condition

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Answer 1 Answer 2 Answer 3 Answer 4 Answer 5 Answer 6 Answer 7


It can be clearly seen from the many plausible solutions that were createdby participants in the M condition that detailed analysis of the explanationsgiven by participants of how they came to their solutions is needed. In termsof categorizing the comments received (if at all) and the expressions used

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for their explanations, the task of unifying the varied explanations was ex-tremely complex and based on subjective interpretation of the oftentimesintuitive and vague statements left behind by the participants. Thus, assign-ing preferred Gestalts to different solutions without a large margin of errorwas difficult.

3.3.2 Second pretest

The second pretest was run from 19 October 2007 to 7 November 2007.

Participants

Complete data sets were obtained from 44 participants who were acquaintedwith members of the study project. To facilitate data analysis, demographicinformation obtained from the participants was done by selecting a responsefrom a given list of options (instead of free input) for all categories.

The participants ranged in age from below 20 to over 40 years of agewith most participants falling into the 20-25 age group. There were 33 malesand 10 females with 40 right-handed participants, one left-handed participantand one ambidextrous participant. With respect to familiarity with geometricproportional analogies, IQ test-taking history showed that most participantshad never taken an IQ test before, and if they had, this had been the case inthe last 1-5 years. In order to see if a mathematical/logic background wouldhave an effect in terms of familiarity in solving geometric analogy problems,a question was included on area of study. Participants’ background revealedno dominant discipline. Almost half of the participants were from Germany,with the rest spread out from different countries in the world. Students andemployees made up a majority of the participants.(For details, see Appendix B.5.)

Procedure

The procedure was a modified and extended version of the “make” conditionof examples first used in the first pretest. In particular, since the commentanalysis for the first pretest proved to be extremely complicated due to thediversity in explanations given for the construction of the experiments, thesecond pretest was primarily used to test whether a more homogenous clus-tering of the solution comments could be achieved.

Again, participants sat in front of a computer screen and completed thepretest online. They were instructed to complete geometric analogies of theform A:B::C:D by constructing the solution for D using the objects available

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to them in the toolbar. The test consisted of two conditions, the“checkboxes”(Check) and the “radio buttons” (Radio) conditions. In the Check condition,participants were given a multiple selection option to indicate all the oper-ations (a-j) that were applied to the C figure to derive at their solution forthe analogy. In the Radio condition, the participants were asked to choosefrom a number of statements (a-f) describing a possible set of operations toderive at a correct solution and select one that best matched their strategy.

Materials

In alternating order, participants were assigned to either the Check or Radiocondition. Within each condition, participants were presented with one offour pseudo-random combinations of 24 analogies (see Appendix B.3). Intotal, there were 42 examples (six “original” analogies with six modificationseach). For each of the six “original” examples, modifications were madesystematically resulting in example classes, each consisting of the original,three with modifications to the objects in the A (and/or B) source domain(i.e. all figures in the C remain as in the original) plus an additional threewith modifications to the objects in the C target domain. These variations ofthe original with fixed A (and/or B) or fixed C domains were divided equallyinto two mixes (see Appendix B.2) to create two combinations of 24 exampleseach, including the six original examples in all combinations.

The motivation for varying features in the source and target domain,respectively, was to determine which manipulations of features would mostinfluence the interpretation of the analogy in terms of preferred Gestalt prin-ciples. By keeping one part of the analogy constant throughout all examplesin each of the six example classes, the aim was to find out the most robustfeatures and/or influential manipulations.

Following the page on which the participants were asked to create an ownsolution to each analogy problem using the objects in the toolbar of the Anal-ogy Lab, the analogy was again displayed in conjunction with the solutioncreated by the participant. Depending on the condition, the participant wasasked to select either multiple options from a list of nine operations or onestatement from a list of five sentences. The operations and statements weredevised from the most commonly mentioned explanations for the solutionscreated in the first pretest (see Appendix B.4).

In the Check condition, participants were given a list of nine operationsthat could possibly be performed on the object(s), separately or in some com-bination, to arrive at the solution. Participants were asked to select all thatapplied in the creation of their solution. Alternatively, in the Radio condi-tion, participants were given a list of five sentences that represented possible

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sequences of operations performed to arrive at a correct solution, from whichparticipants were asked to select one only, that which most matched their ownsolution strategy. In both cases, an option to add a comment was includedin case none of the operations or statements seemed adequate. It shouldbe noted that one of the Radio button statements was always an incorrectoption to ensure that participants did indeed read through the statements.

In sum, there were two randomized sequences created of each of the twomixes to create four combinations of 24 analogies each. Thus, all participantssolved all the “original” examples (S1 to S6) in addition to three sets of threevariations in either the A (and/or B) (A1 to A3) and the C domains (C1to C3). Each participant was asked to create solutions to 24 analogies andcomment on their solution, either by selecting one or multiple checkboxes orone radio button statement.

Results

The second pretest was prematurely terminated as it became clear that itwould not be possible to achieve a large enough sample size to obtain signif-icant results. Furthermore, the test was deemed too long, time-consuming,monotonous and complicated so that it could not generate sufficient interestamong participants to complete the whole experiment. From the feedbackreceived it became clear that the slight variations made to each of the originalexamples were in fact too small for an experiment participant who insteadof focusing on the details viewed the analogies displayed as somewhat repet-itive and did not give each the full attention necessary to tease apart andappreciate the minute differences. In addition, the task of reading throughand selecting the appropriate checkboxes and radio button statements tookmuch longer than expected, especially since many participants created theirsolutions based on intuitions that they found, as already discovered in thefirst pretest, to be difficult to express in words.

All of the six original analogies were solved by all participants. For theother examples, the number of participants ranged from 22 to 27. The resultslogged in the Analogy Lab were parsed and the clusters manually adjusted.Although an initial attempt was made to classify the solutions according toGestalt principles based on the result of selected checkboxes and radio but-tons, the combinatorial variation proved too great to draw any conclusions.

The following figures show the relative frequencies of the preferred so-lution clusters and their visualization. Due to the limited amount of data,no significant conclusions could be drawn and the distribution of solutionscreated are to be viewed as pure tendencies. (All solutions that were createdby two participants or less, though included in the plots, are not displayed.)

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S1

In the S1 series, the original example was modified to test the effect of addinga third color to counter grouping by similarity (S1A1), changing the initialshape to rectangles that created an illusion of being more closely connectedand thus highlighting movement following the Law of Good Continuation(S1A2), highlighting positioning of different parts of a bigger object followingproximity (S1A3) and respective changes to the C domain, i.e., adding a thirdcolor in the C domain (S1C1), changing the shapes in the C domain to createthe illusion of connectedness (S1C2) and highlighting positioning of differentparts of a bigger object (S1C3).

In sum, it could be argued that the stability of the preference for move-ment can be manipulated by different feature changes and may not be asrobust throughout the variations of the original example in this class.

OriginalFor the original example in the S1 series, three solutions were created most

frequently.

: ::: ?4 0 4 12 4 2

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3. Group C into single unit and switch colors

The variations were created to gain a better understanding of the sep-aration into different objects (whether this was influenced by positioning,similarity/dissimilarity of color, shape, etc.) and if perceived movement ofobjects was affected by whether the shapes touched or indicated a clear di-rection of movement.

S1A1

Variation to test influence of additional dissimilarity of color

: ::: ?4 5 4 0 4 11

Solutions:

1. Group C into two separate objects and replace top part correspondingto A with B

2. Group C into single unit (according to proximity), switch colors (whiteto grey) and rotate entire unit 180 degrees

3. Group C into single unit and switch colors (white to black and blackto grey)

The preference order remained unchanged despite the introduction of athird color.

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S1A2

Variation to test influence of connectedness of shapes

: ::: ?4 0 4 11 4 2 4 4

Solutions:

1. Move black circle to top right

2. Group C into two separate objects and switch positions of top andbottom

3. Group C into single unit and rotate entire unit 180 degrees

4. Group C into single unit and reflect along diagonal axis

Rectangles replaced the circles in A and B. It was reasoned that the tworectangles seemed more connected to each other, forming a line which couldbe followed, thus enhancing the effect of movement of the black object.

Although there was no clear preference, it is noteworthy that the simplecolor-switch solution was not among the top three solutions.

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S1A3

Variation to test influence of position

: ::: ?4 1 4 0 4 3

Solutions:

1. Group C into two separate objects and switch positions of top andbottom

2. Group C into single unit and reflect entire unit along horizontal axis


A straight vertical line of four circles replaced the circles in A and B.It was reasoned that a sense of two contrasting halves on top of each otherwould be created.

The preference for the position switch and reflecting the unit along avertical axis located along the midline of C support the intended purpose ofthe variation.

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S1C1

Variation to test influence of color

: ::: ?4 3 4 5 4 0

Solutions:


2. Group C into two separate objects, move black circle to top right byswitching with grey circle in that position

3. Group C into single unit and reflect entire unit along diagonal axis

Similar to S1A1, an additional color was introduced. This time, however,it was in C. This resulted in an increased preference for the rotation solution.

S1C2

Variation to test influence of connectedness of shapes

: ::: ?4 0 4 2 4 1

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Solutions:


2. Group C into two separate objects, move black rectangle to top right


Similar to S1A2, rectangles replaced the circles, albeit in the C domain.The effect was a slight enhancement of the rotation solution, but also of themovement solution.

S1C3

Variation to test influence of similarity of shapes

: ::: ?

4 0 4 1

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Solutions:

1. Group C into single unit and switch colors


A diagonal line of circles replaced the four circles in the C domain tocreate a line of continuity. This resulted in increased occurrence of the color-switching strategy. The second most often created solution involving rotationdoes not yield a figure different from that in C.

S2

In the S2 series, the original example was modified to study the effect ofchanging the color across the analogy of the central object (S2A1), varyingthe size of the center (S2A2), changing within the analogy the color of thecentral object (S2A3), offsetting the center (S2C1), countering the color andposition of the central object (S2C2) and manipulating the symmetry aroundthe center (S2C3) in order to determine exactly which rule was applied inarriving at the most preferred solution.

In sum, it should be noted that having a black and central circle hada persistent effect as the object to focus making changes on when creatingsolutions to the analogy. Attention seems to automatically be drawn to theobject demonstrating the attribute of centrality and “blackness”.

Original

?: :: :

1 0

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Solution: Retain black square.

Although there is only one solution to this unambiguous analogy, it shouldbe noted that there are four different rules that could apply to arrive at thesame solution, i.e.,

1. Retain black objects

2. Remove white objects

3. Retain central objects

4. Remove outer objects

Variations were created to test which of these rules applied and how eachcould be emphasized by manipulation of the objects. It is striking that thestark contrast created by the black square against the white circles makesthe square such a focus of perception.

S2A1

Variation to test influence of color and centrality

?: :: :

1 0 2 1

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Solutions:


2. Remove black objects

Given the two competing features of interest of the central black square,the modification was created to contrast centrality and color (black). Thepreferred solution was to retain the central object, regardless of color.

S2A2

Variation to test influence of centrality

?: :: :

1 0 3 1

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Solutions:


2. Remove grey objects

Another line of shapes and an additional color was introduced to furtherhighlight the centrality of the black square. Although the preferred solution isto attribute centrality solely to the black square, solution cluster 3 1 showsthat the white circles can also be seen as creating a group with the blacksquare if given an additional color and/or shapes.

S2A3

Variation to test influence of color and position

?: :: :

1 0 1 1

Solutions:

1. Retain black objects and move downward

2. Retain top objects and move downward

The solution for this variation required manipulation on the non-centrallylocated black shapes. Still, the preference was for keeping the black circlesand shifting them down to the center — analogous with the black square inC — instead of viewing the black circles as the top objects and moving thosedown.

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S2C1

Variation to test influence of color, centrality and position

?: :: :

1 0 4 0

Solutions:

1. Retain most central objects


A modification of the C domain was created by extending the figure withadditional objects of different shape and color. Again, the preference to keeponly the central black square was strongest.

S2C2


?: :: :

1 1 2 0

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Solutions:



By modifying the object in C to have as its central object not a blackshape, but a white circle, the same shape and color as in the previous analogy,the features of central position and color were pitted against each other. Thetendency was to prefer to retain the most central object, rather than removethe white objects.

S2C3


?: :: :

1 0 2 0

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Solutions:



Despite adding more of the same shape and color objects as those removedin the previous modifications, the solution of keeping the central black objectwas still preferred.

S3

In the S3 series, the original example was modified to test the effect of pittingreflection against rotation as a preferred operation. A triangle was usedsince the directionality of the vertex could reveal which of the two operationswere applied. The variation involved using different colored triangles (S3A1),contrasting colored triangles (as a group in S3A2 and separate in S3A3),contrastingly angled triangles (as a group in S3C1 and separate in S3C3)and addition of triangles (S3C2).

In sum, it could be said that the manipulations with different colored andangled triangles made visible the contrast between the reflection and rotationsolutions. It appears that there is no clear preference for one over the other,even when the arrangement of the shapes would call for it.

Original

?: :: :

2 3 2 1

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Solutions:


2. Group C into single unit and reflect along horizontal axis

Since the figure remains the same even after possible reflection, the vari-ations created were an attempt to distinguish the operations applied.

S3A1

Variation to test influence of separation of objects by color

?: :: :

2 2 2 0

Solutions:

1. Reflect only left object along horizontal axis

2. Rotate only left object 180 degrees

Different colors for the same object were used in the AB domain to high-light the change in only one of the objects. The asymmetry of the trianglereflection along a horizontal axis and rotation made it easier to see whichoperation was used. In terms of preferred solution, however, both were usedwith almost equal frequency.

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S3A2

Variation to test influence of grouping of objects

?: :: :

2 0

Solution: Group C into single unit and rotate 180 degrees /reflect alongvertical axis

Contrasting colors were used for the triangles to bring the movement ofthe figure as a whole to the fore and this was also the only solution to becreated. It cannot be distinguished, however, which operation was used.

S3A3

Variation to test influence of separation of objects by color

?: :: :

2 0

Solution: Group C into two separate objects and rotate each 180 degrees

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Contrasting colors were again used to further demonstrate rotation ofeach separate object, which was the most commonly created solution.

S3C1

Variation to test influence of color

?: :: :

2 2

Solutions: Group C into single unit and rotate 180 degrees/reflect alonghorizontal axis

An additional object in a different color was added in the C domain.Although only one solution was highly preferred, it cannot be clarified whichoperation was used more often.

S3C2

Variation to test influence of grouping by color

?: :: :

3 5 3 3

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Solutions:



In contrast to S3C1, the figure in the C domain appears different afterrotation and after reflection, making clear that for this analogy, the preferredoperation is the rotation of the group.

S3C3

Variation to test influence of separation by color

?: :: :

2 0 2 5 2 3 2 1

Solutions:



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3. Group C into two separate parts and rotate each triangle 180 degrees

4. Group C into two separate parts and rotate grey triangle 180 degrees

In contrast to S3C2, the asymmetric figure comprising only two trianglesdecreases the rotation preference slightly with the reflection solution also be-ing applied. Furthermore, other solutions involving separate rotation of theobjects is considered more than in S3C1.

S4

Similar to the S3 series, in the S4 series, reflection and rotation were measuredagainst each other by manipulation of color (S4A1 and S4A3), addition ofobjects (S4A2 and S4C1) and both (S4C2) and changes in the distancesbetween objects (S4C3).

Given the circles and squares, which in reflection and rotation about theirown centers remain unchanged in shape, the manipulations could highlightthe exclusive use of one operation over the other (namely rotation) as reflectedin the preferred solutions.

Original

?: :: :

2 1 2 2

Solutions:

1. Group C into single unit and rotate 180 degrees

2. Group C into single unit and reflect along vertical axis

This example class aims at teasing apart differences in the application ofreflection and rotation.

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S4A1

Variation to test influence of addition of color switching

?: :: :

2 2 2 1

Solutions:

1. Group C into single unit, rotate 180 degrees and switch colors

2. Group C into single unit, reflect along vertical axis and switch colors

By adding the need to switch colors, preference for the reflection solutionwas slightly increased.

S4A2

Variation to test influence of addition of shape

?: :: :

2 1 2 0

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Solutions:


2. Group C into single unit and rotate clockwise 90 degrees

The more complex figure in the AB domain in the variation was intendedto increase the perception of rotation as a solution. As the solution preferenceindicates, this was indeed the case with reflection no longer being as highlypreferred.

S4A3

Variation to test influence of grouping by color and shape

?: :: :

2 0

Solution: Group C into single unit and reflect along horizontal axisBy eliminating the possibility of rotation due to the asymmetric triangle

rotation of the figure in A, the reflection solution became the obvious highlypreferred solution.

S4C1

Variation to test influence of addition of shapes

?: :: :

4 0

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Solution: Group C into single unit and rotate 180 degrees /reflect alongvertical axis

By adding more objects, a direct object switching solution was sup-pressed. However, although there is only one preferred solution, the op-erations used is not clear from just the visual representation.

S4C2

Variation to test influence of separation by color and shape

?: :: :

2 2 2 1

Solutions:



This variation was again an example of visualizing the rotation versusreflection solutions with a different composition of shapes for the figure in C,with the rotation being slightly more preferred than the reflection.

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S4C3

Variation to test influence of separation by position

?: :: :

2 0 2 1 2 4

Solutions:



3. Group C into two separate parts and switch positions

By creating more distance between the two objects in C, it was intendedto emphasize an object switch solution, which was not as obvious in S4C2.Indeed, this solution 2 4 appeared as a possible candidate by this modifica-tion.

S5

In the S5 series, the running example was modified to see the effects ofchanges to the analogy in order to distinguish the direction (up versus down)of the movement of objects and to which object the movement would beapplied. This was done by addition of black circles (S5A1), changing of theobject to rectangles which suggest a directionality in movement (S5A2 inthe AB domains, S5C3 in the C domain and multiple addition in S5A3) andremoving the difference among the object shapes (S5C1 and S5C2).

It is clear from the solutions that the downward movement is a prominentfeature of the analogy. The salience of the black circle is again obvious in this

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running example with participants tending to apply changes to this objectrather than another.

Original

Running example

?: :: :

2 5 2 0

Solutions:

1. Group C into two separate objects and move white square down/switchobjects

2. Group C into two separate objects and move black circle up

This class contains modifications to the running example. For the orig-inal example, as had been the result previously, both the solution to movethe black circle up toward the square and that to move the white squaredown toward the circle were fairly equally preferred. The modifications werecreated to stress one of the directions.

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S5A1

Variation to test influence of movement by multiple objects

?: :: :

6 1 6 0

Solutions:

1. Group C into two separate objects and move white squares down

2. Group C into two separate objects and move black circles up

By adding more black circles in the B domain it was intended to stressthe downward movement. The result, however, indicates that additionalobjects did not necessarily influence the good continuation of the downwardmovement or that of the black circles sufficiently strongly. Both solutionswere still chosen fairly equally.

S5A2

Variation to test influence of downward movement by single object

?: :: :

2 1

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Solution: Group C into two separate objects and move white square down

With the similar intention to accentuate the downward movement byusing rectangles that formed a continuous line of movement, S5A2 succeededin making the downward movement of the white square more prominent.

S5A3

Variation to test influence of downward movement by multiple objects

?: :: :

6 0

Solution: Group C into two separate objects, repeat three times and movewhite squares down

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As in S5A2, the rectangles accentuated the downward movement in con-trast to the circles which had not done so in S5A1. One consideration is thestrong effect of the black circle, which seemingly becomes a focal point ofmanipulation in creating a solution.

S5C1

Variation to test influence of similarity of objects

?: :: :

2 3

Solution: Group C into two separate objects and move top white squaredown

By removing considerations of the shape of the moving objects, the down-ward movement created by the highly focal black circle is simply applied tothe top white square.

S5C2

Variation to test influence of color and similarity of objects

?: :: :

2 0 2 3

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Solutions:

1. Group C into two separate objects and move top black circle down

2. Group C into two separate objects and move black circles down

Viewed in conjunction with S5C1, the results for the solution preferencessuggests again that the movement of the black circles is taken into consider-ation more than if they did not appear in the C domain.

S5C3

Variation to influence movement by shape

?: :: :

2 5 2 1 2 3

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Solutions:

1. Group C into two separate objects and move black rectangle up

2. Group C into two separate objects and move white square down

3. Group C into two separate objects and move black rectangle down

A downward-pointing rectangle was added to further enhance the effectof the downward movement for black objects. However, the solution createdby moving the rectangle further downward (2 3) was less preferred than themoving of the rectangle in the opposing direction upward (2 5) and also thatof moving the square down (2 1) involving continuation of the movement ina downward direction.

S6

Similar to the S4 series, the rotation versus reflection distinction for figurescomprising multiple objects is investigated in the variations of the originalexample in the S6 series. The rotation solution is made more obvious byconnecting the shapes comprising a whole figure (S6A1 and S6A2) and ma-nipulating the color and shape distribution among the multiple objects (allremaining variations).

In sum, it would seem from the preferences of the solution in this classthat the solution involving grouping together of all objects into a whole androtation of that group is heavily preferred when multiple objects are present.

Original

?: :: :

4 0 4 2

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Solutions:

1. Group C into single unit and rotate counterclockwise 90 degrees

2. *Group C into single unit and rotate clockwise 90 degrees

Although there is only one preferred correct solution, it should be notedthat there are three different rules that could apply to arrive at the samesolution, i.e.,



3. Group C into two separate parts and switch colors of the bottom twoobjects

The S6 series investigated the effect of modifications on the preference ofrotation of the figure in C as a whole versus reflection along diagonal axisand possible color switching strategies. As solution cluster 4 0 suggests, it isnot clear which operation was used.

(As for solution 4 2, it is considered an incorrectly created solution,though it did occur three times.)

S6A1

Variation to influence grouping

?: :: :

4 0 4 3

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Solutions:


2. Group C into two separate parts and switch colors of bottom two ob-jects/retain all shapes and rotate color filling in counterclockwise di-rection

The intention of the modification was for the connectedness of the fiveinstead of four circles to give rise to the preference for the rotation solution,which was the most preferred solution. Again, it is not clear whether adiagonal reflection was applied as the two operations cannot be distinguishedfrom each other by mere visualization of the solution. Although only createdby two participants, solution cluster 4 3 is shown here to point out that thecolor switching solution was also considered.

S6A2

Variation to test influence of grouping by color and shape

?: :: :

4 0

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Solution: Group C into single unit and rotate counterclockwise 90 degrees

By using all squares that touch each other to create a connected figurein the AB domains, the reflection strategy was eliminated. In this way, itcan be clearly seen that the rotation solution was the most prevalent of thepreferred solutions.

S6A3

Variation to influence separation by color and shape

?: :: :

4 5 4 2

Solutions:

1. Retain shapes for each separate object and rotate black color filling 90degrees

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2. Separate C into separate parts and switch colors (black to white andwhite to black)

By adding the grey color and thereby focusing attention on the changesof the contrasting black and white objects, the color switch strategy wasenhanced. In the 4 5 solution cluster, the top black circles were consideredextraneous (and remained the same color as the grey did in the AB domains)and only the bottom objects were modified; while in the less common 4 2solution cluster, the black-white color switching was applied to each object.

S6C1

Variation to test influence of separation by shape

?: :: :

4 3 4 0

Solutions:



The modification in C was to include a figure which was symmetrical inshape along the diagonal axis. The asymmetry of the colors seems to haveaffected the preference for the rotation solution, while the diagonal reflection/color moving solution was slightly enhanced.

S6C2

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Variation to test grouping by color

?: :: :

4 1 4 3

Solutions:



The same effect as in S6C2 was found despite the asymmetrical shape ofthe figure in C.

S6C3

Variation to test influence of grouping by shape and color

?: :: :

4 0

87

Solution: Group C into single unit and rotate counterclockwise 90 degrees

By removing all color discrepancies, the rotation/diagonal reflection so-lution was clearly preferred.

Categorization of comments

As mentioned, the sample size was too small to determine any significantsimilarities between the explanations that participants gave for the creationof their solutions. Furthermore, the data to be analyzed was so disorganizedthat it was not possible to do a systematic classification of the operationsused, as reported by the participants. In the Check condition, the numberof checkboxes and their possible combinations made it impossible to achievecommonality among the 44 participants. As for the Radio condition, theconstraint placed on the freedom of expression by the pre-given statementmay have caused unease among participants who were not certain as how toverbalize their intuitions on their analogy-solving strategy.

Given the inability to distinguish between competing Gestalt principles atplay in the analogies from pure visualization of the solutions, it is of utmostimportance to analyze the comments given by the participants so as to furtherclassify the solution clusters into“Gestalt” clusters, which would be used forthe formalization of the analogies.

In sum, the conclusion drawn for the final test from the second pretest,whose primary purpose was to decide between the use of checkboxes andradio buttons for the facilitation of data analysis, was to return to voluntarycommenting with explicit instructions as to the vocabulary of operations tobe used.

3.3.3 Mini tests: HIT and Technologietag

Participants

The participants were visitors of two public events, the Technologietag 20075

(TT) and the Hochschulinformationstag6 (HIT), who volunteered to takepart in the experiment. In total there were 112 participants, from which 103usable datasets were collected (50 from the TT and 53 from the HIT). Theparticipants’ age ranged across all age groups - from “younger than 20” to

5The Technologietag 2007 was an exhibition for people interested in computer scienceand technology (http://project.informatik.uni-osnabrueck.de/tt2007/).

6The Hochschulinformationstag is an annual exposition day of the study programs atthe University of Osnabruck intended for pupils of the local schools (http://www.zsb.uni-osnabrueck.de/hit.html).

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“over 40” years. Fifty participants were female, 52 were male, and one didnot specify gender. Seventy-four participants (72% of total number) wereGerman; the others were from various countries worldwide. The participantswere not paid for their contribution to the experiment.

Materials and design

For the present study we created 21 analogies composed of circles, squares,triangles and rectangles in three different colors - white, black and grey. Fromthese 21 analogies three were so-called original analogies, from each of whichsix variations were derived. We varied each original analogy three timeswithin the source domain, and three times within the target domain. Thusin the end we had three original analogies and eighteen variations. Figure3.19 shows an analogy with two variations. All analogies were ambiguousand allowed for different plausible solutions. The variations were created insuch a way that different Gestalt laws were emphasized to a different de-gree and therefore meant to trigger different solutions. We focused on fourGestalt principals: proximity, similarity, symmetry and good continuation.The extent of varying the analogies was limited to position, shape, number,and color. Figure 3.19 (top) shows an example of a variation in the sourcedomain. Adding new elements to the B domain of the source creates a newnotion: guided by the Gestalt law of good continuation humans might per-ceive the circles to be arranged in a descending order. The second variationin Figure 3.19 (bottom) adds a new element to the target domain, causing achange in perception, which is this time triggered by the Law of Proximity:the new element (white circle) creates a new relation by means of proximity.

In the experiment we used a software tool especially developed for thispurpose: the Analogy Lab7 (Figure 3.20) is a web-based framework for con-structing solutions to geometric proportional analogies via drag and drop. Italso provides a clustering algorithm to classify the solutions constructed bythe participants. For the construction of the solutions a toolbar with all thenecessary geometric objects was available and there was a trash can symbolfor throwing away undesired objects (Figure 3.20). The objects in A, B andC were not movable. There was neither a time limit to construct the solu-tions, nor a limit to the number of objects in the toolbar that could be used.A button was available for proceeding to the next page. In the experiment,the Analogy Lab was running on standard laptop computers (two at the TTand three at the HIT) with a 15” screen and a resolution of 1024x768 pixelsor above.

7http://mvc.ikw.uos.de/labs/cc.php

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Figure 3.19: An analogy, a variation in the source domain and a variation inthe target domain

Figure 3.20: Screenshot of the Analogy Lab while creating a solution, withadditionally labeled toolbar and trash can

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Figure 3.21: Screenshot of the Analogy Lab while writing in the optionalcomment box

Procedure

Participation in the experiment would begin as the participants sat down infront of one of the laptops at the site of the experiment. The Analogy Labbrowser window was always open showing the starting page of the experimentbefore a participant started taking part. The experiment took place in a semi-controlled environment: an experimenter was present at all times, but peoplepassed by freely in the vicinity of the experiment site.

The experiment consisted of solving a sequence of analogy problems, andeach solution consisted of two phases: the construction phase and the com-ment phase. At the beginning, the participants went through a one-examplepractice session. In the construction phase (Figure 3.20) participants saw thefirst three (A, B, [source domain] and C [target domain]) objects of an anal-ogy and solved the problem by constructing the object D in order to completethe analogy. The input was made by drag and drop with a standard mouse.The object could be dragged to the trash can symbol if not desired. Once theparticipants were satisfied with their solution, they clicked on the OK buttonto proceed to the second phase (Figure 3.21), where participants were askedfor optional comments on their constructed solutions. The participants wereable to see what they had constructed while they were commenting. Theywere asked to give a short description of the steps of reasoning they hadmade in the process of constructing the solution using the keyboard. In or-der to facilitate and shorten the process of commenting, some keywords were

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suggested above the comment box, such as: rotate, mirror, remove/changeobjects, shapes, colors, position etc. Again a button click was required, thistime for proceeding to the next analogy.

The stimuli were presented in a pseudo-randomized manner. The setof stimuli was divided into three combinations: each combo contained oneout of the three original analogies, and two variations of each of the threeoriginal analogies. One of the three combos was randomly selected for eachparticipant. Thus each participant had to solve seven analogies in total.

As soon as a participant had finished the test, a message on the screenindicated the end of the test and gave the subject a feedback statement ontheir performance. The feedback statement was based on the comparison ofthe sum of their time needed to complete the individual analogies (excludingthe time it took them to comment) with the average sum of the other par-ticipants’ times, and on the comparison of a participant’s choice of solutionswith the average choice solutions for the particular analogies. The averagechoice solution was determined by the mean distribution of the answers basedon the performance of the previous participants.

Results

For each participant the constructed solutions for the analogy tasks werestored in the experiment log file together with the comments entered and thetime that was needed for the construction. The complete log file was thenparsed and the data was clustered to identify groups of solutions. The differ-ent solutions were then analyzed and compared to find preferred solutions.The comments were evaluated with the aim of identifying the transformationsused by the participants in the process of creating the solutions, and further-more to determine the strategies for solving geometric analogies. There were31 to 37 participants per combo. The answers for one original analogy andfor its respective variations of the source and target domain were compared.We will focus here on analogy “A1Original” as well as two of its variationsand we give a complete overview of the results for the remaining analogies inAppendix C.

The results for the original analogy A1Original are depicted in Figure3.22: 93% of the participants constructed the solution shown with the letter(a) while the rest created other solutions, labeled (b) and (c) in the figure.87% of the participants who constructed solution (a) provided a comment de-scribing their opinion on how this solution was created. The analysis of thecomments revealed that the participants applied different transformations toconstruct the same solution. The participants explained their solutions withthe following transformations: 38% of the participants argued that they left

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Figure 3.22: Original analogy A1Original

the middle object intact, 27% stated that they removed the outer objects,another 27% removed the white circles, whereas 8% stated that the blacksquare should stay.Figure 3.23 shows the results for analogy A1S3, one of the variations of the

analogy A1Original. In this variation the source domain has been altered bymoving the black circles in A from the center to the top while the rest of theanalogy was kept unchanged. Here 44% of the participants constructed thesolution (a), 38% solution (b) and the rest created sundry solutions. All par-ticipants who constructed solution (a) provided comments, from which thefollowing transformations were identified: 80% removed the white objects,13.3% kept the center object and 6.6% flipped the figures in the center. Forsolution (b) 84% of the participants provided a comment and described thefollowing transformations: 55% kept the first row objects, 15% rotated alongthe Z-axis in a 3D manner, 15% moved the upper row one position down.

Figure 3.24 shows the results for A1T3, which is another variation ofA1Original. Here the target domain has been altered by inserting additionalelements. In this case, 50% of the participants constructed solution (a), 35%solution (b), 10% solution (c) and the rest created miscellaneous solutions.76% of the participants who constructed solution (a) commented on it andfrom their descriptions the following transformations were identified: 85%kept the center object and 15% removed all types of circles. All participantswho constructed solution (b) argued that they took the white figures away.66% of those who created solution (c) provided comments and stated thatthe outermost circles had to be removed.

The results for the remaining variations of the original analogy A1Original

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Figure 3.23: Results for a variation of the source domain (A1S3)

Figure 3.24: Results for a variation of the target domain (A1T3)

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are summarized in Figure 3.25. The analogy tasks are depicted together withthe preferred solutions. For each solution, the number of participants whocreated that solution as well as the transformations that were described inthe comments are given.The results for the analogies A2Original and A3Original and their variationsare documented in Figures C.1 and C.2 in Appendix C.

Discussion

By varying the number or the features of the elements in the analogies wecould distinguish which features of the elements were more salient in theprocess of solving. In A1Original the saliency of the central black objects ispronounced, with resulting removal of the remaining objects. Changing thecolor of the central object, as in A1S1 (in the source domain) or in A1T2 (inthe target domain), still leads to a pronounced saliency of the centeredness,suggesting that position is a more important feature than color. However,the robustness of color increases when a third color is added, as in A1T1,A1T2 and A1T3. Here the ambiguous third color (grey) is perceived, bymeans of applying the Gestalt Law of Similarity, as belonging to the groupof dark objects. This might be explained by the hypothesis that if too manyfeatures are present in an analogy, then humans apply Gestalt principles inorder to group objects and thereby reduce the cognitive load.

This can also be seen in the second analogy (A2Original and its varia-tions), in which the solution of the analogy is reached more often by applyingthe Law of Proximity to group separate objects and then to rotate them asa whole. That can be seen in all the variations. Another observation isthat rotation is preferred to reflection. Although it was not possible to dis-tinguish between these two transformations by analysis of the constructedsolutions, the additional comments of the participants in our experimentprovided more insight and suggested that the preferred transformation is ro-tation. The commenting feature of this experiment let us identify that theLaw of Good Continuation influenced the perceptions of the third analogy(A3Original and its variations).

Conclusions

We designed the geometric proportional analogies so that they have multi-ple plausible solutions. We had the participants construct solutions on theirown. This is different from most other setups, where participants selectfrom pre-defined solutions, as in standard intelligence tests. Our method

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Figure 3.25: Results for A1Original and its variations

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has many advantages. Letting the participants construct their own solu-tions does not constrain their reasoning processes to a limited set of answers,which is particularly important for ambiguous analogies such as the ones usedin our experiment. Moreover, having the freedom of constructing solutionscauses the participants to reflect more cautiously on which transformationsare necessary for them to complete the analogy, in contrast to selecting frompre-defined solutions, which might be accomplished by only considering su-perficial features and not the relational structure.

Another important feature of our setting was that the participants hadthe possibility to provide comments. The participants’ comments were cru-cial in revealing how they actually experienced the analogy-making process.The analysis of the comments confirmed that there are several strategies forsolving geometric proportional analogies, as discussed above. The commentsrevealed very helpful information that could not have been derived solely byanalysis of the constructed solutions.

The data of these comments showed that the participants often appliedtransformations to several objects at a time, which indicates that participantsformed groups of objects. These groupings can be explained by Gestalt-basedperception: most of the time, objects with common color or shape (Law ofSimilarity) or nearby objects (Law of Proximity) have been grouped together.

The number of objects in an analogy and the number of colors and shapesdetermine the complexity of an analogy problem. The observations we maderevealed that more complex analogies lead to a greater variety of createdsolutions. A possible explanation for this effect is the increased cognitive load:The strategies that different individuals employ for the purpose of copingwith this load differ from each other. Varying an analogy by introducingnew objects or new features of an object, in particular in the target domain,increases the cognitive load imposed on the participants in the process ofsolving. A good strategy to cope with an increased cognitive load is to applyGestalt principles to group objects and thereby reduce the complexity of theproblem. Thus, when different Gestalt principles are applied, then differentsolutions arise, as observed in our experiment.

3.4 Final experiment

3.4.1 Participants

238 participants completed the online experiment over the course of threeweeks. Participants were recruited from among the experimenters’ friends,the pool of cognitive science students at the University of Osnabruck, and a

97

number of online fora.Participants ranged in age from below 20 to over 40, with over 60% of all

participants falling into the 21-30 age group. 85 female and 146 male partic-ipants took part in the experiment. 205 participants were right-handed, 16left-handed, and 14 ambidextrous. Half of all participants had never taken anIQ test before. Most participants (62%) were German, followed by 24% In-donesians. 46% of participants were students, 32% were employees. Over halfof all participants did not indicate their area of expertise — this is probablydue to the high percentage of cognitive science students participating in thestudy, for whom there was no item “cognitive science” as opposed to“naturalsciences”, “computer science”, and “humanities”. For detailed distributions ofage, IQ history, nationality, expertise, and occupation, see Appendix D.1.

3.4.2 Procedure

The procedure was a modified version of that employed in the pretests, takinginto account the results from all pretests. In particular, the second pretesthad yielded a high number of participants who aborted the experiment beforereaching the end. We believe this is due to the pretest’s long duration, whichwe attributed to two main factors:

1. The high number of analogies in the second pretest (42). This wassolved by reducing the number of analogies to 30.

2. The necessity of choosing a radio button or (more than one) checkbox tocomment on one’s solutions. This considerably increased the time thatparticipants had to invest in the solving of one analogy. This was solvedby removing the radio buttons/checkboxes and making commentingoptional (as in the first pretest).

Again, participants sat in front of a computer screen and completed thepretest online within the framework of the AI Analogy Lab 8. They wereinstructed to solve geometric analogies of the form A:B::C:D. Their taskwould be to complete D. Instructions were modified so as to incorporate somesuggestions of vocabulary that might be used in the optional commenting(e.g. rotate, mirror, remove, change objects / shapes / colours / positionsetc.”). These suggestions were repeated above the comment box after theconstruction of every analogy.

Before the actual test began, participants completed an information formto provide information about their gender, nationality, handedness, occu-

8see Chapter 3.2.1 for documentation

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Solution order Analogies123 S1OR S2A2 S3A3 S4C2 S5C4132 S1A1 S2A3 S3C2 S4C3 S6OR213 S1A2 S2C1 S3C3 S5OR S6A1231 S1C1 S2C2 S4OR S5A1 S6A3312 S1C2 S3OR S4A1 S5A3 S6C1321 S2OR S3A1 S4A2 S5C3 S6C2

Table 3.1: Solution distribution in the choice condition

pation, area of studies (if applicable) and when they had last taken an IQtest.

The test consisted of three blocks of ten analogies each. The first andthird block were in the choice (C) condition and the second block was in themake (M) condition. C and M conditions were as in the first pretest (seeSection 3.3.1). The only difference was that commenting in the M conditionwas explicitly made optional. We hoped that this would further decrease thenumber of premature experiment dropouts.

A progress bar on the top of the screen displayed participants’ progressin the experiment.

3.4.3 Materials

Participants were presented with 30 analogies, randomized for presentationposition (and, consequently, for condition). We ensured that analogies wereequally distributed over the C and M conditions. Thus, each participant saweach analogy, albeit in a different order. Appendix D.2 contains the full setof stimuli. Analogy nomenclature is as follows: the first number after theS indicates the analogy class. “OR” indicates that this analogy is this class’original analogy, that which is being varied. Analogies containing variationsof the AB domain have an “A” in their name, those with variations in theC domain a “C”. For example, analogy S3A2 belongs to analogy class 3 andcontains a variation of S3OR’s AB domain.

The solution order for the C condition was pseudo-randomized over analo-gies in the following way. For each analogy, the three most frequent solutionsaccording to the data from the second pretest were selected (when available).The solutions over the whole analogy set were then distributed over the sixdifferent ordering possibilites based on their frequency ranking. Distributionof solution order is shown in Table 3.1. For example, solution order 231means that the solution in position 1 is the second most frequent, followed

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: :: :

Figure 3.26: S2Original pretest (top) vs. S2Original final test (bottom)

by the third most frequent, followed by the most frequent one.The number of analogies was reduced compared to the second pretest,

which had consisted of 42 different analogies falling into six different classes.Thus, while an analogy class in the second pretest consisted of one original,three variations of the AB domain and three variations of the C domain (7analogies per class in total), we reduced this to the original and two variationseach in the final test (5 analogies per class in total). Those variations withthe least ambiguity in solutions according to the second pretest were excludedfrom the final test.

In addition, the C domain of the S2 class’ original analogy was modified(see Figure 3.26). This was due to the fact that in both pretests, as well asin the mini tests from the Technologietag and the Hochschulinformationstag,over 99% of all participants chose the same solution. This solution (singleblack square) could be reached both by retaining the black objects as wellas by retaining the central object. To test which of these operations wasmore prominent, the objects’ colors in the C domain were switched. Thus,retention of black objects would lead to one solution (two black circles), whileretention of the central object would lead to another (single white square).A dummy solution (single white circle) was created to fit the schema of threesolutions per analogy. Other analogies with dummy solutions are S3A3.

A further change in stimuli concerns class S5 (see Figure 3.27). Thedownward pointing rectangle in S5C3 was intended to facilitate the “goodcontinuation” (absolute downward movement) solution. For the final test anadditional analogy S5C4 was created that was identical to S5C3, with theexception that the black rectangle pointed upwards. We hoped to find a

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Figure 3.27: S5C3 (top) vs. S5C4 (bottom)

preference for the proximity solution in comparison to S5C3.Appendix D.3 contains a classification of all solutions in terms of a) the

transformations that, when applied to the figure in C, yield the solution andb) the Gestalt laws we assume to be in play when the application of the re-spective transformations is salient. This classification, which we constructedprior to conducting the experiment, reflects our own judgment about the pos-sible ways to get to the different solutions. Its validity was tested by compar-ing it with the results from participants’ optional commenting. Figure 3.28shows the distribution of solutions according to applicable transformations.

3.4.4 Results

The number of datasets obtained ranged between 75 and 85 for each analogyin the M condition, and between 153 and 163 in the C condition.

In what follows, plots of relative frequency for each solution in each anal-ogy are shown, alongside the analogy itself and any additionally createdsolutions in the M condition. Solutions that were created by less than 5% ofparticipants were not taken into account.

The same color over C and M plots indicates the same solution. Thesolution numbers in the C plot are coupled to the solution position in theanalogy’s solution set as presented to participants. For example, in analogyS1OR, the bars in the C condition solution plot are labelled as “2=n53”,“3=n54”, and“1=n52”. 2,3, and 1 indicate the solution’s position as presentedin the analogy, which is displayed above the plot.

Due to the fact that there are too many factors that potentially influencethe perception of analogies and the investigation of what these factors are

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

rotation partially

rotation / switch colour

switch colour

switch position

object movement

reflection

reflection / switch colour

rotation

deletion of non-central objects

deletion of non-central objects / colour switch

deletion of non-top group / object movement

deletion of white objects

Overview of transformations:colour saliency / object movement

deletion of grey objects

deletion of non-black objects

Figure 3.28: Distribution of solutions according to applied transformations

would have well exceeded the scope of the CoUGAR project, we we werenot able to systematically vary the analogies so as to make them comparableacross analogy classes. This also had the effect of making statistical analysisof solution preferences impossible. Therefore, if we speak of e.g. the datareflecting a preference for rotation over reflection, we are speaking not ofan actual statistical effect, but rather of a tendency. Analogy results arelisted and discussed individually and are only compared to other analogiesin cases of minimal feature variation. At the end of each analogy section thetransformations and Gestalt laws associated with solutions 1 to 3 are listed.

S1

S1OR

102

2=n53

3=n54

1=n52

S1ORC

0.0

0.2

0.4

0.6

0.8

n53

n52

n54

n56

n55

4_1

1_0

1_1

4_3

4_5

4_6

S1ORM

0.0

0.2

0.4

0.6

0.8

n55

n56

Ambiguity in this analogy stems from the possibility of perceiving theanalogy in one or many of the following ways:

1. The whole figure in A is perceived as being rotated.

2. All objects are perceived as switching their color.

3. The black circle is perceived as moving upwards.

4. The objects are perceived as being reflected over a diagonal axis goingfrom the upper left to the lower right.

5. The figure is perceived as consisting of an upper and a lower group,which exchange positions.

Data from the C condition indicates that the rotation solution is preferred(i.e. all objects in C are grouped together according to the Law of Proximity),followed by solutions 3 and 1. Thus, in this analogy rotation is preferred overcolor switching (according to the Law of Similarity) and object movement(according to the Law of Good Continuation - GC).

The M condition reflects the preferences revealed in the C condition andadditionally yielded solutions corresponding to 4.) and 5.).

Solutions: transformations and Gestalts

1. Object Movement – GC

2. Rotation – Proximity

3. Color switch – Similarity

103

S1A1

2=n35

1=n33

3=n34

S1A1C

0.0

0.2

0.4

0.6

0.8

n35

n33

n34

4_26 0_0

4_15

4_21

4_25 4_6

4_7

4_9

4_12

4_13

4_17

4_18 4_2

4_20

4_23

4_24

4_27 4_3

4_5

4_8

S1A1M

0.0

0.2

0.4

0.6

0.8

4_26

In this variation we varied the AB domain by introducing an additionalcolor, with the aim of testing whether this would lower the preference for(binary) color switching.

This was not the case. Instead, among the two most preferred solutionsin the C condition is solution 2, which is created by switching the color of allblack objects to grey and of all white objects to black. Indeed, this was thepreferred solution in the M condition, and was created twice as often as thenext preferred solution. Thus, instead of decreasing the preference for colorswitching, this variation in fact increased it.

The M condition also yielded a novel solution, created by a combination ofobject movement and global color switching. This solution stands in contrastto solution 1, where the two circles unaffected by the black circle’s movementretain their white color. In the novel solution not only does the affected whitecircle change its color to grey, but rather all white circles do.


1. Object movement – GC



104

S1A2

3=n37

2=n36

1=n39

S1A2C

0.0

0.2

0.4

0.6

0.8

n36

n37

n38

n40

n39

0_0

1_0

4_12

4_13

S1A2M

0.0

0.2

0.4

0.6

0.8

n38

n40

In this variation we varied the AB domain by changing the shape of theobjects. Instead of circles, the analogy contained rectangles. The purpose ofthis was to invoke the image of a “line” consisting of two shapes, along whichthe shapes could move. Based on this, we predicted a higher preference forthe movement solution.

This was the case in the M, but not in the C condition. Here there wasstill a strong preference for the rotation solution. The relative frequency ofmovement solution preference in the C and M conditions are overall not sodifferent (32% vs. 38%). It is the difference between the rotation solutionpreference which is striking (57% in the C condition vs. 25% in the M con-dition). This difference might be due to the fact that 19% of participantsin the M condition constructed a color switch solution (n38) which was notavailable as a choice in the C condition. Thus, rotation may have been the“second best solution” for a large part of participants in the C condition.

The overall comparison of frequency of movement solution constructionbetween S1A2 (average of C and M conditions: 34%) and S1OR (average ofC and M conditions: 25%) provides support for the prediction that variationof the AB domain such as to facilitate perception of object movement “alonga line” increases preference for the movement solution.


105

1. Position switch – Similarity



S1C1

3=n44

1=n46

2=n42

S1C1C

n=162

0.0

0.1

0.2

0.3

0.4

0.5

0.6

n44

n46

n41

n43

n42

n45

4_10

4_11

4_20

4_22

S1C1M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

3=n46

1=n46

2=n42

S1C1C

0.0

0.2

0.4

0.6

0.8

n44

n46

n41

n43

n42

n45

4_10

4_11

4_20

4_22

S1C1M

0.0

0.2

0.4

0.6

0.8

n44 n41

n43

3=n46

1=n46

2=n42

S1C1C

0.0

0.2

0.4

0.6

0.8

n44

n46

n41

n43

n42

n45

4_10

4_11

4_20

4_22

S1C1M

0.0

0.2

0.4

0.6

0.8

n44 n41

n43

Analogous to S1A1, the C domain was varied by introducing a new color.The goal, as in S1A1, was to test whether (in the M condition) this wouldlead to a lower preference for color switching.

This seems to be the case — solution n43, corresponding to the colorswitch solution (black turns white, white turns black) was created in only14% of the cases, as opposed to 22% in this class’ original analogy. Again,there was a strong preference for the rotation solution in the C condition(62%) which decreased substantially in the M condition (36%). As in S1A2,we attribute this decrease to the solution construction freedom in the M con-dition, which, if absent, seems to trigger the “second best solution” choice.The solution construction freedom in the C condition leads to the construc-tion of two new solutions: the color switch solution, mentioned above, andthe solution corresponding to 5.) as described in section 3.4.4. Here, there isa similarity-based grouping of the C domain into an upper and lower group,which switch their position to yield solution n41.


106


2. Reflection – Similarity


S1C2

2=n51

3=n50

1=n48

S1C2C

0.0

0.2

0.4

0.6

0.8

n48

n51

n50

n49

n47

4_0

4_1

4_10

4_13

4_14

S1C2M

0.0

0.2

0.4

0.6

0.8

n49

Analogous to S1A2, the circles in the C domain were replaced by rectan-gles, analogous to S1A2. A higher preference for the movement solution waspredicted.

Compared to S1OR (25% movement solution), this prediction was notconfirmed — the movement solution on average in the C and M conditionswas chosen in 25% of all cases. However, there is a difference of 10% inmovement solution preference between the C and M conditions. Why this isthe case is not entirely clear.

Rotation solution preferences display the same pattern as above: highpreference in the C condition (52%), with a decrease in the M condition(31%). Participants in the M condition additionally constructed the solutioncorresponding to the switch of upper and lower group of circles. Preferencefor the color switch solution was comparable to the original S1 analogy.



107



Summary of S1 series In this series we attempted two things: to decreasethe preference for color switching by introducing a new color, and to increasethe preference for object movement by employing rectangles instead of circlesto create the image of a“line”along which objects can move. While the formerdid not seem to work out as planned, the data indicates that the latter did.

S2

S2OR

3=1_

0

2=n6

4 1

S2ORC

0.0

0.2

0.4

0.6

0.8

1_0

n64

2_1

2_2

3_0

S2ORM

0.0

0.2

0.4

0.6

0.8

The variations in this series are of the old, not the new S2OR9. All varia-tions were aimed at teasing apart the application of the following four trans-formations.

1. Non-black objects are deleted.

2. Non-central objects are deleted.

9see Figure 3.26 for the difference; see section 3.4.3 for the explanation of this difference.

108

3. White objects are deleted.

4. Outer objects are deleted.

The results of S2OR show a clear preference for the retention of the centralobject over the retention of the black objects10.


1. DUMMY

2. Object deletion (non-black) – Similarity

3. Object deletion (non-central) – Similarity

S2A2

1=1_

1

2=3_

0 3

S2A2C

0.0

0.2

0.4

0.6

0.8

1_1

3_0

0_0

1_0

1_2

2_0

5_0

5_1

5_2

S2A2M

0.0

0.2

0.4

0.6

0.8

An additional “layer” of grey circles was added to the top and bottomof the figure in A, with the purpose of forcing participants to decide for asolution corresponding to the deletion of either grey or non-central objects

10“Retention of black objects” is used synonymously with“deletion of non-black objects”.Analogous with “retention of central objects” and “deletion of non-central objects”. Whileparticipants’ comments never contained the negative formulation (i.e. “deletion of non-central/non-black objects”), we use this expression nevertheless for the sake of uniformity.

109

(solutions 2/3 and 1 respectively). As in S2OR, centrality seems to be themore prominent feature - there was a clear preference for the solution corre-sponding to the retention of the central object as opposed to the deletion ofthe grey objects.


1. Object deletion (non-central) – Proximity

2. Object deletion (grey) – Similarity

3. Object deletion (grey) – Similarity

S2A3

1=n59

3=n61

2=n60

S2A3C

0.0

0.2

0.4

0.6

0.8

n59

n61

n60

n58

0_0

n62

3_0

S2A3M

0.0

0.2

0.4

0.6

0.8

n58

This variation was aimed at teasing apart application of non-black objectdeletion and combined deletion of outer objects with color switch. In addi-tion, there was a solution corresponding to the downward movement of thetopmost object.

This time, there was a clear preference for retaining the black object, fol-lowed by the solution corresponding to downward movement of the topmostobject. Retaining the central object, which was the preferred transformationin the previous analogies, was applied in the C and M conditions in only19% of the cases (as opposed to 79% in S2OR). This is a drastic difference.

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However, solution 3 not only requires retaining the central object, but also inaddition switching that object’s color. It seems reasonable to conclude thatthe application of more than one operation is more complex and thus lesspreferred. Therefore, although centrality seems to be an important aspectin geometric analogy perception, the number of transformations that haveto be applied to reach a certain solution might be claimed to be the moreconstraining factor in solution construction.

A novel solution was also created by 9% of participants: this solutionis arrived at by the application of two transformations, retention of blackobjects followed by downward movement of retained objects.



2. Object deletion (non-central) and color switch– Proximity

3. Object deletion (non-top group)/movement – GC

S2C1

1=4_1

3=n57

2=1_0

S2C1C

0.0

0.2

0.4

0.6

0.8

1_0

3_1

n57

3_4

4_1

2_1

3_5

3_6

2_0

3_3

4_0

7_0

S2C1M

0.0

0.2

0.4

0.6

0.8

3_1

3_4

This variation of the C domain is designed to yield different results forthe application of all four of the transformations mentioned in section 3.4.4.

111

Thus, deletion of non-central objects yields solution 2, deletion of white ob-jects solution 1, deletion of non-black objects solution 3. The restriction tothree solutions in the C condition did not allow presentation of the fourth so-lution (that results from application of outer object deletion). This solutionwas created, however, in the M condition (solution 3 1).

Interestingly, the results of this analogy in the C and M conditions do notreflect the same preferences. The data from the M condition shows a clearpreference for deletion of non-central objects (solution 2, constructed 49%),followed by deletion of outer objects (solution 3 1, constructed 19%). In theC condition, on the other hand, solution 2 was very dispreferred. In thiscondition, white object deletion was most preferred, followed by non-blackobject deletion.

Why is this the case? First of all, participants might simply be lazy.Solution 2 is the one that contains least objects and is consequently fastestto construct. Thus, when confronted with a number of intuitively sensiblechoices, they may have simply chosen the “easiest” solution in the M condi-tion. In the C condition, on the other hand, the effort for choosing a solutionis equal for each solution. This is a factor that was not controlled for - herewe may be seeing the effects.

A second reason for this distribution of solution preferences might be thenature of the C domain, which initially looks quite complex. It contains threedifferent colors and two different shapes. Of the five objects it contains, onlytwo are equal. Thus, it might be that participants were overwhelmed by theanalogy’s complexity and simply chose the first option presented to them,which happened to be that corresponding to the deletion of white objects.We are potentially seeing here the effects of not randomizing solution positionand of not controlling for analogy complexity in terms of number and kindof shapes.

However, if we disregard the laziness hypothesis as an explanation of thedata in the M condition, but take it as is, it falls in line with the previousdata exhibiting a clear preference for retaining the central object.


1. Object deletion (white) – Similarity



112

S2C2

3=n6

3

1=2_

0

2=1_

0

S2C2C

0.0

0.2

0.4

0.6

0.8

n63

1_0

2_0

2_4

1_2

2_6

0_0

2_3

2_5

S2C2M

0.0

0.2

0.4

0.6

0.8

In this variation of the C domain, the position of the black object waschanged, and a new color was introduced. Thus, the application of whiteobject deletion, non-black object deletion, and non-central object deletionleads to different results.

The obtained results fit in well with the other data from the S2 series:solution 3, corresponding to the retention of the central object, is highlypreferred.


1. Object deletion (white) – Similarity



Summary of S2 series This series was aimed at determining whetherobject position (centrality) or color (blackness) were more salient, which weexpected to see directly mirrored in the deletion/retention of certain ob-jects. Results generally indicated that object centrality is very prominent,thus leading to central objects’ retention. The only factor that seems tobe stronger than centrality is analogy complexity in terms of the amount oftransformations that have to be applied to reach a certain solution. Where

113

reaching a solution required not only retaining the central object (the gen-erally preferred strategy), but additionally performing a color switch, thissolution was very dispreferred.

S3

S3OR

2=n5

3=2_

6

1=2_

5

S3ORC

n=153

0.0

0.1

0.2

0.3

0.4

0.5

0.6n5 2_6

2_5

0_0

2_1

1_0

2_2

2_3

S3ORM

0.0

0.1

0.2

0.3

0.4

0.5

0.6

This analogy is ambiguous in that one can perceive the relation betweenA and B to be one of the following:

1. Rotate the triangle.

2. Rotate the whole figure.

3. Reflect the figure horizontally.

The results indicate a preference for global rotation, i.e. for perceivingthe figure as a whole, presumably proximity-based, and rotating this whole.Second most preferred is the reflection solution, followed by the dispreferredpartial rotation solution. Results are similar in the C and M conditions.


1. Rotation – Similarity


3. Reflection – Symmetry

114

S3A1

2=2_

2

3=2_

7

1=2_

6

S3A1C

0.0

0.2

0.4

0.6

0.8

2_2

2_7 n1 0_0

2_4

2_6

1_0

2_0

2_1

2_5

S3A1M

0.0

0.2

0.4

0.6

0.8

In order to inhibit perception of the figure as a whole, and rather facilitateperception of individual objects, we introduced a new color in AB.

This led to a strong decrease in the global rotation solution (here solution1), while preference for partial rotation increased (solution 2).





115

S3A3

1=n2

2=2_0 3

S3A3C

0.0

0.2

0.4

0.6

0.8

n2 n3 0_0

2_0

S3A3M

0.0

0.2

0.4

0.6

0.8

n3

In order to further enhance perception of the individual objects, we con-trasted white and black instead of white and grey objects, in AB. In addition,objects in A were aligned such that applying global rotation would not leadto the constellation in B. This indeed led to a clear preference for individuallyrotating the objects in C, yielding solution 1.




3. DUMMY

116

S3C2

1=3_

6

3=n4

2=3_

7

S3C2C

0.0

0.2

0.4

0.6

0.8

3_6 n4 3_7

3_1

0_0

3_0

3_3

3_4

3_5

5_0

S3C2M

0.0

0.2

0.4

0.6

0.8

To further test whether the figure in C was being perceived as a wholeor as individual objects, we introduced an additional white triangle, whichwould lead to a different solution if rotated individually (solution 2) insteadof as part of the whole figure (solution 1).

Both the C and M conditions supported the preferences obtained in S3OR.Global rotation (53% - S3OR: 61%) is preferred over reflection (33% - S3OR:29%) and partial rotation (11% - S3OR: 7%).





117

S3C3

2=2_5

1=2_2

3=2_3

S3C3C

n=158

0.0

0.1

0.2

0.3

0.4

2_5

2_2

2_3

2_1

2_4

2_6

0_0

2_0

2_7

S3C3M

0.0

0.1

0.2

0.3

0.4

2=2_5

1=2_2

3=2_3

S3C3C

0.0

0.2

0.4

0.6

0.8

2_5

2_2

2_3

2_1

2_4

2_6

0_0

2_0

2_7

S3C3M

0.0

0.2

0.4

0.6

0.8

2_1

2_4

For the purpose of blocking perception of the figure as a whole and insteadfacilitate perception of the individual objects, we introduced a new color inC (analogous to S3A1). In addition, the shape of the square was changed toa triangle - this was done to see whether the white shape would be rotatedas well (which is obviously not visible when using squares).

Solution 1 corresponds to the global rotation solution, solution 2 to thatof partial rotation (rotation of triangles initially “standing upright”), andsolution 3 to the individual rotation of all triangles. The same ranking ofsolution preferences was obtained in both C and M conditions.

There was a preference for solution 2, indicating that object orientationis a salient feature in perception. This was even stronger than the preferencefor global rotation, which is surprising given the findings above. In additiona solution was created which was arrived at by rotating only the grey triangle(solution 2 4). However, this occurred much less frequently than the solutioncorresponding to the rotation of either all or only upward pointing triangles.Thus, one might conclude that object orientation is a more salient featurethan object color when grouping according to the Law of Similarity.

However, solution 2 may also be construed as a reflection solution, arrivedat by reflecting the figure in C along a horizontal axis. The just mentionedconclusion then is not valid.



118



Summary of S3 series This series shows that participants prefer groupingof all objects and performing global rotation on the resulting group whenpossible. A similar observation was already made for the S1 series, whererotation of the whole figure was a highly preferred strategy. This is quiteintriguing, since grouping is generally considered to be a complex, costlyprocess. However, these results suggest that the opposite may be the case:global grouping may be the default, with splitting being a costly process onlyto be applied if certain features make individual objects particularly salient(as in e.g. S3C3).

S4

S4OR

3=2_

0

1=2_

1 2

S4ORC

0.0

0.2

0.4

0.6

0.8

2_0

2_1

2_2

n12

0_0

S4ORM

0.0

0.2

0.4

0.6

0.8

This series was aimedat further testing the preference of rotation vs. reflection, including also theswitching of object position.

The different applicable operations in this analogy are:

1. The figure is reflected along a central vertical axis.

2. The whole figure is rotated by 90 degrees to achieve a “laying” position.

119

3. The objects switch position / The whole figure is rotated by 180 degrees.

Results show a preference for solution 3 — this may be arrived at both byglobal rotation or object position switch. While the reflection solution wasalso constructed in a quarter of the cases, the 90 degrees rotation was highlydispreferred.





S4A1

2=n6

3=2_1

1=2_5

S4A1C

0.0

0.2

0.4

0.6

0.8

n6 2_1

2_5

2_6

0_0

1_0

2_0

2_3

2_4

3_1

S4A1M

0.0

0.2

0.4

0.6

0.8

2_6

The colors in the B domain were switched. This had the effect thatsolution 1, corresponding to global rotation was only a partial solution. Theonly “correct” solutions left over were solution 2 and 3. The former wasreached by the combination of global rotation with additional object colorswitch, or by object position switch with additional color switch. The latterwas reached by vertical reflection with additional object color switch. Also,an additional partial solution was created (solution 2 6), which was reachedby simply reflecting over the vertical midline axis.

Although participants chose and constructed the “correct” solutions 60%of the time (again with a preference for global rotation/object position switch),

120

there was a surprisingly high amount of partial solutions in both the C andM conditions (19%). This might be attributed the the analogy’s complexity,arising from the high number of different shape/color combinations presentin the AB domain - no two objects are the same. This supports the findingsfrom analogy S2C1.





S4A2

3=2_

1

2=2_

3

1=2_

5

S4A2C

0.0

0.2

0.4

0.6

0.8

2_1

2_3

2_5

0_0

2_0

2_2

2_4

S4A2M

0.0

0.2

0.4

0.6

0.8

In this variation, an additional white circle was added in the AB domain,with the purpose of facilitating a 90 degrees rotation solution. This indeedwas the case. 90 degrees rotation and reflection were equally preferred.





121

S4C2

1=n7

2=2_5 3

S4C2C

0.0

0.2

0.4

0.6

0.8

n7 2_5 n8 0_0

1_0

2_1

2_3

2_4

4_0

S4C2M

0.0

0.2

0.4

0.6

0.8

n8

In this variation, the circle in C was replaced by a triangle. This has theconsequence that global rotation and object position switch lead to differentresults, which was not the case in the previous S4 analogies.

Results in both C and M conditions indicate a preference for global ro-tation. Reflection was also a preferred transformation (solution 2). Thesolution corresponding to object position switch (n 8) was constructed in9% of the cases. Thus it is safe to say that rotation is more preferred thanposition switch. This again supports the hypothesis that (assuming low com-plexity is preferred over high complexity) grouping is in fact cognitively lesscomplex than splitting.





122

S4C3

1=n9

3=n1

1

2=n1

0

S4C3C

0.0

0.2

0.4

0.6

0.8

n9 n11

n10

0_0

1_0

2_1

2_2

2_3

2_5

2_6

S4C3M

0.0

0.2

0.4

0.6

0.8

Similar to S4C2, in this variation the shapes in C are not directly adjacent.The intention behind this was to test whether this would facilitate perceptionof the individual objects as opposed to a whole figure. If so, solution 2 shouldbe preferred.

The results indicate nothing of the sort, but instead were congruent withthe results obtained from S4C1. This may show that the Law of Proximitydoes not apply stronger to directly adjacent objects than to objects thatstand slightly apart.



2. Position switch – Proximity


Summary of S4 series This series further investigated rotation prefer-ences. Further support for the hypothesis that global rotation is a generallypreferred solution strategy was provided. In addition, these data show thatrotation is preferred over object position switch. However, a potential prob-lem may be that the analogies in general are not controlled for the transfor-mations that lead to their solutions. The effect this has on our analogy set

123

is that there are 21 different solutions that can be arrived at by the applica-tion of global rotation, but only three that are reached by switching objects’positions. This may have biased participants to construct rotation solutionswhen possible, i.e. it might be that once they began employing the rotationstrategy, they became “blind” to the alternatives.

S5

S5OR

1=n2

3

2=n2

2 3

S5ORC

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n23

n22

0_0

1_0

2_1

S5ORM

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ambiguity in this analogy comes about by potentially perceiving the anal-ogy in the following ways:

1. The upper object moves towards the lower object.

2. The circle moves towards the square.

3. The circle moves downwards.

While there seems to be a slight preference for the “upper object movestowards lower object” solution, the absolute downward circle movement solu-tion was highly dispreferred, indeed was not created at all in the M condition.Since pretests had already indicated that this solution was not among the

124

preferred, variations of the analogy aimed at facilitating absolute downwardmovement.





S5A1

1=n1

3

3=n1

4 2

S5A1C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n14

n13

6_5

0_0

2_0

6_2

6_7

S5A1M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

By varying B such that a “line” of circles is formed, we hoped to facilitateperception of absolute circle downward movement. Thus, solution 2 shouldhave been chosen more often than the corresponding solution 3 in the original.

This was not the case, however (3% chosen in S5OR, 4% in S5A1). Again,it was not created in the M condition. The preference for solution 3 was evenhigher, though.

In addition, there is an incongruence between solution preferences in theC and M conditions - while solution 1 is preferred over solution 2 in the Ccondition, this preference is reversed in the M condition. It is not clear whythis is the case. One hypothesis is that it might be an effect of solution

125

position in the C condition: instead of reasoning about the solution beforechoosing, it may be that participants see the first solution and try to makeit “fit” the analogy, thus leading to a higher preference for the first solution.





S5A3

2=n1

5

1=n1

6 3

S5A3C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n15

n16

6_6

0_0

1_0

2_0

2_1

2_2

6_2

6_7

S5A3M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

To further enhance perception of absolute downward movement, the cir-cles in AB were replaced by rectangles, analogous to S1A2 and S1C2.

This again had no effect in the M condition, though there is a slightincrease in preference for the absolute downward solution in the C condition(6%). However, the overall preference for solution 2, the “upper towardslower object” solution, became even clearer.



126



S5C3

1=n1

9

2=n1

7

3=n1

8

S5C3C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n19

n17

n18

2_0

2_8

S5C3M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Analogous to S5A3, the circle in C was replaced by a downward slantedrectangle to facilitate absolute downward movement perception.

Although this led to the construction of the absolute downward solutiontwice in the M condition, it was chosen even fewer times than in the original.Preference for the“upper towards lower object movement”solution was foundonce again.





127

S5C4

3=n2

1

1=n2

0

2=2_

4S5C4C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n21

n20

0_0

1_0

2_4

S5C4M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

In this variation, the rectangle in the C domain was slanted upwardsfrom left to right instead of downwards, as was the case in the previousanalogy. We thus expected a higher preference for solution 1 (“rectanglemoves towards square”) than for the corresponding solution 2 in S5C3, aswell as a lower preference for the absolute downward movement solution 2than the corresponding solution 3 in S5C3.

This was not the case. In addition, the preference for the “upper towardslower object” solution seems robust.





Summary of S5 series This series aimed at varying the perception of“lines” created by slanted rectangles and lined up circles. We hypothesizedthat the more salient the line perception, the more often the absolute down-ward movement would be constructed, in line with the Law of Good Contin-uation.

128

However, this was not the case. One reason may be that the spatialregion that needs to be used for the absolute downward movement solutionlies outside the region used by the A, B, and C domain. Thus, the absolutedownward movement requires outside-the-box thinking.

In general, object position seems to be a more salient feature than objectshape, i.e. preference for solutions corresponding to the movement of objectsbased on their position (“upper towards lower”) was higher than for solutionscorresponding to the movement of objects based on their shape (“circle movestowards square”).

S6

S6OR

1=n3

1

2=n3

2

3=4_

3

S6ORC

0.0

0.2

0.4

0.6

0.8

n31

n32

0_0

1_0

2_0

4_1

4_3

S6ORM

0.0

0.2

0.4

0.6

0.8

Similar to the S4 series, this series aimed at investigating the preferenceof global rotation over individual object movement or feature switch.

The results of the original analogy S6OR show a strong preference forglobal rotation (solution 1), with the solution corresponding to the groupingof the four objects into two groups (upper right and lower left), with objects inthe lower left undergoing a color switch (solution 2) being much less preferred.However, the latter was constructed much more often in the M condition(17%) than in the C condition (6%). This might again be due to a solutionposition effect in the C condition.


129




S6A1

2=n2

5

1=4_

3 3

S6A1C

0.0

0.2

0.4

0.6

0.8

n25

4_3

0_0

1_0

3_0

4_1

4_2

S6A1M

0.0

0.2

0.4

0.6

0.8

To test whether rotation preference could be further increased, we addeda fifth circle to the middle of the figure in the A and B domain, with allcircles now being directly adjacent. We intended this to increase perceptionof the figure as a whole, this in turn increasing preference for global groupingand thus global rotation. While there seems to be no difference in the Ccondition as compared to the original, the rotation solution was constructedin 89% of cases, as opposed to 77% in the original.

The color switch solution (solution 1) was much dispreferred in the Mcondition (4%) as opposed to the original. Thus, while our hypothesis wassupported in the M condition, it was not in the C condition (which mightalso be due to the solution position effect hypothesis put forward previously).





130

S6A3

3=n2

6

1=n2

7 2

S6A3C

0.0

0.2

0.4

0.6

0.8

n26

n27

4_9

0_0

1_0

4_0

4_1

4_3

4_8

S6A3M

0.0

0.2

0.4

0.6

0.8

To increase perception of two groups as described in the original, weintroduced a new color in the AB domain, to facilitate visual discriminationof these two groups. We predicted a higher preference for the color switchsolution.

This indeed seems to be the case. Solution 1, which is arrived at byswitching the color of all black and white objects, was highly preferred inboth conditions.





131

S6C1

2=n2

8

3=4_

5 1

S6C1C

0.0

0.2

0.4

0.6

0.8

n28

4_5

0_0

4_0

4_2

4_3

4_4

S6C1M

0.0

0.2

0.4

0.6

0.8

To further increase perception of two groups instead of one in the Cdomain, we varied the objects’ shape — the upper group consisted of circles,the lower group of squares. We predicted a higher preference for the colorswitch solution over the global rotation solution.

While results indicate that global rotation is still highly preferred, it isless so than in the original. Conversely, there seems to be a slight increasein color switch preference. Thus, the prediction may be said to have playedout, albeit not strongly.





132

S6C2

3=n2

9

2=4_

6

1=n3

0

S6C2C

0.0

0.2

0.4

0.6

0.8

n29

4_6

n30

4_10 3_

0

4_0

4_11 4_

3

4_5

4_7

5_0

S6C2M

0.0

0.2

0.4

0.6

0.8

Complementary to S6C2, we varied the C domain such that the lower leftgroup contained objects of the same color (white) than of the same shape.We predicted this to result in an increased preference for object positionswitch.

This does not seem to be the case. Global rotation is still highly preferred.Solutions: transformations and Gestalts


2. Position switch – Proximity


Summary of S6 series The goal of this series was to investigate the ro-bustness of global rotation as solution strategy. There seems to be a slightinfluence of shape on grouping preferences, as demonstrated in S6C1, whilecolor grouping in this case does not seem to be very salient.

3.4.5 Comment classification

The detailed comparison of comment classification and solution classificationis listed in Appendix D.4.

133

Chapter 4

Closing discussion

Clemens Bauer, Judith Degen, Irena Dorceva, Maxim Haddad,Martin Schmidt, Rolf Stollinski, Kae Sugwara, Martesa Tandra &Radomir ZugicInstitute of Cognitive Science, University of Osnabruck

In this report presents our research on human strategies to solve geomet-ric proportional analoges and documents the findings from experimental in-vestigation of human analogy perception, inspired by Gestalt psychologicalconsiderations.

We conducted a number of experiments — two pretests, two mini-experimentsat TdT and HIT, and a final experiment — to investigate solution prefer-ences participants have and solution strategies participants employ in geo-metric analogy-solving. Pretest results showed that, while we can extractpreference tendencies, there are a) many diverse solution strategies leadingto one and the same solution, and b) many different solutions to one and thesame analogy that can be explained in terms of the same Gestalt law. Thismade analogy stimulus construction and variation a very complex problem,indeed resulting in an uncontrolled distribution of transformation (solutionstrategy) possibilities over the set of analogies in the final experiment. Thus,our main result, that participants tend to group globally and apply rotationwhen possible, is to be taken with a grain of salt. That rotation was thepreferred solution strategy may simply be due to the high number of possi-ble rotation solutions compared to solutions that may have been reached byapplication of other transformations. In short, there may have been strongpriming effects towards rotation. This is one point one could further improvein future experiments. It requires elaborate analysis of possible object group-

134

ings as well as of kind and amount of transformations that can be applied to agiven scene in order to control for available solution strategies over analogies.

A related point is that it is not clear what effect a variation in color,shape, size, etc. really has on scene perception. Since we did not vary onlycolor, only shape, or only number of objects in any given analogy, we couldnot systematically investigate the contribution of each of these features. Thiswould require a large-scale investigation of each feature in isolation.

In addition, there may have been effects of solution position in the Choicecondition. That is, in some cases the first solution (read from left to right)was preferred, even though it was not constructed (or constructed less fre-quently) in the Make condition. To factor this out, solution position shouldbe randomized in future experiments. Also, a dummy solution should beincluded in the solution set, to control for participants who are not properlyengaging in the task.

Until the end, the problem persisted of how to collect useful commentsfrom participants about their solution strategies. This is obviously an im-portant part of the experiment, since often the only way to determine whichtransformations participants applied to reach a given solution is by looking attheir comments. We tried optional commenting with and without keywords,choosing one of six pre-formulated comments, and commenting by checkingone or more keywords (see Section 3.3.2 for a more precise description of thelatter two methods). In the final experiment, we decided to use free optionalcommenting with keywords as support. However, this was not optimal, asmany participants chose not to comment. This is a difficult problem, whichseems to be based partly on the difficulty of accessing one’s own perceptualprocesses.

A further big project is to develop a complexity measure for geometricanalogies. For the project itself it would have been interesting to investi-gate whether correlations between analogy complexity and either number ofcreated solutions or number of wrong/partial solutions could be found. Anattempt at developing such a measure based on superficial analogy featureswas made in the course of the project. The features initially taken into ac-count were a) number of objects in the scene, b) number of different colorsand shapes involved, and c) number of permutations of shapes and colorsin the objects. These features were deemed insufficient to develop an ade-quate complexity measure — it is necessary to take into account e.g. relativedistance of objects to one another, the number of possible object groupings,whether the number of objects in A and C is the same (which facilitatesobject mapping), how many different transformations may lead from A to B,to name just a few. This is certainly a very interesting task and should befurther investigated.

135

Bibliography

W. Bechtel and G. Graham, editors. A Companion to Cognitive Science.Wiley-Blackwell, 1999.

P.W. Cheng and K.J. Holyoak. Pragmatic reasoning schemas. CognitivePsychology, 17(4):391–416, 1985.

M. Dastani and R. Scha. Language for gestalts of line patterns. Journal ofMathematical Psychology, 47:429–449, 2003.

T. G. Evans. A heuristic program to solve geometric analogy problems.Technical Report Technical Report: AIM-46, Massachusetts Institute ofTechnology, 1962.

Gentner. Structure-mapping: A theoretical framework for analogy. CognitiveScience, 7(2):155–170, 1983.

D. Gentner, M.J. Rattermann, and K.D. Forbus. The roles of similarity intransfer: Separatincl-retrievability from inferential soundness. COGNI-TIVE PSYCHOLOGY, 25(52):524–575, 1993.

Holyoak K.J. & Kokinov B.N. Gentner, D. The Analogical Mind: Perspectivesfrom Cognitive Science. MIT Press, 2001.

M.L. Gick and K.J. Holyoak. Schema induction and analogical transfer.Elsevier, 1983.

W. Gordon. Synectics: The development of creative capacity. Harper andRow, New York, 1961.

D. Hofstadter and C.D. Dennett. The Mind’s I: Fantasies and Reflectionson Self and Soul 0A1

4ed. Basic Books, 1981.

K. Holyoak and P. Thagard. Analogical mapping by constraint satisfaction.Cognitive Science, 13:295–355, 1989.

136

Indurkhya. Modes of analogy. En Analogical and Inductive Inference, pages217–230, 1989.

W. Kahler. Gestalt Psychology. Liveright, New York, 1929.

M.T. Keane. On adaptation in analogy: Tests of pragmatic importanceand adaptability in analogical problem solving. The Quarterly Journal ofExperimental Psychology Section A, 49(4):1062–1085, 1996.

K. Koffka. Principles of gestalt psychology. Psychological Bulletin, 19:531–585, 1935a.

K. Koffka. Principles of Gestalt Psychology. Harcourt, New York, 1935b.

B. Kokinov and French. Computational models of analogy-making. EnL. Nadel, ed. Encyclopedia of Cognitive Science. London, pages 113–118,2003.

A.B. Markman. Constraints on analogical inference. Cognitive Science, 21(4):373–418, 1997.

W. Metzeger. Certain implications in the concept of gestalt. AmericanJournal of Psychology, 40:162–166, 1928.

E. Mullally and O’Donoghue. Geometric proportional analogies in topo-graphic maps: Theory and application. En Cambridge, UK, pages 95–108,2005.

L. Novick. Analogical transfer, problem similarity, amd expertise. Journal ofexperimental psychology. Learning, memory, and cognition, 14(3):510–520,1988.

U. Schmid, J. Wirth, and K. Polkehn. A closer look at structural similarityin analogical transfer. Cognitive Science Quarterly, 3(1):57–90, 2003.

D. Schon. Displacement of concepts. Humanities Press, New York, 1963.

A. Schwering and et al. Using gestalt principles to compute analogies ofgeometric figures. In CogSci2007 Austin: TX, 2007.

B.A. Spellman and K.J. Holyoak. Pragmatics in analogical mapping. Cogni-tive Psychology, 31(3):307–346, 1996.

M. Wertheimer. Gestalt Theory. The Gestalt Journal Press, New York,March 1924. Reprint (1997).

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M. Wertheimer. Experimentelle Studien uber das Sehen von Bewegung.Zeitschrift fur Psychologie, 61:161–265, 1929.

M. Wertheimer. Productive Thinking. Harper and Row, New York, 1954.

138

Appendix A

First pretest

A.1 List of analogies

Red numbers 1 - 3 indicate the numbers of the answers in the multiple choicetest. Those circled in red are considered the “correct” solutions in the caseof the unambiguous analogies.

2 31

A1-1

1 2 3

A1- 2

1 2 3

A1- 3

2 31

A1- 4

139

1 2 3

A2 - 1

1 2 3

A2 - 2

2 31

A2 - 3

1 2 3

A2 - 4

2 31

U1 - 1

1 2 3

U1 - 2

140

1 2 3

U1 - 3

21

U1 - 4

3

1

U2 - 1

2 3

1 2

U2 - 2

3

21

U2 - 3

3

141

1

U2 - 4

2 3

A.2 Analogy combinations

Combo 1: U2c + A2c + A1m + U1mCombo 2: U1c + A2c + A1m + U2mCombo 3: U2c + A1c + A2m + U1mCombo 4: U1c + A1c + A2m + U2m

A1 includes A1-1 to A-1-4, analogous for A2, U1, and U2. ‘c’ indicates“choose” condition, ‘m’ indicates “make” condition.

A.3 Participant information

AgeFifty-four (54) participants indicated their age. The participants ranged from19 to 48 years of age. Mean age was 27.17 and median age was 26 (13.0%).

Age group

below 20 120-24 1825-29 1930-34 1135-39 340-44 –

45 and above 2

142

Gender

Of the 54 participants who indicated theirgender, there were 30 males (53.6%) and24 females (42.9%).

female

male

NA's

Gender

Handedness

There were 51 right-handed participants(91.1%) and two (2) ambidextrous partic-ipants (3.6%). No participant indicatedleft-handedness and three participants didnot select their handedness.

ambidextrous right

NA's

Handedness

IQ test historyWith respect to familiarity of geometric proportional analogies,participants were asked about the last time they took an IQ test.Most of the participants (26 participants or 46.4%) had nevertaken an IQ test before, while 5.4% of the participants (or 3 partic-ipants) had taken one recently within the past year. The rest hadtaken an IQ test either 1-5 years ago (12 participants or 21.4%),5-10 years ago (5 participants or 8.9%) or more than 10 years agoprior to 1997 (7 participants or 12.5%). Three participants didnot select an answer.

Time of last IQ test

never 26less than 1 year ago 3

1-5 years ago 125-10 years ago 5

more than 10 years ago 7never < 1 yr 1−5 yrs 5−10 yrs > 10 yrs

0.0

0.1

0.2

0.3

0.4

143

Area of studyIn order to see if a mathematical/logic background would havean effect in terms of familiarity in solving analogy problems, aquestion was included on area of study. Only 41 participantsgave a response. The subject areas were diverse and furthermoreinput was not uniform. Four participants (7.1%) were studyingor lecturing in linguistics and there were three students each incomputer science and statistics (5.4% respectively). Other studiesincluded biology, cognitive science as well as disciplines in thehumanities and natural sciences.

Area of study

Linguistics 4Computer science 3

Statistics 3Biology 2

Cognitive science 2Other 27

NationalityAside from half (50%) of the participants being German (28), thenationalities of the participants were varied. Five participants(8.9%) came from Mexico, followed by four (7.1%) from the USA,as well as three (5.4%) from Macedonia and two (3.6%) from Ser-bia. The rest of the nationalities were split among Europe (6participants), Asia (3 participants) and one from South Africa.Four participants did not indicate their nationality.

Nationality

German 28Mexican 5American 4

Macedonian 3Serbian 2Other 10

DE MX US MK RS Other

0.0

0.1

0.2

0.3

0.4

0.5

144

OccupationLooking at the responses of the 52 participants that indicated theiroccupation, more than half were students (60.7%), including highschool, university and graduate studies. Five other participantsadditionally worked in the education sector (either as an employeeor teaching) and two participants (3.6%) were apprentices.

Occupation

Student 34Education (other than student) 5

IT 2Banking 2

Apprentice 2Employee 2

Other 5

A.4 Test environment

Two language versions (German and English) were set up to pool participantsfrom within Germany as well as abroad. All 16 examples (A1-1 through A1-4, A2-1 through A2-4, U1-1 through U1-4 and U2-1 through U2-4) used wereidentical in both versions. Participants were randomly assigned to one offour analogy combinations (Combo1, Combo2, Combo3 and Combo4).

Language version

Of the 56 participants, 21 (37.5%) optedfor the English version and 35 (62.5%) forthe German version.

English

German

Language version

145

Randomized combination assignment

Despite some effort to achieve a quasi-randomized assignment of the four differ-ent analogy combinations, automatic as-signment was unbalanced.

1

2

3

4

Combination

A.5 Results by analogy

A1-1-D A1-2-D A1-3-D A1-4-D A2-1-D A2-2-D A2-3-D A2-4-D

Answer 2

Answer 3

Answer 4

Answer 5

Answer 6

Answer 7

Answer 8

Answer 9

Answer 10

Answer 11

Answer 12

8 10 2 7 2 11 9 11

15 5 6 7 8 1 0 4

1 1 1 5 1 2 1

2 1 1 1 2 1

1 1 1 1 1 2

1 8 1 1

2 1

1 1

1

1

1

0

25

50

75

100

1 2 3 4 5 6 7

A1-1

Make Choice

0

25

50

75

100

Answer 1 Answer 3 Answer 5 Answer 7

A1-2

Make ChoiceMake Choice

Make Choice

0

25

50

75

100

A1-4

0

25

50

75

100

A2-2

2 31

A1-1

6

7

Good continuation

Proximity/Similarity

Proximity Good continuation Similarity

Proximity

146

9

5


0

25

50

75

100

1 2 3 4 5 6 7 8 9

0

25

50

75

100

1 2 3 4 5 6 7 8 9

A1-4

Make Choice

0

25

50

75

100

1 2 3 4 5 6

A2-1

Make Choice

0

25

50

75

100

1 2 3

0

25

50

75

100

1 2 3

A2-2


0

25

50

75

100

1 2 3 4

A2-3


2 31

A1- 4

4

6Proximity

Symmetry

Similarity

Proximity/Symmetry

5


0

25

50

75

100

1 2 3 4 5 6 7 8 9

0

25

50

75

100

1 2 3 4 5 6 7 8 9

A1-4

Make Choice

0

25

50

75

100

1 2 3 4 5 6

A2-1

Make Choice

0

25

50

75

100

1 2 3

0

25

50

75

100

1 2 3

A2-2


0

25

50

75

100

1 2 3 4

A2-3


1 2 3

A2 - 1

6

Proximity

Proximity

147

ProximityGood continuation Similarity

6 Similarity

0

25

50

75

100


A2-4

Make Choice

1 2 3

A2 - 4

Proximity

ProximityGood continuation

5


Answer 2

Answer 3

Answer 4

Answer 5

Answer 6

Answer 7

Answer 8

Answer 9

Answer 10

Answer 11

Answer 12

8 10 2 7 2 11 9 11

15 5 6 7 8 1 0 4

1 1 1 5 1 2 1

2 1 1 1 2 1

1 1 1 1 1 2

1 8 1 1

2 1

1 1

1

1

1

0

25

50

75

100

1 2 3 4 5 6 7

A1-1

Make Choice

0

25

50

75

100

1 2 3 4 5 6 7

A1-2


Make Choice

0

25

50

75

100

A1-4

0

25

50

75

100

A2-2

1 2 3

A1- 2

Proximity

Good continuation


Answer 5

Answer 6

Answer 7

Answer 8

Answer 9

Answer 10

Answer 11

Answer 12

2 1 1 1 2 1

1 1 1 1 1 2

1 1 1

2 1

1 1

1

1

1

0

25

50

75

100

1 2 3 4 5 6 7

A1-1

Make Choice

0

25

50

75

100

1 2 3 4 5 6

A1-2


Make Choice

0

25

50

75

100

A1-4

0

25

50

75

100

A2-2

148

Proximity

0

25

50

75

100

1 2 3 4 5 6 7 8 9

0

25

50

75

100

1 2 3 4 5 6 7 8 9

A1-4

Make Choice

0

25

50

75

100

1 2 3 4 5 6

A2-1

Make Choice

0

25

50

75

100

1 2 3

0

25

50

75

100

1 2 3

A2-2


0

25

50

75

100

1 2 3 4

A2-3


1 2 3

Good continuation

Symmetry

Proximity

Good continuationGood continuation

2 31

A2 - 3

0

25

50

75

100

1 2 3 4 5 6 7 8 9

0

25

50

75

100

1 2 3 4 5 6 7 8 9

A1-4

Make Choice

0

25

50

75

100

1 2 3 4 5 6

A2-1

Make Choice

0

25

50

75

100

1 2 3

0

25

50

75

100

1 2 3

A2-2


0

25

50

75

100

1 2 3 4

A2-3


4 Proximity

Good continuation

Good continuation

149

Make Choice

0

25

50

75

100

1 2 3 4 5 6 7 8 9 10 11 12

A1-3


Proximity/Similarity

6

Proximity

Similarity

1 2 3

A1- 3

8

Similarity

Similarity

U1-1-C U1-2-C U1-3-C U1-4-C U2-1-C U2-2-C U2-3-C U2-4-C

Answer 1

Answer 2

Answer 3

0 0 0 1 4 1 0 0

1 21 0 0 0 34 31 35

20 0 21 20 31 0 4 0

0

25

50

75

100

1 2 3 4

U1-1


0

25

50

75

100

1 2 3 4 5

U1-3

Make Choice

0

25

50

75

100

U1-4

Make Choice

2 31

U1 - 1

4 Symmetry

Symmetry

150

0

25

50

75

100

1 2 3

U1-2


0

25

50

75

100

1 2 3 4

0

25

50

75

100

1 2 3 4

U1-4

Make Choice Make Choice

1 2 3

U1 - 2 Similarity

U1-1-C U1-2-C U1-3-C U1-4-C U2-1-C U2-2-C U2-3-C U2-4-C

Answer 1

Answer 2

Answer 3

0 0 0 1 4 1 0 0

1 21 0 0 0 34 31 35

20 0 21 20 31 0 4 0

0

25

50

75

100

1 2 3 4

U1-1


0

25

50

75

100

1 2 3 4 5

U1-3

Make Choice

0

25

50

75

100

U1-4

Make Choice

1 2 3

U1 - 3Similarity/Proximity

151

0

25

50

75

100

1 2 3

U1-2


0

25

50

75

100

1 2 3 4

0

25

50

75

100

1 2 3 4

U1-4

Make Choice Make Choice

21

U1 - 4

3

Similarity/Good continuation


Answer 1

Answer 2

Answer 3

0 0 0 4.7619047619 11.428571429 2.8571428571 0 0

4.7619047619 100 0 0 0 97.142857143 88.571428571 100

95.238095238 0 100 95.238095238 88.571428571 0 11.428571429 0

0

25

50

75

100

1 2 3 4

U2-1


0

25

50

75

100

1 2 3

U2-3


1

U2 - 1

2 3

Proximity

4 Symmetry

152

Good continuation0

25

50

75

100

1 2 3

U2-2


0

25

50

75

100

1 2 3 4

U2-4


1 2

U2 - 2

3

Proximity


Answer 1

Answer 2

Answer 3

0 0 0 4.7619047619 11.428571429 2.8571428571 0 0

4.7619047619 100 0 0 0 97.142857143 88.571428571 100

95.238095238 0 100 95.238095238 88.571428571 0 11.428571429 0

0

25

50

75

100

1 2 3 4

U2-1


0

25

50

75

100

1 2 3

U2-3


21

U2 - 3

3

Proximity/Good continuation

0

25

50

75

100

1 2 3

U2-2


0

25

50

75

100

1 2 3 4

U2-4


1

U2 - 4

2 3

A.6 Feedback

The following two sections comprise a summarized evaluation of the feedbackthat the subjects gave during the pretest.

153

Pros

• Instructions were very understandable and precise

• The vast majority knew exactly what to do at all times

• Embedded exercise examples & trash can received a positive feedback

• Apparently, the usability of the resources toolbar was quite easy andintuitive

• All in all, no severe technical problems occurred (except for the screenresolution issue, see Cons below)

• Commenting their solutions helped some subjects to clarify their takenactions (during the construction phase) or to realize mistakes after-wards

• A lot of people considered the first part to be easier than the secondpart (which was our intention when we decided to let the subjects tothe ”multiple-choice task” first)

• With respect to the ambiguous analogies, some subjects indeed realizedthat there were several plausible solutions in contrast to the unambigu-ous analogies(Question: Is this really a positive feedback or just a confirmation ofour intentions?)

• Some subjects considered the analogies to have different levels of diffi-culty(Question: Once again, rather a positive feedback for us or simply aconfirmation of our intention to have examples that differ in their com-plexity?)

Cons

• In the end, the experiment was not entirely compatible with resolutionsbelow 1024x768 / 1280x720 (usage of arrow keys / space key necessaryto display certain buttons)

• The analogy images seem to load very slowly when using slow internetconnections like a 56k modem or ISDN (see general improvements belowfor a possible solution)

154

• Some subjects wanted for some feedback showing them how they per-formed throughout the experiment (similar to typical IQ tests => Note:providing exact percentages is impossible due to our ambiguous exam-ples)

• Subjects needed significantly less than 30 minutes to complete the ex-periment (the corresponding remark on the introduction page was ob-viously overstated)

• Certain analogy examples and its variations were used too often incomparison to other examples => even a small number of subjects re-peatedly pointed out that some examples were quite familiar to themwhile they proceeded in the experiment (see general improvements be-low for a possible solution)

• A very small number of participants felt bored when they had to readagain and again certain instructions like ”Please comment on your so-lution for D” etc.

• Some participants had severe problems to comment on what they justdid in the construction step before => that is, they did not knowexactly what to write as they considered their solutions to be somewhatlogical or obvious (this might be one possible reason why a rather highnumber of subjects did not comment their solutions at all)

• It was suggested to display the analogies in front of a visible grid suchthat the exact object positions become more obvious than it is the casenow without a grid

• Some subjects skipped back some pages during the experiment e.g.to correct their constructions (see general improvements below for apossible solution)

A.7 Suggestions for improvement

The following section comprises a listing of general improvements that onehas to think of for the ”real” experiment.

• Comment box: message alert if no text was entered

• Disable back / forward buttons to prevent the subjects from skippingback and forth the pages: 1) for all browsers, 2) corresponding mousebuttons

155

• Return key issue (information page): suppress return key, otherwiseform is submitted immediately

• Implement ”nice” looking buttons instead of ”ugly” local links

• Consistent position of button: always in same location at bottom leftof screen

• Load all images before showing whole page (to circumvent gradual ap-pearance of images when using modem / ISDN connection)

• Implement example-specific toolbars

• Make sure: equal size for all geometric elements

• Participant results to be stored in separate files

• Information form: values to be same for both language

• If possible: prevent shifting of colons (different browsers = unfortu-nately different positions of colons)

• Disallow multiple trials with different browsers

• Enable formal (real) randomization (to allow for a more sophisticatedstatistical analysis compared to the current one)

• Session management for unstable internet connections (continue wherelast left off)

• Hide address bar in IE popup window

• Snapshot images as explanation (especially on exercise pages)

• Flash animated instructions

• Set cookie after entering welcome page

• Enable scrolling in popup window (could resolve the resolution issue)

• Make sure: Everything is visible even when using low screen resolutions(could be problematic as we would lose space on each page)

Remark: Some suggestions for improvement obviously become null andvoid provided that the real experiment is conducted offline (or online, but inan controlled environment)

156

Appendix B

Second pretest

B.1 List of analogies

S1Original

: ::: ?

S1A1

: ::: ?S1A2

: ::: ?S1A3

: ::: ?

S1C1

: ::: ?S1C2

: ::: ?S1C3

: ::: ?

157

S2Original

?: :: :

S2A1

?: :: :

S2A2

?: :: :

S2A3

?: :: :

S2C1

?: :: :

S2C2?: :: :

S2C3

?: :: :

S3Original

?: :: :

S3A1?: :: :

S3A2?: :: :

S3A3?: :: :

S3C1?: :: :

S3C2?: :: :

S3C3?: :: :

158

S4Original

?: :: :

S4A1?: :: :

S4A2?: :: :

S4A3?: :: :

S4C1?: :: :

S4C2?: :: :

S4C3?: :: :

S5Original

?: :: :

S5A1

?: :: :

S5A2?: :: :

S5A3

?: :: :

S5C1

?: :: :

S5C2

?: :: :

S5C3

?: :: :

S6Original

?: :: :

S6A1?: :: :

S6A2?: :: :

S6A3?: :: :

S6C1?: :: :

S6C2?: :: :

S6C3?: :: :

159

B.2 Stimuli combinations

There were a total of six original examples with six variations which can begrouped into two sets of three AB fixed and three C fixed, respectively. Inaddition to all original examples, the variation sets (A & C) were dividedequally to create two different combinations of 24 examples each.

Mix 1 Mix 2

S1 C S1 AS2 A S2 CS3 C S3 AS4 A S4 CS5 C S5 AS6 A S6 C

B.3 Example randomization

Two combinations each of the two mixes were created to give four combina-tions each in the Checkboxes and Radio buttons conditions. All exampleswithin the combinations were completely randomized.

Combo 1 Combo 2 Combo 3 Combo 4S1C3 S2A1 S2Original S4C1S3C3 S2A2 S3A3 S1A2S4A2 S1Original S4Original S5A3S2A3 S1C3 S4C3 S2OriginalS5Original S1C1 S5Original S4C2S4A1 S2Original S3A2 S5OriginalS6A2 S4A1 S2C2 S1OriginalS4Original S3C3 S3Original S4OriginalS1Original S4A3 S4C2 S6C3S6A3 S5Original S6C1 S3A2S3C1 S5C2 S6Original S3A3S1C1 S4A2 S3A1 S6C2S5C1 S3Original S1A3 S2C3S2Original S5C3 S4C1 S2C2S5C2 S1C2 S1A2 S5A1S4A3 S6A1 S6C3 S5A2S2A1 S2A3 S2C3 S2C1S6Original S5C1 S5A3 S6OriginalS5C3 S3C2 S5A2 S3Original

160

S3Original S4Original S5A1 S1A1S6A1 S3C1 S2C1 S6C1S2A2 S6A2 S6C2 S1A3S1C2 S6Original S1A1 S4C3S3C2 S6A3 S1Original S3A1

Thus, each participant was given a set of 24 examples, in either the Check-boxes or the Radio buttons condition exclusively, from one of the mixes withall examples in a randomized order.

B.4 Randomization of Check/Radio options

The order in which the Check and Radio options were presented was ran-domized for each combination. The final options (j in Check, f in Radio) inboth of the conditions was the commenting option. Option e in the Radiobuttons comments was always an incorrect solution.

Checkboxes Radio buttons

S1a. Switch colors a. The colors of the objects are switched.b. Move object(s) down b. The figure in C is rotated by 180◦.c. Rotate by 180◦ c. The figure in C is flipped twice.d. Flip along diagonal axis d. The black object moves upwards.e. Flip along horizontal axis e. The black object moves downwards.f. Switch positions f. None of the aboveg. Move object(s) uph. Rotate by 90◦

i. Remove objectj. Other

S2a. Black object(s) remain a. The central objects remain.b. Move object(s) downwards b. The black objects remain.c. Grey object(s) remain c. The white objects are removed.d. Remove white object(s) d. The outer objects are removed.e. Remove black object(s) e. The white and black objects switch colors.f. Remove grey object(s) f. None of the aboveg. Central object(s) remainh. Remove outer object(s)

161

i. Remove central object(s)j. Other

S3a. Rotate white object(s) a. The figure in C is flipped horizontally.b. Rotate grey object(s) b. The figure in C turns 180◦.c. Flip along vertical axis c. The triangle in C is flipped.d. Rotate triangle(s) d. The figure in C is flipped vertically.e. Flip along horizontal axis e. The square in C is rotated by 90◦.f. Rotate left object(s) f. None of the aboveg. Rotate right object(s)h. Rotate object(s) by 180◦

i. Rotate object(s) by 90◦

j. Other

S4a. Move object(s) upwards a. The figure in C is flipped.b. Move object(s) downwards b. The figure in C turns 180◦.c. Move object(s) around c. The objects in C change positions.d. Flip along vertical axis d. The figure in C is reflected vertically.e. Flip along horizontal axis e. Grey objects turn black.f. Rotate object(s) by 90◦ f. None of the aboveg. Change positionsh. Switch colorsi. Rotate object(s) by 180◦

j. Other

S5a. Change position a. The circle moves downwards.b. Move square(s) upwards b. The circle moves upwards towards the

square.c. Rotate object(s) by 180◦ c. The square moves downwards towards

the circle.d. Change shape d. The square moves upwards.e. Add object(s) e. Gravity pulls the circle.f. Move circle(s)/rectangle(s) upwards f. None of the aboveg. Move circle(s)/rectangle(s) downwardsh. Remove object(s)i. Move square(s) downwardsj. Other

162

S6a. Move square(s) diagonally downwards a. The figure in C turns 90◦.b. Change positions b. The white square moves downwards.c. Move circle(s) diagonally upwards c. The white square changes position

with the lower circle.d. Flip along diagonal axis d. The objects switch color.e. Rotate object(s) by 90◦ e. The square moves upwards.f. Switch colors f. None of the aboveg. Change shapeh. Move circle(s) diagonally downwardsi. Move square(s) diagonally upwardsj. Other

B.5 Participant information

Age

Age group

below 20 420-25 1726-30 931-40 11

above 40 3

<20 20−25 26−30 31−40 >40

Age

0.0

0.1

0.2

0.3

Gender

There were 33 males and 10 females. female

male

Gender

163

Handedness

There were 40 right-handed participants,one left-handed participant and one am-bidextrous participant.

ambidextrous

left

right

Handedness

IQ test history

Time of last IQ test

never 19less than 1 year ago 3

1-5 years ago 125-10 years ago 5

more than 10 years ago 4

never <1 1−5 5−10 >10

IQ

0.0

0.1

0.2

0.3

0.4

Expertise, nationality & occupation

Area of study

Humanities 7Natural sciences 5Computer science 4Engineering 3Social sciences 3Languages 2Maths 1Other 9

Nationality

German 21Macedonia 4

USA 4Indonesia 3

Switzerland 2Other 7

Occupation

Student 18Employee 14

Self-employed 6Unemployed 3

Other 3

164

Appendix C

Mini tests: HIT andTechnologietag

165

Figure C.1: Results for ”A2Original” and variations

166

Figure C.2: Results for ”A3Original” and variations

167

Appendix D

Final test

D.1 Demographics

<10

11−

15

16−

20

21−

25

26−

30

31−

40

>40

Age

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

less_than_1 1−5 5−10 more_than_10 never

IQ

0.0

0.1

0.2

0.3

0.4

Expertise

Engineering 40Natural sciences 36Computer science 23Humanities 13Other 77NA 49

Nationality

Germany 148Indonesia 57USA 8Italy 4Switzerland 3Canada 2Other 16

Occupation

Student 110Employee 77Self-employed 19Civil servant 8Pupil 7Unemployed 6Other 11

168

D.2 Stimuli

S1OR

S1A1

S1A2

S1C1

169

S1C2

S2OR

S2A2

S2A3

170

S2C1

S2C2

S3OR

S3A1

S3A3

S3C2

171

S3C3

S4OR

S4A1

S4A2

S4C2

S4C3

172

S5OR

S5A1

S5A3

173

S5C3

S5C4

S6OR

S6A1

174

S6A3

S6C1

S6C2

175

D.3 Solution classification

AnalogyGestalt Transformation

S1OR GC object movement

S1A1 GC colour saliency / object movement

S1A2 similarity switch position

S1C1 similarity switch colour

S1C2 GC object movement

S2OR DUMMY DUMMY

S2A2 proximity deletion of non-central objects

S2A3 similarity deletion of non-black objects

S2C1 similarity deletion of white objects

S2C2 similarity deletion of white objects

S3OR similarity rotation partially

S3A1 proximity rotation

S3A3 similarity rotation

S3C2 proximity rotation


S4OR symmetry reflection






S5A1 GC object movement




S6OR proximity rotation

S6A1 similarity switch colour








S1C1 similarity reflection


S2OR similarity deletion of non-black object

S2A2 similarity deletion of grey objects

S2A3 proximity deletion of non-central objects / colour switch

S2C1 proximity deletion of non-central objects

S2C2 similarity deletion of non-black objects


S3A1 similarity rotation partially


S3C2 similarity rotation partially

S3C3 symmetry reflection

Solution 1

Solution 2

176


S4A1 proximity rotation, switch colour

S4A2 symmetry reflection


S4C3 proximity switch position






S6OR similarity switch colour




S6C2 proximity switch position


S1OR similarity switch colour

S1A1 proximity rotation, switch colour




S2OR similarity deletion of non-central objects

S2A2 proximity deletion of grey objects

S2A3 GC deletion of non-top group / object movement

S2C1 similarity deletion of non-black objects

S2C2 proximity deletion of non-central objects

S3OR symmetry reflection


S3A3 DUMMY DUMMY


S3C3 similarity rotation partially


S4A1 symmetry reflection / switch colour














Solution 3

177

D.4 Solution analysis

178

Choose condition

solution # of participants per

solution cluster ID (same ID was used in the make condition)

D1

58/161

(36%) n33

D2

59/161

(37%) n35

D3

44/161

(27%) n34

Make condition


solution

# of comments per

solution

transformation(s)

derived from comments

# of

participants per

transformation

D1

12/77

(15%)

3/32 (9%)

1 discarded switch colours 2

D2

24/77

(31%) 13/32 (41%) switch position 13

D3

12/77

(15%)

5/32 (16%)

1 discarded rotation + switch colours 4

novel 30/77

(39%)

11/32 (34%)

3 discarded view next page(s) for details

total # of comments: 32

Analogy

S1A1

total # of discarded comments: 5/32 (15.6%)

Novel solutions (make condition)


solution cluster ID

transformation(s)


# of participants

per

transformation

1/30 4_2

1/30 4_3

1/30 4_5

2/30 4_6switch colours + switch

position 1

2/30 4_7

1/30 4_8

2/30 4_9 switch colours + rotation 1

1/30 4_12

Analogy

S1A1

1/30 4_13

179



solution cluster ID

transformation(s)


# of participants

per

transformation

2/30 4_15 switch colours 1

1/30 4_17

1/30 4_18

1/30 4_20

switch colours 1

2/30

4_21reflection 1

1/30 4_23

reflection + switch

colours 1


Analogy

S1A1

2/30 4_25rotation + switch colours

+ switch position 1



solution cluster ID

transformation(s)

derived from comments # of participants

4/30 4_26

*

1/30 4_27

Analogy

S1A1

blank 2/30 0_0

* Remark: clusters 4_26 and 4_27 basically entail the same solutions, but due to the shifted black circle in 4_27 two distinct clusters

were generated in the end!

180

Choose condition



D1

18/161

(11%) n39

D2

52/161

(33%) n36

D3

91/161

(56%) n37

Make condition


solution

# of comments per

solution

transformation(s) derived

from comments

# of participants

per

transformation

D1

4/77

(5%) 1/34 (3%) switch position 1

object movement 1

switch position 2

reflection 2

rotation 2D2

29/77

(38%)

12/34 (35%)

4 discarded

switch colours 1

rotation 3

switch position + rotation 1

reflection 2D3

19/77

(25%)

8/34 (24%)

1 discarded

switch position 1

novel 25/77

(32%) 13/34 (38%) view next page(s) for details


Analogy

S1A2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/25 1_0

1/25 4_12 rotation 1

15/25 n38 switch colours 8

5/25 n40 reflection 4

1/25 4_13

Analogy

S1A2

blank 2/25 0_0

181

Choose condition



D1

43/162

(27%) n46

D2

17/162

(10%) n42

D3

102/162

(63%) n44

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

switch position 1

D1

17/76

(22%) 4/26 (16%)

switch colours 3

D2

3/76

(4%) 2/26 (8%) reflection 2

switch colours +

position 1

switch position 2

reflection 2D3

27/76

(36%) 10/26 (38%)

rotation 5

novel 29/76



Analogy

S1C1




solution cluster ID

transformation(s)


# of participants

per

transformation

1/29 4_10 reflection 1

1/29 4_11


1/29 4_22

switch colours +

position 1

switch colours 1

12/29 n41

switch position 1

11/29 n43 switch colours 5

Analogy

S1C1

2/29 n45

182

Choose condition



D1

36/163

(22%) n48

D2

84/163

(52%) n51

D3

43/163

(26%) n50

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

switch colours 4

object movement +

switch colours 1

switch position 1 D1

24/75

(32%)

10/30 (33%)

2 discarded

reflection 2

reflection 3

D2

23/75

(31%) 5/30 (17%)

rotation 2

D3

15/75

(20%) 8/30 (27%) switch colours 8

novel 13/75



Analogy

S1C2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/13 4_0 rotation 1

1/13 4_1


1/13 4_13

1/13 4_14 rotation + switch colours 1


Analogy

S1C2


183

Choose condition



D1

2/161

(1%) 1

D2

30/161

(19%) n64

D3

129/161

(80%) 1_0

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

0/77

(0%) 0/28 (0%)

deletion of white objects 4

D2

16/77

(21%)

8/28 (28%)

1 discarded deletion of non-black

objects 3

D3

58/77

(75%)

19/28 (68%)

1 discarded

deletion of non-black

objects 18

novel 3/77

(4%)

1/28 (4%)



Analogy

S2OR




solution cluster ID

transformation(s)


# of participants

per

transformation

1/3 2_1

1/3 2_2

Analogy

S2OR

1/3 3_0

184

Choose condition



D1

123/156

(79%) 1_1

D2

18/156

(12%) 3_0

D3

15/156

(9%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1 68/82

(83%) 29/34 (85%)

deletion of non-central

objects 29

D2

5/82

(6%) 2/34 (6%) deletion of grey objects 2

D3

0/82

(0%) 0/34 (0%)

novel 9/82

(11%)

3/34 (9%)



Analogy

S2A2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/9 1_0


objects 1

1/9 1_2

1/9 2_0

1/9 5_0

1/9 5_1

1/9 5_2 add objects 1

Analogy

S2A2

blank 3/9 0_0

185

Choose condition



D1

71/157

(45%) n59

D2

31/157

(20%) n60

D3

55/157

(35%) n61

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation


objects + object

movement

1

D1

28/81

(35%)

13/34 (38%)

2 discarded deletion of non-black

objects 10

D2

14/81

(17%) 5/34 (15%)


objects + switch colours 5

D3

25/81

(31%)

10/34 (29%)

1 discarded

deletion of non-top

group + object

movement

9

novel 14/81

(17%)

6/34 (18%)



Analogy

S2A3


Choose condition



D1

71/157

(45%) 4_1

D2 31/157

(20%) 1_0

D3

55/157

(35%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

5/83

(6%) 2/47 (4%) deletion of white objects 2

D2 41/83

(50%) 24/47 (51%)


objects 24

D3

6/83

(7%)

4/47 (9%)

1 discarded


objects 3

novel 31/83

(37%)

17/47 (36%)



Analogy

S2C1


186



solution cluster ID

transformation(s)


# of participants

per

transformation

1/31 2_0

2/31 2_1

16/31 3_1deletion of non-central

objects 9

1/31 3_3


objects 1

2/31 3_5


objects + object

movement

1


objects 1

1/31 4_0

Analogy

S2C1

1/31 7_0

Choose condition



D1

29/163

(18%) 2_0

D2 23/163

(14%) 1_0

D3

111/163

(68%) n63

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

4/75

(6%) 2/34 (6%) deletion of white objects 2

D2 6/75

(8%) 4/34 (12%)


objects 4

D3

55/75

(73%)

25/34 (73%)

3 discarded


objects 22

novel 10/75

(13%)

3/34 (9%)



Analogy

S2C2


187



solution cluster ID

transformation(s)


# of participants

per

transformation

2/10 1_2

1/10 2_3 deletion of grey objects 1

3/10 2_4

1/10 2_5deletion of white objects

+ switch colours 1

2/10 2_6

Analogy

S2C2

blank 1/10 0_0

Choose condition



D1

12/153

(8%) 2_5

D2

47/153

(31%) n5

D3

94/153

(61%) 2_6

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

reflection 1

D1

5/85

(6%) 4/37 (11%)

rotation 3

reflection 5

D2

51/85

(60%) 22/37 (59%)

rotation 17

reflection 7

D3

22/85

(26%)

10/37 (27%)

2 discarded rotation 1

novel 7/85



Analogy

S3OR




solution cluster ID

transformation(s)


# of participants

per

transformation

1/7 1_0


1/7 2_2

1/7 2_3

Analogy

S3OR

blank 2/7 0_0

188

Choose condition



D1

13/159

(8%) 2_6

D2 74/159

(47%) 2_2

D3

72/159

(45%) 2_7

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

2/79

(3%) 1/30 (3%) reflection 1

reflection 4

rotation 2D2

43/79

(54%)

14/30 (47%)

1 discarded partial rotation 7

reflection 8

rotation 1D3

23/79

(29%) 13/30 (43%)

partial rotation 4

novel 11/79

(14%)

2/30 (7%)



Analogy

S3A1




solution cluster ID

transformation(s)


# of participants

per

transformation

1/11 1_0

1/11 2_0

1/11 2_1

2/11 2_4 rotation 1

1/11 2_5

3/11 n1

Analogy

S3A1

blank 2/11 0_0

Choose condition



D1

138/155

(89%) n2

D2

9/155

(6%) 2_0

D3

8/155

(5%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

reflection 12

D1

75/83

(91%)

30/33 (91%)


D2

1/83

(1%)

1/33 (3%)

1 discarded

D3

0/83

(0%) 0/33 (0%)

novel 7/83

(8%)

2/33 (6%)



Analogy

S3A3


189



solution cluster ID

transformation(s)


# of participants

per

transformation

6/7 n3

Analogy

S3A3 blank 1/7 0_0

Choose condition



D1

89/161

(56%) 3_6

D2 20/161

(12%) 3_7

D3 52/161

(32%) n4

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

reflection 7

rotation 8D1

36/77

(47%) 16/36 (46%)

reflection + rotation 1

D2 5/77

(6%)

5/36 (14%)

2 discarded partial rotation 3

D3 26/77

(34%) 12/36 (32%) reflection 12

novel 10/77



Analogy

S3C2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/10 3_0 rotation 1

3/10 3_1

1/10 3_3

1/10 3_4

1/10 3_5 rotation 1

1/10 5_0 rotation + switch colours 1

Analogy

S3C2

blank 2/10 0_0

190

Choose condition



D1

57/158

(36%) 2_2

D2

70/158

(44%) 2_5

D3

31/158

(20%) 2_3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

24/80

(30%)

14/38 (37%)


reflection 16

D2

30/80

(38%)

18/38 (47%)


D3

11/80

(14%) 2/38 (5%) reflection 2

novel 15/80



Analogy

S3C3




solution cluster ID

transformation(s)


# of participants

per

transformation

1/15 2_0


5/15 2_4 partial rotation 1


1/15 2_7

Analogy

S3C3

blank 1/15 0_0

191

Choose condition



D1

44/157

(28%) 2_1

D2 2/157

(1%) 2

D3

111/157

(71%) 2_0

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

14/81

(17%) 9/43 (21%) reflection 9

D2 0/81

(0%) 0/42 (0%)

rotation 15

reflection 11


62/81

(77%)

34/43 (79%)

1 discarded

rotation + reflection 1

novel 5/81



Analogy

S4OR




solution cluster ID

transformation(s)


# of participants

per

transformation

2/5 2_2

2/5 n12

Analogy

S4OR blank 1/5 0_0

Choose condition



D1

27/159

(17%) 2_5

D2

96/159

(60%) n6

D3

36/159

(23%) 2_1

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

7/79

(9%) 1/37 (2%) rotation 1

rotation + switch colours 9

switch colours + position 7

reflection + switch colours 2

change objects/shapes +

switch colours 4

colour saliency + object

movement 1

D2

47/79

(59%) 24/37 (65%)

switch objects 1

reflection + switch colours 2


switch colours + position 1 D3

14/79

(18%) 7/37 (19%)

switch colours 3

novel 11/79

(14%)

5/37 (14%)



Analogy

S4A1


192



solution cluster ID

transformation(s)


# of participants

per

transformation

1/11 1_0

1/11 2_0

1/11 2_3

1/11 2_4reflection + switch

colours 1

5/11 2_6reflection + retain

colours 1

1/11 3_1

Analogy

S4A1

blank 1/11 0_0

Choose condition



D1

20/161

(12%) 2_5

D2

69/161

(43%) 2_3

D3 72/161

(45%) 2_1

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

9/77

(12%) 2/36 (6%) reflection 2

D2

29/77

(38%)

13/36 (36%)

2 discarded rotation + switch colours 11

rotation 19

D3 34/77

(44%)

21/36 (58%)

1 discarded switch position 1

novel 5/77



Analogy

S4A2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/5 2_0

1/5 2_2

1/5 2_4

Analogy

S4A2

blank 2/5 0_0

193

Choose condition



D1

105/158

(66%) n7

D2

49/158

(31%) 2_5

D3 4/158

(3%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

rotation 10

D1

46/81

(57%)

21/41 (51%)

1 discarded reflection 10

D2

21/81

(26%) 15/41 (37%) reflection 15

D3 0/81

(0%) 0/41 (0%)

novel 14/81

(17%)

5/41 (12%)



Analogy

S4C2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/14 1_0

1/14 2_1

1/14 2_3

1/14 2_4

* 1/14 4_0 switch position 1

7/14 n8 switch position 3

Analogy

S4C2

blank 2/14 0_0

* One participant provided two solutions at a time. Only the left figure is a novel solution.

The right figure matches D1 and is thus crossed-out in this context.

194

Choose condition



D1

95/155

(61%) n9

D2

20/155

(13%) n10

D3

40/155

(26%) n11

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

rotation 10

D1

42/83

(50%) 16/37 (43%)

reflection 6

D2

8/83

(10%)

6/37 (17%)


reflection 10

D3

25/83

(30%)

13/37 (35%)


novel 8/83

(10%)

2/37 (5%)



Analogy

S4C3




solution cluster ID

transformation(s)


# of participants

per

transformation

1/8 1_0

1/8 2_1

1/8 2_2

1/8 2_3

1/8 2_5

1/8 2_6switch colours +

position 1

Analogy

S4C3

blank 2/8 0_0

195

Choose condition



D1

80/156

(51%) n23

D2

71/156

(46%) n22

D3

5/156

(3%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

object movement 8


45/82

(55%)

19/29 (66%)

4 discarded switch objects 1

object movement 5


33/82

(40%)

9/29 (31%)

2 discarded object movement + rotation 1

D3

0/82

(0%) 0/29 (0%)

novel 4/82

(5%)

1/29 (3%)



Analogy

S5OR




solution cluster ID

transformation(s)


# of participants

per

transformation

1/4 1_0

1/4 2_1

Analogy

S5OR

blank 2/4 0_0

196

Choose condition



D1

81/154

(53%) n13

D2

6/154

(4%) 2

D3

67/154

(43%) n14

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

object movement 2

rotation 3D1

28/84

(33%) 7/33 (21%)

reflection 2

D2

0/84

(0%) 0/33 (0%)

object movement 8

switch position 6

switch colours +

position 3 D3

50/84

(60%)

25/33 (76%)

7 discarded

switch colours 1

novel 6/84



Analogy

S5A1




solution cluster ID

transformation(s)


# of participants

per

transformation

1/6 2_0

1/6 6_2

2/6 6_5 rotation 1

1/6 6_7

Analogy

S5A1

blank 1/6 0_0

197

Choose condition



D1

64/157

(41%) n16

D2

83/157

(53%) n15

D3

10/157

(6%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

object movement 2

switch position 1

add objects 2 D1

20/81

(25%)

7/28 (25%)

1 discarded

rotation 1

object movement 5

switch position 5

add objects 3 change objects/shapes + switch

colours 1 D2

50/81

(62%)

18/28 (64%)

3 discarded

gravity (object attraction) 1

D3

0/81

(0%) 0/28 (0%)

novel 11/81



Analogy

S5A3


* Remark: clusters 2_0 and 2_2 are obviously the same. Error during clustering?



solution cluster ID

transformation(s)


# of participants

per

transformation

1/11 1_0

*

1/11 2_0

1/11 2_1

*

1/11 2_2

1/11 6_2

add objects 1

4/11 6_6

switch position 1

1/11 6_7switch colours and

objects 1

Analogy

S5A3

blank 1/11 0_0

198

Choose condition



D1

86/159

(54%) n19

D2

71/159

(45%) n17

D3

2/159

(1%) n18

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

object movement 14

D1

46/79

(58%) 20/32 (63%)

switch position 6

object movement 8

D2

29/79

(37%) 10/32 (31%)

switch position 2

D3

3/79

(4%) 1/32 (3%) object movement 1

novel 1/79



Analogy

S5C3




solution cluster ID

transformation(s)


# of participants

per

transformation

Analogy

S5C3 1/1 2_0

gravity (object

attraction) 1

199

Choose condition



D1

72/156

(46%) n20

D2

3/156

(2%) 2_4

D3

81/156

(52%) n21

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

object movement 10


32/82

(39%)

15/35 (42%)

2 discarded switch position + rotation 1

D2

1/82

(1%) 1/35 (3%) object movement 1

object movement 12

switch position 5

switch colours 1 D3

47/82

(57%) 19/35 (52%)

retain position 1

novel 2/82



Analogy

S5C4




solution cluster ID

transformation(s)


# of participants

per

transformation

1/2 1_0

Analogy

S5C4 blank 1/2 0_0

200

Choose condition



D1

145/160

(91%) n31

D2

10/160

(6%) n32

D3

5/160

(3%) 4_3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

60/78

(77%)

28/37 (75%)


switch colours 2

rotation 3

D2

13/78

(17%) 7/37 (19%)


D3

1/78

(1%) 1/37 (3%) rotation 1

novel 4/78

(5%)

1/37 (3%)



Analogy

S6OR




solution cluster ID

transformation(s)


# of participants

per

transformation

1/4 1_0

1/4 2_0

1/4 4_1

Analogy

S6OR

blank 1/4 0_0

201

Choose condition



D1

21/163

(13%) 4_3

D2

139/163

(85%) n25

D3

3/163

(2%) 3

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

rotation 1

D1

3/75

(4%) 2/38 (5%)

switch colours 1

rotation 31

switch position 1

D2

67/75

(89%)

35/38 (92%)

2 discarded reflection 1

D3

0/75

(0%) 0/38 (0%)

novel 5/75



Analogy

S6A1




solution cluster ID

transformation(s)


# of participants

per

transformation

1/5 1_0

1/5 3_0

1/5 4_1 rotation 1

1/5 4_2

Analogy

S6A1

blank 1/5 0_0

202

Choose condition



D1

37/159

(23%) n27

D2

7/159

(4%) 2

D3

115/159

(73%) n26

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

15/79

(19%) 9/31 (29%) switch colours 9

D2

0/79

(0%) 0/31 (0%)

D3

54/79

(68%) 19/31 (61%) switch colours 19

novel 10/79



Analogy

S6A3




solution cluster ID

transformation(s)


# of participants

per

transformation

1/10 1_0

1/10 4_0 partial rotation 1

1/10 4_1


1/10 4_8

3/10 4_9 object movement 1

Analogy

S6A3

blank 2/10 0_0

203

Choose condition



D1

1/158

(1%) 1

D2

122/158

(77%) n28

D3

35/158

(22%) 4_5

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

0/80

(0%) 0/41 (0%)

D2

56/80

(70%) 30/41 (73%) rotation 30

switch colours 2

reflection 1

rotation 1



18/80

(22%)

9/41 (22%)

1 discarded

partial rotation 1

novel 6/80



Analogy

S6C1




solution cluster ID

transformation(s)


# of participants

per

transformation

1/6 4_0 rotation 1

1/6 4_2object movement /

gravity 1

1/6 4_3

1/6 4_4

Analogy

S6C1

blank 2/6 0_0

204

Choose condition



D1

14/161

(9%) n30

D2

23/161

(14%) 4_6

D3

124/161

(77%) n29

Make condition


solution

# of comments per

solution

transformation(s)


# of

participants per

transformation

D1

3/77

(4%) 1/39 (3%)

partial rotation + switch

colours 1

D2

5/77

(6%)

2/39 (5%)


rotation 32

D3

60/77

(78%) 33/39 (84%)

partial rotation 1

novel 9/77

(12%)

3/39 (8%)



Analogy

S6C2




solution cluster ID

transformation(s)


# of participants

per

transformation

1/9 3_0

1/9 4_0

1/9 4_3

1/9 4_5 rotation 1

1/9 4_7


1/9 4_11

Analogy

S6C2

1/9 5_0

205

Investigating Experimentally Problem Solving Strategies in ...

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