=Paper=
{{Paper
|id=Vol-1869/paper-4
|storemode=property
|title=Investigating Representational Dynamics in Problem Solving
|pdfUrl=https://ceur-ws.org/Vol-1869/paper-4.pdf
|volume=Vol-1869
|authors=Benjamin Angerer,Cornell Schreiber
}}
==Investigating Representational Dynamics in Problem Solving==
https://ceur-ws.org/Vol-1869/paper-4.pdf
Investigating Representational Dynamics in Problem
Solving
Benjamin Angerer Cornell Schreiber
Institute of Cognitive Science Department of Philosophy
University of Osnabrück Research Platform Cognitive Science
benjamin.angerer@uos.de University of Vienna
cornell.schreiber@univie.ac.at
Abstract
Successful problem solving relies on the availability of suitable mental
representations of the task domain. In more complex, and potentially
ill-defined problems, there might be a wide variety of representations
to choose from and it might even be beneficial to change them during
problem solving. To explore such dynamics on the representational level,
we developed a complex spatial transformation and problem solving
task. In this task, subjects are asked to repeatedly mentally cross-fold a
sheet of paper, and to predict the resulting sheet geometry. Through its
deliberate under-specification and difficulty, this task requires subjects
to find new and better fitting representations – ranging from visuospatial
imagery to symbolic notions. We present an overview of the task domain
and discuss various ways of representing the task as well as potential
dynamics between them.
1 Introduction
Often, the difficulties of problem solving lie not only in how to perform heuristic search, but start with how to
understand a given task [Van88]. In the study of problem solving, task understanding is typically conceptualised
as “setting up” one’s mental representation of the problem – its goals, constituents and possible operations –
and considered a preparatory phase before the actual problem solving activity ensues in a subsequent solution
phase [SH76, Vos06]. However, there is empirical evidence which suggests considerable interaction between
task understanding and problem solving. For example, evidence suggests that pertinent phenomena such as
insight, analogy, and transfer can be explained best in terms of changes of one’s representation during problem
solving [GW00, KOHR99, KRHM12]. Furthermore, developmental studies have shown that the use of different
solution strategies, potentially employing distinct problem representations, might “overlap” during problem solving
[Sie02, Sie06]. This suggests that representational dynamics, i. e. ongoing changes to how one represents a given
task, might play an essential role throughout problem solving.
However, to date systematic investigations of representational dynamics, mapping out problem solving activity
on the “representational level”, are still missing. Given the predominant focus on the research of heuristic search,
tasks are usually designed to constrain subjects to well-defined problem spaces [Goe10]. Considering how closely
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A.M. Olteteanu, Z. Falomir (eds.): Proceedings of ProSocrates 2017: Symposium on Problem-solving, Creativity and Spatial
Reasoning in Cognitive Systems, at Delmenhorst, Germany, July 2017, published at http://ceur-ws.org
29
such problem spaces resemble their task environment, representational dynamics are thus already precluded
by task design and presentation. In contrast, we propose the investigation of tasks that elicit representational
dynamics as a regular part of problem solving. In order to make progress in solving such a task, subjects are
continuously challenged to acquire more knowledge about – and potentially change their perspective towards –
the task domain.
To this end, we present a complex spatial transformation and problem solving task in the domain of iterated
paper folding. In the following, our focus lies on a first description of the task domain (Sect. 3). With this, we
establish a prerequisite for systematic representational-level analyses of problem solving in the targeted domain.
As a first step towards such analyses, we conclude with a discussion of various ways of representing the task
mentally and the potential dynamics between those (Sect. 4), and a few final remarks discussing further aspects
of this task which are beyond the scope of this introductory paper (Sect. 5).
2 Task: Iterated Paper Folding
The task we want to discuss in this paper consists of two subtasks which are presented consecutively, i. e. only
after completion of the first subtask the second one is revealed:
1. Imagine cross-folding a sheet of paper and inspect the folded sheet,
2. Draw 2D sketches of the forms of edges on each side of the folded paper.
In subsequent iterations of the task, the number of times the sheet is to be cross-folded (and its sides to be
sketched) is incremented – resulting in multiply-folded sheets (“folds”). Hence, while the subtasks stay the same
in all iterations, the complexity of the folds to be made is increased with each iteration.
Figure 1: Cross-folding a sheet of paper five times (by alternating between perpendicular folding directions).
This task has several characteristics that make representational dynamics an essential part of human subjects
solving it successfully1 . As a prerequisite for this, it is important to note that the domain of iterated paper folding
is such that problem solvers will find a variety of generally distinct representations effective, each with their
merits and weaknesses. For instance, the domain can be approached visuospatially (as a case of mental imagery)
or analytically (as case of spatial or even symbolic reasoning), but also more intricate combinations of these
approaches are feasible (see Sect. 4). Moreover, depending on one’s prior knowledge, and the experience gathered
in successive iterations, one’s conceptual understanding of the domain’s underlying principles (see Sect. 3) permits
the development of increasingly efficient representations. Taken together, these characteristics provide ample
opportunity for representational dynamics.
In the initial engagement with the task, the noted variety of representational possibilities is likely to be most
apparent, since the task is posed as an ill-defined problem [Ree15, Sim73, Rei65]: Besides the instructions for
the first subtask being deliberately vague, no further verbal or graphical hints are provided. Thereby the task
instructions do neither “prescribe” the use of specific representations or procedures, nor do they state a precise
goal. To form a productive understanding of the task, subjects have to explore the task domain, potentially
considering different kinds of more or less suitable representations and procedures [BV02, Goe10]. At the outset,
problem solving activity will thus likely be relatively idiosyncratic, taking place in incomplete or incoherent, and
dynamically changing problem spaces based on the subject’s prior experiences with and familiar knowledge about
paper folding.
While at later stages problem solving activity should become substantially more coherent, the reason to still
expect sustained dynamics lies with the task’s iterative procedure. Since with each iteration the complexity of the
folds increases substantially, generating them gets progressively more challenging. Thus, simply applying the
1 Presuming that from the start one would apply a form of representation that is well-suited for the complexity of later iterations,
and assuming the availability of unlimited processing capacities, it would theoretically be possible to solve our task without changes
in how it is represented – but both of these are not the case for typical human subjects (owing to both, the capacity limits of mental
imagery, and a lack of detailed knowledge about the consequences of iterated folding).
30
representations and procedures developed so far to the new challenges will not suffice. To counter these increasing
cognitive demands, subjects will instead have to find more efficient ways of representing and manipulating
cross-folds – furthering their understanding of the task domain in the process, and effectively changing the
problem spaces they are operating in.
Related work
While the task presented has been newly developed and there is to our knowledge no work directly concerning
iterated cross-folding, there is a vast amount of work on other types of paper folding. For instance, there are
psychological investigations of the mental folding of cube nets, and unfolding of mutliply-folded sheets with
holes punched into them [HHPN13, SF72]. There is also work on the qualitative modelling of these kinds of
tasks [Fal16], as well as mathematical and algorithmic descriptions of more complex paper folding such as
Origami [DO07, IGT15]. Particularly noteable with respect to our work is a study of origami folding tasks which
investigated how people reconceptualise the task and its constituents over the course of the study, thus equally
emphasising possibilities for conceptual and representational change [TT15].
3 Overview of the Task Domain
In the following we provide a systematic overview of the domain of iterated paper folding in terms of its task-
relevant entities and their regular interrelation, which subjects might represent in some form or another when
engaging with the task. With this, we establish an objective point of reference for conducting and discussing
representational-level analyses.
The overview is divided in three parts: First, Sect. 3.1 explains the procedural details of how to fold (illustrating
the extent of under-specification in the first subtask). Then, Sect. 3.2 describes the forms the sheet’s sides assume
after being folded, and which are closely related to the sketches asked to be drawn for the second subtask. Finally,
Sect. 3.3 describes the relations holding between these sides and between consecutive folds, respectively.
3.1 Folding Procedures
Cross-folding can be defined as folding a sheet such that it is halved in middle by each fold and such that consecutive
creases are perpendicular to each other. Ignoring the sheet’s size, thickness, and aspect ratio – the n-th cross-fold
Fn is uniquely determined by:
(a) the initial spatial orientation of F0
(b) the number of times folded (n)
(c) the folding procedure used
A folding procedure is determined by how exactly the sheet is being folded, most importantly the direction in
which the fold is made. Initially (for F0 !F1 ), there are 8 directions in which we can fold, two of which yielding
identical folds (Fig. 2). Since cross-folding requires successive creases to be perpendicular, from F1 onwards there
are only 4 directions that can be chosen.2
When combining folding directions arbitrarily, there are 22n+1 different folding procedures for F0 !Fn . Of
special interest, however, are procedures in which folding directions are combined systematically, such as alternating
between the same two perpendicular directions (e. g. Fig. 1). Using such a procedure limits the number of
possibilities to 32 (8 directions ⇥ 4 perpendicular directions).
Alternatively, a cross-fold can also be achieved by introducing a 90 -rotation-step between two foldings in
the same direction (e. g. Fig. 3). For such a procedure, there are only 16 possibilities (8 folding directions ⇥ 2
rotation directions).
Even though all of these procedures yield a cross-fold and hence fulfil the first sub-task equally, it is important
to distinguish between them. Since the cross-folds produced by them differ in certain respects, they are relevant
when trying to identify subjects’ representations and evaluating their performance in the second subtask.
2 Whereas folding in one of the other 4 directions would yield a parallel fold.
31
Le ge n d
vertically, tow ards, to right horizontally, tow ards, dow n
vertically, backw ards, to right horizontally, backw ards, dow n
vertically, tow ards, to left horizontally, tow ards, up
vertically, backw ards,to left horizontally, backw ards, up
Figure 2: Eight possibilities of turning F0 into F1 .
+ + + +
Figure 3: A folding procedure which rotates the sheet 90 clockwise before folding horizontally-towards-up (rotation
steps not depicted)
3.2 Fold Forms
Describing the consequences of folding on the sheet, we can identify the occurrence of regular schematic forms,
which all cross-folds have in common.
Depicting each side of a cross-fold two-dimensionally (as required by the second subtask) five basic forms can
be observed. They are the forms of all folds from F2 onwards, namely: Two uncreased rectangles, the crease itself
(I), a single creased edge (V ), a double V (DV ), and a nested V (N V ).
Figure 4: An exemplary illustration of F2 , its six sides, and their positions: rectangles (front, back), I (right), V
(bottom), DV (left), and NV (top).
The rectangles, I, and V (re-)appear unchanged in all folds, but DV and N V only retain their basic shapes,
with every DV 2 consisting of two sub-figures side-by-side, and every N V 2 consisting of two nested sub-figures.
As the number of folds increase, they show more complex creasing of edges, hence they are referred to as a fold’s
two complex sides.3
Figure 5: N V0 to N V5 (left to right), as produced by the procedure shown in Fig. 1.
Looking at F1 and F2 , it might seem that while there are many different folding procedures (see Sect. 3.1),
3 Hence, the remaining description of our task domain will be mainly concerned with those two sides.
32
they always bring forth the exact same folds, only in different spatial orientations. Yet, with higher fold numbers
certain differences start to appear.
3.2.1 Left-handed & Right-handed Folds
Beginning with F3 folds are chiral, i. e. they are no longer identical with their mirror images and we have to
distinguish left- and right-handed variants. For instance, the folding procedure shown in Figures 1 and 5 produces
folds of alternating chirality.
Figure 6: The achiral N V2 next to left- and right-handed variants of N V3 .
3.2.2 In-folding & Out-folding
Starting with F4 , different folding directions produce yet another distinction: Depending on whether a new fold
Fn encompasses the Vn 1 with the N Vn 1 or the other way around, we distinguish out- and in-folds (denoted by
a superscript I and O). Folding procedures with fixed direction and rotation will always yield out-folds, whereas
non-rotating or alternately-rotating procedures can produce both kinds.
Figure 7: From left to right: Left-handed N V4O , N V4I , right-handed N V4O , N V4I .
3.2.3 Mixed Folds
Furthermore, depending on the folding direction either an in- or an out-fold is produced each time we fold. This
means that if allowing arbitrary combinations of folding directions, folds cannot only be “purely” in- or out-folded,
but they can be mixed folds, with an ever increasing number of possible combinations of in- and out-folding.
This results in a total number of 2n+1 fold variants for Fn (as opposed to an upper bound of 32 variants if we
disallowed arbitrary folding directions, and hence mixed folds).
Figure 8: Four different N V5 , from left to right: Out-fold, mixed (in-folded out-fold), mixed (out-folded in-fold),
in-fold.
3.3 Regular Relations
While in theory, every detail of how a fold will look like and how its sides relate to each other can be derived
from the chosen folding procedure, these details can be relatively hard to see. Hence, in the following section, we
describe the most important of these relations holding in the general case.
3.3.1 Within-fold Regularities
While folds can vary in their orientation, and in the detailed structure of their sides etc., the spatial relations
holding between a fold’s individual sides are the same for all cross-folds (see Table 1).
33
Table 1: An overview of the spatial relations between the sides of a fold.
V DV NV
I ? k ?
V ? k
DV ?
(k = spatially opposite, ? = perpendicularly adjacent)
3.3.2 Between-fold Regularities
In an equally general manner, we can describe the relations holding between the four non-rectangular sides of two
consecutive folds Fn and Fn+1 (for n 2). Table 2 presents these relations as rules of how to manipulate each
side of Fn two-dimensionally in order to generate Fn+1 ’s sides from them.
Table 2: An overview of the generative relations between the sides of consecutive folds.
In+1 : Vn+1 : DVn+1 : N Vn+1 :
new fold(In ) align(Vn , N Vn ) fold(DVn )
3.4 Summary
While there are many more advanced facts about the task domain4 , above we have introduced the domain’s
basic properties: (a) The different procedures that can be used to make cross-folds, (b) the forms a sheet’s sides
assumes after being cross-folded in different ways, and (c) the regular relations holding between a fold’s sides.
Following from these basic properties, the large number of possible cross-folding procedures, with its intricate
distinctions in higher iterations (chirality, in-/out-folding etc.), as well as the overall increasing complexity of
higher folds, present particular challenges to subjects which will potentially give rise to representational dynamics.
Additionally, the distinctions presented here are also relevant when trying to identify the representations subjects
might use in solving the task.
4 Discussion: Representational Dynamics
In the following, we discuss which representational dynamics can be expected in the task. We present how people
have been shown to solve mental folding tasks in general, using different varieties of representations. On this basis,
we point out potential representational dynamics that can ensue within and between these kinds of representations
in the domain of iterated paper folding.
Generally, research on spatial transformation distinguishes between two kinds of solution approaches which
presume largely different kinds of representations. As indicated earlier, there are visuospatial approaches, i. e.
imagining a 3D object, transforming it, and then “seeing” the result, and analytic approaches, i. e. understanding
and solving the problem based on explicit domain knowledge [HHPN13]. Furthermore, visuospatial and analytic
approaches are usually conceived as the poles of a continuum, allowing for mixed approaches [GF03].
When faced with the repeated challenges of the task, subjects will have to change their perspectives towards the
problem several times, thus furthering their understanding of it, and taking advantage of the merits of different
approaches. It is thus necessary to discuss each approach in some detail.
4.1 Visuospatial Approaches
In terms of a visuospatial approach, the mental imagery of paper folding is typically conceived of as the mental
analogue of folding physically [SF72]. The physical process of folding is a non-rigid transformation, i. e. folding
an object affects its individual parts differently [HHPN13]. In cross-folding, for instance, a single act of folding
affects a sheet’s sides in distinct ways (cf. Table 2). Consequently, in order to mentally form a visuospatial
representation of a fold as whole, subjects will likely require multiple repetitions of the same transformation of a
sheet, while variably attending to its individual sides.
4
PnFor instance, some readers might be interested that the number of nestings in a fold’s N Vn corresponds to the partial sum
1 bic
i=0
2 2 (sequence A027383 in the OEIS [OEI17]).
34
The transformation of a visuospatial representation is usually understood as akin to sensorimotor trans-
formations, i. e. an analog transformation of one visuospatial representation into another one mediated by a
motor process [Iac11, MK09]. While this transformation is described as analogous to the physical act of folding,
visuospatial representations often already leave out many physical details which are irrelevant to the task (such as
aspect ratio or size). Crucially, in such a process information on the parts and their spatial interrelation is merely
implicit. Presuming a certain everyday familiarity with the activity of cross-folding, visuospatial approaches have
the advantage that subjects can perform them without much explicit knowledge about the task domain. Their
effectiveness, however, is limited to spatially rather simple or very familiar complex objects [BFS88].
Consequently, when attempting to mentally cross-fold for the first few times, it might be possible to fold F1
and F2 in this manner. However, with increasing complexity of the folds, even attending to their visuospatial
representations side-by-side will eventually become too demanding, and subjects have to find other approaches.
Consider for example Fig. 9, where a DV2 is being transformed into an N V3 (as part of folding F3 from F2 ).
Depending on one’s familiarity with cross-folding, this might already be a very advanced transformation and at
the boundary of what a typical subject is able to achieve in a purely visuospatial manner.
Figure 9: F2 !F3 , attending to the transformation of DV2 into N V3 .
4.2 Analytic Approaches
Analytic approaches to mental folding are more straightforward cases of problem solving or spatial reasoning.
To this end, the mental object is construed as a symbolic representation of interrelated parts. As opposed to
visuospatial approaches, spatial information is not implicit in such representations, but has to be made explicit
as a semantic relation between symbolic entities. In cross-folding, subjects might represent a fold in terms of
the regularities presented in Sect. 3. For instance, F2 can be represented as a nested structure, comprising six
schematic 2D forms (cf. Sect. 3.2) which stand in regular spatial relations (cf. Table 1). Additionally, each of
those forms can be further decomposed into a spatial configuration of edges. Given such a representation, a
successor fold can be realised as a rule-based construction, i. e. a symbol-by-symbol translation according to the
between-fold regularities (cf. Table 2).
Crucially, the feasibility of an analytic approach depends on the subject’s explicit knowledge of the task domain.
Thus, analytic approaches are unlikely to be adopted in the initial phase of the task when subjects are still
exploring the ill-defined problem. Only after they have gathered sufficient knowledge, such as the sheet’s basic
forms and their spatial relations, analytic approaches might start to occur. For example, subjects might utilise
their knowledge that some schematic forms are the same for all Fn , i. e. the rectangles, Is and V s. Beyond that,
the transformation of an N V and V into a DV also lends itself to an analytic solution, since it does not involve
complex visuospatial manipulations besides aligning two sides of the previous fold (Fig. 10). While analytic
approaches thus allow to avoid otherwise complex operations, the lack of visuospatial representations during
problem solving can also lead to paradoxical situations: For instance, aligning two wrong sides would yield a
symbolic representation of a physically impossible fold state.
As it gets successively more demanding to represent the sides of higher-numbered folds in terms of simple
derivations of the basic forms, ultimately an analytic approach requires an even more economical, syntactic way
of representing sides. This could be achieved by encoding the number of nestings of edges on the open end of a
side. For instance, one could use 0 to encode a simple open edge, 1 for a looped edge, 2 for two nested looped
edges etc. So [0,0] would represent a single V , and [0,0,0,0,0,0,1,1,4,1] an N V5O (such as on the right of
Fig. 11). Based on such a syntax, one could formulate a recursive procedure which – using the sides of F2 as base
cases, and the regularities in Table 2 as transformation rules – can generate the complex sides of any Fn .
35
∥
Figure 10: Aligning two existing figures of spatially opposite sides (V3 , N V3 ) to form DV4 .
4.3 Mixed Approaches
For the most part however, neither visuospatial nor analytic approaches will likely be used exclusively for any
longer period of working on the task. We can rather expect both of them being employed, making use of their
respective merits, and with various forms of dynamics taking place between them.
A straight-forward case of interaction would be to generate solutions with one of the approaches, but to employ
the other one for checking the results. For instance, since the visuospatial transformation illustrated in Fig. 9 is
already quite difficult, subjects might be well advised to check their results explicitly against relevant knowledge.
On the other hand, a potentially inadequate posit from a symbolic construction, as in the paradoxical case above,
can be verified and corrected with the help of visuospatial manipulations.
But there are more intricate forms of dynamics, as well. Notably, explicit knowledge can be utilised variously
in order to scaffold or augment complexity-bounded visuospatial thinking. For example, instead of one holistic
visuospatial representation of the sheet, subjects could use visuospatially less demanding 2D representations
of each side, while maintaining the correct spatial relations between them explicitly (cf. Table 1). And even
within a visuospatial 2D representation of one of the fold’s sides, advanced domain knowledge could also allow
decomposing a complex side into simpler sub-figures, which by themselves are easier to deal with visuospatially
(folded, rotated etc.) once more (Fig. 11).
Finally, the recognition of perceptually analogous features of different folds (requiring visuospatial representa-
tions) might lead to the identification and rule-like representation of general features of cross-folding. For instance,
noticing the perceptual similarity between successive N V s (cf. Fig. 5) might lead to the rule-like hypothesis that
all N V s are composed of two nested sub-figures.
Figure 11: Transforming a DV4 into an N V5 , colours marking the different sub-figures.
4.4 Summary
Above we have outlined an account of representational dynamics for the proposed task of iterated paper folding.
According to this, how subjects approach the task can be expected to vary, dependent on the difficulty of its
current stage and the subject’s explicit knowledge of the task domain.
While in the initial stages of the ill-defined problem, subjects will likely be successful with visuospatial
approaches, in later, more difficult iterations, subjects will profit from changing to more analytic approaches.
Therefore, it becomes imperative to gather explicit knowledge of the task domain. However, for the most part
subjects will likely follow mixed approaches – as for the relative merits and weaknesses of both visuospatial
imagery and knowledge-based constructions. It might thus be more apt to conceive of their development as
mutually dependent.
5 Final Remarks
We have presented a new spatial transformation and problem solving task of iterated cross-folding and outlined
an analysis of the task domain.
Furthermore, we have discussed how subjects can approach the task domain with a variety of representational
forms, each of which can be subject to representational dynamics by themselves, but with dynamics also occurring
between these varied approaches [SSB13]. However, the systematic task description provided may have given
the impression that subjects actually show equally systematic behaviour when approaching the given task. Yet,
according to our prior experience with this task such an assumption would be problematic. For instance, we
36
hardly touched upon the varied roles both perceptual and structural cross-domain analogies can play in scaffolding
representations [CPS12, Dun01]. In a similar way, metaphors can also play an important role in changing the
task domain’s conceptualisation [Ami09]. An overall more naturalistic discussion, describing more ephemeral
aspects of working on this task (including “task-extrinsic” aspects such as mind-wandering) is provided in [Sch15].
Regarding the theory of problem solving, the extent of representational dynamics observable in a task such as
the one presented here might lead us to doubt the notion of stable problem representations in general. Yet, the
question of the potential changeability of problem representations has, for example, also lead to several proposed
extensions of problem space theory which try to address the problem by the assumption of additional search
processes [KBVK14, KD88, SK95].
Ultimately, in order to make progress in these questions and advance theory, we would need wider-ranging
representational-level analyses of subjects solving the presented task, and tasks similar to it. We hope that the
task domain and first analyses we presented here have set a good starting point for future endeavours of this kind.
Acknowledgements
We thank Stefan Schneider for his invaluable help during cognitive task analysis.
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