Rethinking Understanding FATHULLA

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Rethinking our understanding of diagrams

KAMARAN FATHULLA

Abstract

The role and value of diagrams in advancing human knowledge throughout

history is evident in the literature. However, persistent evidence from the re-

search and practitioner communities from across a range of disciplines points

to difculties and problems with understanding and using diagrams. Whether

it is at the application level or as an abstract denition, our existing frame-

works for understanding diagrams reduce the full meaning of diagrams to a

single perspective or emphasis. This paper advocates placing multi-aspectual

human functioning in a central role in the development of an enriched meta-

approach to understanding diagrams. Two such aspects, namely, the spatial

and symbolic, are essential to the understanding of diagrams across applica-

tions. The philosophical underpinnings and practical contributions of this ap-

proach to dealing with long standing problems of diagrams are discussed.

Keywords: symbolic (Sy); spatial (Sp); mapping (M); redundant Sp; main

SySpM

1. Introduction

Understanding diagrams plays an important role in our sense making of the

world and solving problems. Almost every discipline or study makes use of

diagrams and other forms of visual aids (Novick and Hurley 2001; Vekiri

2002; Pisan 1995). Ambrose et al. (1999) investigated difculties students

have with physics diagrams and found that, although some of the difculty

may be attributed to lack of understanding of subject matter, a signicant pro-

portion of the problems were to do with how students interpret diagrammatic

formalisms. A recent study by Ornek, Robinson, and Haugan (2008) attributes

the difculties to the different graphical representations students have to dealwith, and, furthermore, the need to convert from one representation to another.


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124 K. Fathulla

Scaife and Rogers (1996) argue that, despite the vast body of empirical studies

in the eld, little is known about why and how graphics facilitate learning. The

authors found many shortcomings in empirical studies on graphical representa-

tion, such as assumptions merely based on intuition, lack of integration oftheoretical concepts, and non-systematic analyses. Uesaka, Manalo, and Ichi-

kawa (2007) argue that despite the uses of diagrams in solving math problems,

research and educational practice reports indicate that students lack spontane-

ity in their use of diagrams. Diezmann (2000) and Paquette et al. (2006) report

that generating an appropriate diagram is problematic for many students. They

explored primary-aged students difculties in generating diagrams for novel

problems. Their results suggest that there is a need for a methodology in the

use of diagrams. The specic areas of concern related to the ambiguity of dia-

grams and their dynamic features. Ambiguity is often related to the potential of

one spatial feature being used to express a number of different semantics. A

report by The Royal Statistical Society expressed concern at the extremely

poor quality of graphs and diagrams submitted in recent years by candidates at

all levels of the Societys examinations (Oliver 1998). Anastopoulou, Sharples,

and Baber (2003) point to dynamics of diagramming as yet another source of

confusion and difculty for students. Problems with understanding UML dia-

grams have been discussed in Evans (1998) and Li, Hosking, and Grundy (2007).

2. Existingwaysofunderstandingdiagrams

Two inuential approaches dominate the underlying thinking behind many of

the existing frameworks of understanding diagrams, and are summarized in

Table 1.

These two approaches have inuenced many lines of inquiry into the area of

diagrams and their understanding in a fundamental way. They polarize the

nature of the topic. Examples of these poles include internal versus external

debate, abstract versus application, etc. Whatever the nature of the poles may

be, the overriding assumptions are that diagrams are understood in terms ofone single perspective, as shown in Table 2.

There is emerging dissatisfaction with the potential of these ways of under-

standing diagrams (Norman 2000; Kupla 2003; see Bresciani and Eppler in a

Table 1. Inuential frameworks of understanding diagrams

Source Emphasis

Peirce (CP), semiotics What we do with diagramsBertin (1983), semiology of graphics Structure of diagrams


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Rethinking our understanding of diagrams 125

managerial context). Bishop (1994) argues that the centuries-held assumption

that a drawing is a drawing is a drawing is progressively shown to be invalid.Horn (2001) claims that our current ways of understanding diagrams is one of

confusion.

This paper responds to these calls and advocates a rethinking of our under-

standing of diagrams and proposes a way that runs parallelwith rather than

opposing existing frameworks, but has distinctly different philosophically-

guided underpinnings.

3. Towardsarethinking

The main feature of existing ways of understanding diagrams is to either de-

tach the diagram from its human creator and treat it in purely spatial or graph-

ical terms or reduce the diagram to a property of human cognition. This later

approach ignores the spatial aspect of diagrams. However, there are many

examples that call for the need to think of diagrams in a more integrated way,

i.e., combining spatial and cognitive aspects.

A number of quotes from practitioners in the eld illustrate the point. The

way people use diagrams, irrespective of the application, has been eloquentlydescribed by J. D. Watson, co-discoverer of the structure of DNA and Nobel

prize winner in 1968: drawing and thinking are frequently so simultaneous

that the graphic image appears almost an organic extension of the thinking

process. In Gombrichs discussion of Leonardos creative process, he sug-

gested that: in searching for a new solution Leonardo projected new mean-

ings into the forms he saw in his old discarded Sketches (1966). Similarly, the

famous American architect Daniel H. Burnham says: . . . a noble, logical dia-

gram once recorded will not die, but long after we are gone be a living thing,

asserting itself with ever-growing insistence (quoted in Moore 1921). Simi-larly, Braham (2000) talks about the suffering of diagrams in the context of

Table 2. Poles of the debate

View of diagrams Nature of reduction

Internal

(Human centered)

Cognitive, or

Sensory (visual), or

Psychological, or

Linguistic

External/Abstract Spatial, or

Graphical

Application This can be any single perspective depending on the nature of the

application


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126 K. Fathulla

architecture. This characteristic of diagrams calls for the need for a more

human element in our understanding of diagrams.

The message conveyed by these quotes highlights a softer, human friendly,

side to diagrams as well as their traditional spatial characteristics. It is essentialthat both sides are treated equally and simultaneously for the appreciation and

a good understanding of the full meaning and purpose of diagrams. This type

of understanding requires a radical shift in traditional approaches that are

mainly reductionist in nature and assume a single perspective of what consti-

tutes diagrams. This paper presents one such non-reductionist approach, the

foundations of which are centered on the notion of Aspects.

4. Aspect(s)ledrethink

The dening nature of this approach is underpinned in the notion of aspect,

which is part of Herman Dooyeweerds (1955) philosophical framework. An

aspect is a sphere or modality of meaning within which things, including

diagrams, are meaningful. Aspectual philosophy stipulates that nothing can be

of one aspect alone. This central element of the philosophy forms the corner-

stone from which, in this paper, a proposal for a better understanding of dia-

grams is outlined.

Each aspect has a distinct set of laws that guide and enable functioning.

All 15 aspects, shown in Table 3, together constitute a framework which

Dooyeweerd called the law side of created reality. This is a framework that

enables everything to exist and be. The aspects are what explain the diversity

and coherence of everyday experience. Therefore, each aspect provides us

with a distinct way in which things have become meaningful or make sense.

The philosophical foundations of the notion of aspecthave a number of

distinct features that will inform the vision pursued in this paper:

emphasis on (human) functioning rather than on entities: We function

according to laws that are both determinative and normative. irreducibility of aspects: Aspects are fundamentally irreducible. This

means that no aspect can be derived from another, and that each aspect

must be given proper respect in a situation.

sphere universality : that these aspects, though irreducible, are nevertheless

closely intertwined, such that in each there are echoes of each of the others.

dependency among aspects: The aspects form a sequence, in which the

laws of an aspect depend on those of earlier aspects for their proper func-

tioning, even though they may not be reduced to them.

the notion of a qualifying aspect: That (almost) all human activities, and(almost) all entities are qualied by one aspect, even though the function-


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ing involved in the activity is in fact multi-aspectual. It is the qualifying

aspect that gives an activity or entity its primary meaning, and also pro-

vides the most useful criterion for evaluating whether the activity or entity

is good or impaired.

The notion of enkapsis: What we experience as an entity is often, in fact,

an enkaptic intertwinement of several distinct entities, each of which is

qualied by a different aspect. Enkapsis speaks of what individuality struc-

tures are necessary to the proper understanding of an enkaptic structural

whole, rather than what individuality structures could be part of it in var-ious circumstances.

diagrams and aspects: The notion of qualifying aspects allows us to distin-

guish diagrams from other types of graphical material, such as paintings or

decoration. Both diagrams and paintings are founded in the spatial aspect

(although one might argue that, where color is important, the sensitive as-

pect is also important). But diagrams are qualied by the lingual aspect of

symbolic meaning and communication, while paintings and decoration, on

the other hand, are often qualied by the aesthetic aspect.

The notions ofsphere universality and irreducibility of aspects also inu-ence the proposed approach because it means that the spatial and lingual

Table 3. The fteen aspects and diagrams

Aspect Relevance to diagrams

Numeric Concerned with the number of shapes, etc.

Spatial Concerned with things like shapes in the diagram, spatial arrangement,

connectivity, etc.

Kinematic This concerns the activity of diagramming itself when conveying symbolic

meaning

Physical This concerns the specic physical medium or substrate used to display the

diagram

Biotic This aspect concerns life functions that are a necessary precondition for the

sensory and nervous systems to function well enough for diagram creation or

reading.

Sensitive This concerns perceiving color, texture, etc., and also emotions

Analytic This aspect concerns the activity of distinguishing what is meaningful in a diagram

from its background

Formative The laws of this aspect govern the creation or formation of the symbol structure

being expressed, and also of the method by which a drawing is composed

Symbolic This concerns conveyance of meaning

Social This concerns social impact and norms used when diagramming

Economic This concerns the cost of producing the diagram

Aesthetic This concerns how well a diagram is presented

Juridical This concerns issues such as copyright

Ethical Diagrams should not be offensive


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128 K. Fathulla

aspects of a diagram must be considered separately and yet in relation to each

other. In this context, separately means:

Phenomena of each aspect must be recognized in terms of the aspect and

not subsumed by the other.

Appropriate recognition must be given to the distinct laws of each aspect.

The terminology used for phenomena must, as far as possible, avoid meta-

phor and be that which is appropriate to the aspect. For example, in a box

and arrows diagram, we should use the term item when speaking from

the point of view of the lingual individuality structure and the term box

(or, better, rectangle although it is often a different shape) when speak-

ing from the point of view of the spatial individuality structure.

Although all fteen aspects are important when discussing diagrams, neverthe-less, some of them only come to the forefront only when diagrams are in use

through some human application. However, only two aspects pertain to dia-

grams in themselves regardless of use or context: the symbolic and spatial.

These two aspects are brought together through the notion of enkapsis.

Enkapsis provides precision in our understanding of the nature of diagrams.

Namely, a diagram, as an enkaptic structural whole, comprises two individual-

ity structures, both of which are necessary for a proper understanding of a dia-

gram as a:

Spatially qualied structure, which is subject to the laws of the spatial

aspect, and in which we recognize spatial shapes, arrangements, and

relationships;

Symbolically qualied structure, which is subject to the laws of the lingual

aspect, and in which we recognize symbolic objects such as items, relation-

ships, numbers, etc., and the relationships between them, and we must take

account of their use for purposes of symbolic meaning, which includes

both communication and stimulation of thinking.

The identication of these two aspects is not made on an ad hoc basis but ratheraccording to principled reasoning:

1. They conform to our intuitive understanding of diagrams, i.e., spatial

marks used to convey some meaning.

2. They resonate with what we discussed in the above mentioned quotes. The

symbolic/lingual aspect accommodates the human friendly or soft ele-

ments of diagrams while the spatial aspect covers the traditional view of

diagrams.

3. Their selection is philosophically underpinned.

In the following section, the integration of these two aspects is elaborated.


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5. TheSySpMframework

Full discussion of this framework and its contributions is beyond the scope of

this paper and can be found in Fathulla (2006). Here we briey introduce themost important elements of the framework. The central thrust of this frame-

work is: separating Symbolic from Spatial but allowing for their Mapping

(SySpM). The framework is based on the notion that Sy is distinct and separate

from Sp and is irreducible to it. From this framework individual SySpMs

could be constructed. The term SySpM is used to denote a distinct (particular)

collection of Sy and a distinct collection of Sp and a distinct Mapping between

the two. Synonymous with this term are drawing styles or types of diagrams.

We grouped the Sy and Sp terms into primary and secondary lists. The former

includes Sy and Sp terms that are used in the simplest or basic form of the

SySpM. This category of Sy and Sp is also in all instances of diagrams of the

relevant SySpM. The latter includes terms that are used in complex diagrams

of the relevant SySpM and it also includes things that have to do with the rela-

tionships or interaction between the various Sy things of the relevant SySpM.

Development of each SySpM also includes list of Sp things that do not map

onto any Sy known as redundant Sp things. These could then become available

for other SySpMs to use to give mixed diagrams, list of constraints and why

they occur, and list of events or changes relevant to the SySpM under consid-

eration. Special features of a SySpM use redundant Sp features to bring in

secondary Sp features rather than another SySpM as mixed diagrams. An ex-

ample of this in a box and arrows SySpM is when lines are allowed to cross

other lines. This happens when one line is given a kink or a gap to indicate

clearly that one is passing over or under the other rather than connecting to it,

e.g., electronic circuits. Subtypes are diagrams in which the original and sim-

ple SySpM is constrained/complicated because of specic needs usually asso-

ciated with a type of application. This is achieved by bringing in an extra Sy

constraint that, owing to Mapping, also gives a different Sp feel to the diagram.

There are at least three different ways of Sy subtypes in a box and arrows

SySpM: networks, lists, and trees. For each SySpM there might be specialcases that do not t well. Many spatial applications involve several of such

special cases such as holes, discontinuities, and other irregularities. We need to

identify these and explain the problems they give, that is, what constraints they

break, either spatial (as here) or symbolic.

A SySpM could contain features that are outside the range of its base sym-

bolic types. This recognizes that each SySpM will be able to express only a

subset of the symbol level, not all of it. To express the whole wide range of

things at the Sy level requires several different SySpMs. Within this context,

two types of mixedness are identied. One is when several SySpMs are pres-ent in a diagram but none dominates the overall meaning of the diagram. An


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130 K. Fathulla

Table 4. Types of diagrams generated from the SySpM framework

The SySpM Mapping

Boxes and Arrows Item mapped onto box, relationship

mapped onto arrow

Communicating Similarity A collection of items mapped onto a

collection of shapes

Map of Objects Item location mapped onto icon

position

Set Membership A shape inside a loop mapped onto

member of a set

Bar Charts Magnitude mapped onto length of a

bar

Route Maps Route mapped onto curvilinear line

Contour Maps The set of location with the same

quantitative value mapped onto

closed continuous curve

Surface Coverage Region mapped onto area

example of this is the Napoleons march on Moscow diagram shown below.

This type of mixedness is referred to as True mixed diagrams. The other type

is referred to as Augmented diagrams. This type of mixedness has one

SySpM occupying a primary importance while other SySpMs are added in

and have secondary importance. This type of mixedness occurs when redun-dant Sp features of a SySpM are used to bring in Sy from other SySpMs. For


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example, in a Box and Arrows SySpM thickness of lines could be used to bring

in quantitative value from Bar Chart SySpM.

6. Thegenerativecontributionoftheframework

The SySpM framework is a way of understanding diagrams that can be used to

generate types of diagrams. In Fathulla (2006), eight such types or SySpMs

were developed namely:

1. Boxes and Arrows

2. Communicating Similarity

3. Map of Objects

4. Set Membership5. Bar Charts

6. Route Maps

7. Contour Maps

8. Surface Coverage

These SySpMs are not seen as a nal list but rather to demonstrate the poten-

tial of SySpM as a framework in accounting for a diverse range of types of

diagrams.

7. ApplicationoftheSySpMs

The various SySpMs have been used in the interpretation of four real world

semantically rich diagrams. The full methodology and analysis is found in

Fathulla (2006). Figure 1 shows how to apply the SySpMs to one of those

diagrams.

Table 5 shows an example of how the various SySpMs can be used in the

understanding of Minards well-known and celebrated diagram of Napoloens

march on Moscow (see Figure 2).The potential of using knowledge of the various SySpM to understand dia-

grams goes beyond the task of interpretation. A number of useful insights can

be acquired with this approach.

It becomes possible to identify possible or potential errors or confusions

with diagrams. For example, the thin vertical lines may be confused with splin-

ter or diversions because they look visually similar. However, when considered

from a Boxes and Arrows SySpM, these lines are seen as relationships between

two objects: one is temperature value and the other is size of army.

The small sideway kinks (edges), Figure 3, along the route suggest an errorof display and the need for smoothing out the edges. However, knowing that


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132 K. Fathulla

Figure

1.

(topleft)Aquantied

owchart;(topright)Pressure-vo

lumegraph;(bottomleft)Charles

JosephMinard(17811870),depicting

thefateofNapoleonsarmysmarchonMoscow.Source:CollectionEcoledespontsaveclacoteENPCduoudesdocuments.(bottomright)

Ordinancesurvey(OS)mapwith

OSpermission.


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Table 5. SySpM analysis of the Napoleons march on Moscow (see Figure 1 [bottom left])

Diagram

feature

Description SySpM analysis

Sp SySpM Sy

1 A long straight

horizontal line

is drawn across

the entire length

of the diagram

symbolizing the

division of the

space.

Line Surface Coverage Space divider

2 A number of

vertical lines

Line Boxes and Arrows

where boxes are

nominal entities

that are important

values that should

be linked

Relationship

3 The proximity of the

end of the black

line to the start

of the textured

line is to allow

the comparison

of two extremely

different values.

Proximity of

position

Bar Chart Comparison of

quantitative

values:

422,000

with 10,000

4 Discontinuity of

textured splinter

line from its

black segment

Break or gap

between the

lines at the

start of the

campaign

(use of empty

space)

Variant of Route

Map

Location

5 Distances between

lines joining the

two spaces of

the diagram

Proximity Variant of Boxes

and Arrows

Extreme

changes in

values

6 A pair of X and Y

axis lines on the

bottom surface

of the diagram

depicting

temperature scales

Length Bar Chart sub

type in which the

orientation of the

bar chart is

turned upside

down

Comparison of

quantitative

values

7 The vertical lines

linking the retreat

path to the graph

Lines

Length of bars

Boxes and Arrows

where boxes are

nominal entities

that are important

values that should

be linked

B Ch S S M

Relationship

Discrete value


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134 K. Fathulla

the width of the line is Bar Chart SySpM explains the need for this feature to

help the reader interpret the distinct values along the route.

Towards the end of the campaign, the end of the advance route seems to

have shifted its position suggesting that the army jumped its position south-

ward, see Figure 4 (left panel). This hints at the possibility of an error in the

diagram around Moscow. The apparent error is explained through Route MapSySpM, which allows lines to touch each other. However, this part of the

Figure 2. Minards well-known and celebrated diagram of Napoloens march on Moscow

Figure 3. A kink pointed to by an arrow


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diagram could have been improved with a better, more continuous bend

around that region, see Figure 4 (right panel).

The approach could also help with assessing the quality of diagrams. This

particular diagram has a number of good quality features. The main message of

this diagram is the disastrous fate of the army. Comparison of the thickness of

the white and black line segments (Sp) on the left hand side of the diagram

depicts the armys fate. The comparison of these two extreme magnitudes (Sy)

is expressed using Bar Chart SySpM to show two extreme magnitudes.

The good quality of the diagram also lies in the naturalness with which three

or more SySpMs work together without confusion or ambiguity because they

do so by:

Judicious use of redundant Sp features of each SySpM namely Route Map

and Bar Chart.

Splitting the diagram into two areas with Route Map and Bar Chart in oneand XY Bar Chart in the other.

Examples of this naturalness can be found in the effective linking of the upper

route of the campaign and lower XY Bar Chart using Boxes and Arrows.

Sp features are used redundantly in the same SySpM to bring in another

SySpM. This is illustrated in using the width of a line to bring in the length of

a bar in a Bar Chart SySpM to express size of the army and in using the width

of a bar, which is a redundant Sp feature, to bring in a Route Map SySpM.

8. Conclusion

This paper is a response to the persistence with which the literature cites prob-

lems with our understanding and using diagrams. Existing frameworks that

have inuenced our understanding of diagrams for much of the time are prov-

ing to be insufcient to deal with the range of diversity of diagrams and ap-

plications that make use of them. A philosophically guided framework is pro-

posed based on the notion ofaspects, which states that no entity is of a single

aspect but rather all reality, including diagrams, function with a given set ofaspects. This philosophical approach was developed to work out a richer

Figure 4. (left) The apparent jump; (right) a better rounded route


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136 K. Fathulla

understanding of diagrams based on symbolic and spatial aspects. The pro-

posed SySpM framework offers a richer vocabulary for discussing static and

dynamic features of diagrams across a range of applications.

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Kamaran Fathulla (b. 1961) is a senior lecturer at the University of Northampton . His research interests include understanding diagrams and philoso-

phy. His publications include Frameworks for understanding diagrams: Enriching the debate

(with A. Basden, 2007); Symbolic, spatial, mapping (SySpM): A framework for interpreting

maps (with A. Basden, 2008); Understanding diagrams: A pointer to the development of

diagramming software (2008); and A framework for understanding diagrams in education

(with F. Hameed, 2009).


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