Beyond quality and quantity: Representing empirical structures by embedded typologies

Abstract

In this article, we propose a new approach to the problem of integration in mixed methods research that builds on a representational understanding of empirical science. From this perspective, qualitative and quantitative modeling strategies constitute two different ways to represent empirical structures. Whereas qualitative representations focus on the construction of types from cases, quantitative representations focus on the construction of dimensions from variables. We argue that types and dimensions should be integrated within a joint representation of the data that equally acknowledges qualitative and quantitative aspects. We outline how the proposed representational framework can be used to embed qualitative types in quantitative dimensions using an empirical study on teachers’ epistemological beliefs.

Full text

Frontiers in Education 01 frontiersin.org
TYPE Methods
PUBLISHED 09 February 2023
DOI 10.3389/feduc.2023.1087908
Beyond quality and quantity:
Representing empirical structures
by embedded typologies
MatthiasBorgstede * and CarolineRau
Foundations of Education, University of Bamberg, Bamberg, Germany
In this article, wepropose a new approach to the problem of integration in mixed
methods research that builds on a representational understanding of empirical
science. From this perspective, qualitative and quantitative modeling strategies
constitute two dierent ways to represent empirical structures. Whereas qualitative
representations focus on the construction of types from cases, quantitative
representations focus on the construction of dimensions from variables. Weargue
that types and dimensions should beintegrated within a joint representation of the
data that equally acknowledges qualitative and quantitative aspects. Weoutline how
the proposed representational framework can beused to embed qualitative types
in quantitative dimensions using an empirical study on teachers’ epistemological
beliefs.
KEYWORDS
qualitative and quantitative research, mixed methods, documentary method, discriminant
analysis, cluster analysis, teachers’ epistemological beliefs
1. Introduction
Although qualitative and quantitative research methods adhere to rather distinct assumptions
about the aims and scope of scientic enquiry (Freeman etal., 2007), attempts to integrate both
strategies within a mixed methods approach are eventually gaining attention (Creswell, 2015). Mixed
methods research attempts to combine the strengths of qualitative and quantitative research by using
both strategies within an overarching methodological framework. However, there is no generally
accepted integration framework for mixed methods designs (Fetters and Molina-Azorin, 2017).
Consequently, integration of qualitative and quantitative strategies remains a major challenge for
mixed methods research (Moran-Ellis etal., 2006; Creswell, 2009; O'Cathain etal., 2010; Fielding,
2012; Bazeley, 2016). In some cases, the qualitative and quantitative parts of a mixed methods design
may even appear to betwo dierent studies that are only connected thematically (Yin, 2006;
Bergman, 2011).
In this article, weapproach the problem of integration from a representational perspective that
characterizes qualitative and quantitative research as dierent ways to represent empirical
observations by means of scientic theories (Borgstede and Scholz, 2021). In this view, the semantics
of a scientic theory (or model) is twofold. On the one hand, it consists of the meaning of its
constituting concepts. On the other hand, it consists of the topology that relates these concepts to
one another. Dierent types of models may emphasize either one of these aspects. Following
Borgstede and Scholz (2021), qualitative representations focus on theoretically meaningful concept
formation that is based on the essential properties of the objects under study. e objects may beof
very dierent nature, depending on the focus of the study. is includes inanimate objects, as well
as living organisms, people, groups, interviews, or even abstract concepts and themes. Quantitative
representations, however, are more concerned with the topological structure that relates the concepts
OPEN ACCESS
EDITED BY
Antonio P. Gutierrez de Blume,
Georgia Southern University,
UnitedStates
REVIEWED BY
Vahid Nimehchisalem,
Putra Malaysia University,
Malaysia
Kathryn Holmes,
Western Sydney University,
Australia
*CORRESPONDENCE
Matthias Borgstede
matthias.borgstede@uni-bamberg.de
SPECIALTY SECTION
This article was submitted to
Teacher Education,
a section of the journal
Frontiers in Education
RECEIVED 02 November 2022
ACCEPTED 18 January 2023
PUBLISHED 09 February 2023
CITATION
Borgstede M and Rau C (2023) Beyond quality
and quantity: Representing empirical structures
by embedded typologies.
Front. Educ. 8:1087908.
doi: 10.3389/feduc.2023.1087908
COPYRIGHT
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Borgstede and Rau 10.3389/feduc.2023.1087908
Frontiers in Education 02 frontiersin.org
to one another. In other words, they focus on the dimensions by which
the concepts dier and their geometric properties.
We argue that acknowledging that meaning and topology are just
dierent aspects of the semantics of scientic concepts is the key to a
general framework for integrating qualitative and quantitative research.
Qualitative research captures the essential properties of scientic
concepts by means of abstract typologies. Quantitative research captures
the relations between dierent scientic concepts by means of
geometric spaces.
We develop a corresponding methodological framework that
integrates both perspectives to answer the following questions: (a)
How do qualitative representations (typologies) relate to quantitative
representations (geometric spaces)? (b) How can the former
beproperly embedded into the latter? e focus of our analysis is a
common research problem that arises when the results of a qualitative
study are used to guide quantitative model building and test
construction. Nevertheless, the methodology can easily beadapted to
other mixed methods designs. For example, a study may start with
quantitative questionnaire data and then proceed with a qualitative
investigation to further explore the underlying empirical structure.
Once the qualitative data has been collected, our framework may
beapplied in exactly the same way as if the qualitative study had not
been informed by questionnaire data. In addition to the quantitative
embedding of qualitative types, the resulting geometric space may
be used to revise the original questionnaires and inform further
quantitative inquiry.
In the following sections, wewill rst elaborate on the general
relation between (qualitative) case-based and (quantitative) variable-
based models from a representational perspective (section 2). Wewill
proceed to characterize both approaches with an emphasis on explicit
and implicit model properties (sections 3, 4). Wewill then rene the
representational view to develop a methodological framework for
integrating qualitative and quantitative research (section 5), and
demonstrate the feasibility of our framework by an empirical case
study about teachers’ epistemological beliefs (section 6). Finally,
wediscuss the implications of our analysis with regard to further
methodological developments and possible applications (section 7).
2. Qualitative and quantitative
representations
Borgstede and Scholz (2021) argue that qualitative and quantitative
research employ two dierent, yet compatible, ways to describe
empirical relational structures. Whereas the qualitative strategy uses a
case-based approach to characterize individuals, the quantitative strategy
applies a variable-based approach to characterize attributes and their
functional relations (Ragin, 1987; Rihoux and Ragin, 2009). In many
cases, however, both modeling strategies can beapplied to one and the
same empirical structure. Consequently, qualitative (case-based) models
oen imply a quantitative structure that is distinct from the observed
qualia. Similarly, quantitative (variable-based) models oen imply a
qualitative structure that is distinct from the variables that are used to
represent the data.
To illustrate the relation between qualitative and quantitative
representations, let us imagine a simple empirical structure consisting
of two kinds of objects—a collection of cases and a collection of
observations. e cases may beindividuals, groups, texts or instances of
any other category. e observations may bedistinct behaviors, answers
in a standardized test, utterances in a conversation or any other class of
attributes of the cases under study.1
If the cases are individuals and the observations are the answers in
a competence test, wemight represent the observed empirical structure
by means of a quantitative model involving a single dimension. One of
the most commonly used models for such structures is the Rasch model
(Rasch, 1960). e Rasch model postulates a single quantitative
dimension by which individuals and test items can becompared, such
that higher dierences between an individual and a test item result in a
higher probability of a correct answer. is probabilistic relation makes
some empirical structures (i.e., answer patterns in a test) more likely
than others. Hence, if an observed empirical structure is not too unlikely
given the Rasch model, wecan use the structure to statistically estimate
the corresponding model parameters. is scaling procedure transforms
the empirical structure that consisted of individuals and test items into
a new abstract structure that consists of points in a unidimensional
geometric space – a so-called latent variable. Wehave thus constructed
a numerical representation of an empirical structure. Such
representations are called measurement (Krantz et al., 1971).
Measurement, as described above, is the foundation of any quantitative
science. However, it is important to note that proper measurement has
to begrounded in suitable empirical structures.2
If the cases are, for example, work teams and the observations are
utterances in a discussion, wewould rarely apply a Rasch model to
represent the empirical structure (although, technically, this would
bepossible). In such a case the more intuitive approach would beto group
the dierent utterances according to their semantic similarity and to
group the teams such that they maximally dier with respect to the
semantic content of their utterances. Instead of mathematical model
equations and statistical estimation techniques, such a grouping usually
relies on interpretative acts from the part of the researchers. However, the
result is also a representation of the empirical structure, the main
dierence being that instead of numbers wehave used words, or abstract
concepts, to guide our interpretation of the topics and themes that have
been discussed by the work teams under study. e interpretative act
transforms the empirical structure that consisted of teams and utterances
into a new abstract structure that consists of categories and types. Wehave
constructed a conceptual representation of an empirical structure. Such
representations are commonly called typologies or, if the grouping is only
performed over the topics rather than the cases, category systems or
patterns. Typologies and category systems are the foundation of all
qualitative science. Like measurement, such representations have to
begrounded in empirical observations (Flick, 2014). In light of these two
examples, quantitative and qualitative approaches seem not so dierent
aer all. Both rely on empirical structures consisting of cases and
observations. And both provide means to construct abstract
representations of these empirical structures. However, categories are not
dimensions and typologies are not geometric spaces.
Qualitative and quantitative research can beconsistently interpreted
as specic modeling strategies. ey may even both beapplied to one
1 Mathematically, cases and observations are two dierent kinds of objects that
jointly form a relational structure.
2 In fact, there are many instances of so-called “quantitative” social science
that just define variables ad hoc, i.e., without establishing an empirically grounded
measurement model to begin with. See Michell (1999) for an in depth discussion
of the problems arising from such pseudo-quantitative science.

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Frontiers in Education 03 frontiersin.org
and the same empirical structure. Nevertheless, the kinds of
representations (or models) they produce are substantially dierent. e
qualitative strategy emphasizes the meaning of concepts, whereas the
quantitative strategy emphasizes the respective topology. Consequently,
there is no straightforward way to “translate” a qualitative model into a
quantitative model. For example, using the abstract description of a type
to inspire a collection of test items that are then scaled by a psychometric
model, would not acknowledge the structural dierence between
qualitative and quantitative modeling approaches. Such a procedure
would imply to abandon the typology in favor of a quantitative model,
rather than to incorporate the strengths of both kinds of models.
As an alternative, wesuggest to translate between qualitative and
quantitative models by tracing them back to the empirical structures
they are meant to represent. If an empirical structure allows for both, a
qualitative and a quantitative representation, the common empirical
grounding of the representations ensures that they can becombined in
a joint representation where qualitative types are embedded in
quantitative dimensions.
e key to integration in mixed methods research is thus to
approach the data from a qualitative and a quantitative perspective
simultaneously. Whereas the qualitative perspective emphasizes the
similarity between members of one type and dissimilarity between
members of dierent types, the quantitative structure emphasizes the
gradual transitions between dierent values on a quantitative
continuum. However, any sorting by similarity implicitly presumes that
there must besomething with respect to which the objects dier. And
since objects can bemore or less similar, this something has at least
some properties of a quantitative dimension. On the other hand,
whenever there are gradual transitions between objects on a quantitative
continuum, it is possible to identify some objects that are more alike
with respect to this dimension than others. erefore, any quantitative
dimension allows for grouping of objects by similarity.
It is easy to see the connection between qualitative and quantitative
representations when objects only dier with respect to a single
criterion. For example, in the context of developmental psychology,
we may represent individual change over the life span by means
developmental stages, as proposed by Piaget (1952). Although the
concept of a developmental stage is clearly qualitative, it is inherently
linked to the concept of cognitive ability, which is conceived as a
quantitative continuum. us, although there may be qualitatively
dierent developmental stages, these stages can be located on a
quantitative dimension.
e connection between quality and quantity is somehow less
obvious if the empirical structure is more complex. For example, a
qualitative reconstruction of teachers’ beliefs will most certainly
consider several criteria of similarity simultaneously. By denition,
beliefs are complex conglomerates of attitudes, thoughts and behavioral
dispositions. As a result, the dimensional structure that implicitly
underlies a qualitative typology of teachers’ beliefs is obscured by the
complexity of the eld. A trained qualitative researcher may well identify
relevant similarities and dissimilarities between the cases. However, it is
dicult to construct an underlying quantitative structure from the
typology alone.
e above analysis suggests that a special methodology is needed to
identify implied quantitative dimensions underlying qualitative
typologies. In the following sections, we shall provide such a
methodology. Since our approach requires a profound understanding of
both, qualitative and quantitative research strategies, we will rst
elaborate on the specics of qualitative type formation [with an emphasis
on reconstructing “pure types” as characterized by Weber (1904)] and
of similarity-based quantitative models (particularly cluster analysis and
linear discriminant analysis). Wewill then outline a general strategy for
the embedding of qualitative types in quantitative dimensions.
3. Constructing qualitative types
As outlined above, qualitative research is mainly concerned with the
construction of case-based models. Case-based models abstract from
singular observations to construct a more general descriptive scheme
for the objects under study. For example, if the objects under study are
work teams, each team constitutes a singular case. However, each case
may also beinterpreted as a specic instance of a more general, abstract
type, which is abductively constructed from qualitative categories that
build on comparisons between and within cases. In general, the criteria
for these comparisons are not known a priori, but emerge from an
iterative process of constructing and revising categories (Peirce, 1998;
Schurz, 2008).
ere are various methodological approaches to the construction of
abstract typologies from singular cases (Kluge, 2000). In this article,
wefocus on the documentary method, which analyses qualitative data
with regard to the way how people talk about certain topics, rather than
what they say. e rationale behind this shi of focus is the observation
that sometimes people’s verbal statements seem to contradict their
actions. For example, when asked about sustainable behavior, a person
may state that the environment is extremely important to her.
Nevertheless, she may still fail to implement her stated attitudes in her
actions (e.g., taking a hot bath every day instead of a shower or traveling
by plane rather than by train). e idea behind the documentary
method is that any disparity between what people say and what people
do will become manifest in the way people talk about a topic. ese
dierent modes of dealing with a topic in a conversation are then used
to reconstruct general patterns of orientation (Bohnsack, 2010). For
example, a person who reports a positive attitude toward sustainable
behavior may deal with the topic by emphasizing the political dimension
of sustainability and thereby downplay the role of the individual. A
dierent mode of dealing with the topic would beto point toward other,
supposedly more important or more urgent, problems such as poverty
or war. Both modes point toward dierent patterns of orientation
(“questioning responsibility” vs. “questioning relevance”), which both
reveal that the stated positive attitudes most likely dier from actual
behavior. e documentary method aims to identify such patterns of
orientation to account for the oen-observed mismatch between verbal
statements and actual behavior and to infer what may bethe true
motivating forces of peoples’ behavior.
In general, the patterns of orientation in a specic context all deal
with a common theme – the so-called tertium comparationis. In the
documentary method, the tertium comparationis provides the
interpretative framework for all consecutive analyses. Within this
framework, the cases are interpreted as specic realizations of
qualitatively dierent ways of dealing with the tertium comparationis,
i.e., dierent patterns of orientation. For example, “questioning
responsibility” and “questioning relevance” are two qualitatively dierent
ways to deal with the common theme of “rationalization of unsustainable
behavior.” e patterns of orientation are then further condensed into a
collection of pure types, which together form a qualitative typology that
intends to capture all cases within a common interpretative framework.
A pure type is not just a descriptive category of what has been observed

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Frontiers in Education 04 frontiersin.org
empirically. It is a theoretical abstraction that transcends the singular
cases to form an idealized concept that is reected in the singular cases
but cannot bereduced to them (Weber, 1904). For example, in the
context of sustainable behavior, pures type like the “ignorant hedonist”
or the “cynical fatalist” may becharacterized in such a way that they
corresponds to only few (if any) actually observed cases, and yet capture
an essential qualitative mode of dealing with the topic of sustainability.
e iterative process of between-cases and within-cases comparisons
makes it possible to explicate the pure types in terms of the essential
features by which they dier – the horizons of comparison that emerge
together with the typology (Bohnsack, 2010). ese horizons of
comparison provide the means to distinguish between the individual
cases with respect to the qualities captured by the typology. Like the
tertium comparationis and the types themselves, the horizons of
comparison are not known a priori but are the result of an iterative
interpretative process of comparison. All types show dierent ways of
dealing with the tertium comparationis. e horizons of comparison
identify the essential properties by which the types dier. For example,
the type “ignorant hedonist” may dier from the type “cynical fatalist”
with respect to various horizons of comparison, such as the amount of
self-ecacy or the amount of social orientation.
e documentary method builds on extensive comparisons within
and between cases. ese comparisons ensure that the theoretical
constructions of the researcher are actually grounded in the empirical
material, and that other researchers can retrace their interpretation of
the data. However, due to theoretical samplings strategies (Glaser and
Strauss, 1979), the documentary method only works with a small or
intermediate number of cases, which in turn limits its empirical scope
(Bohnsack, 2010). Furthermore, although the documentary method
provides a highly systematic rationale for qualitative data analysis, it is
not guaranteed that dierent researchers would arrive at the same results
in a specic context. Working in research groups and validating the
individually obtained interpretations and theoretical constructions
against the critical view of the other members of the research group
helps to deal with this problem. However, the results of a Documentary
analysis will never beindependent of the conducting researchers (i.e.,
objective).
e result of a qualitative analysis based on the documentary
method is a theoretically rich typology consisting of an idealized
description of the ways that individual cases deal with a certain theme.
Since all theoretical concepts emerge from the data by means of constant
empirical comparisons, the documentary method is especially useful as
a method of theory construction from observations by means of
inductive and abductive reasoning.
4. Constructing quantitative
dimensions
Whereas qualitative research focuses on case-based models that
abstract from singular observations to idealized typologies,
quantitative research builds on variable-based models that emphasize
the distances of case representations on the dimensions of geometric
spaces. However, since any geometric space allows for the calculation
of distances between arbitrarily positioned objects, it is always possible
to compare objects with respect to their geometric representation and
sort them based on their similarities or dissimilarities. In fact, there
are various quantitative methods that start with the representation of
objects in a geometric space to group them according to their distance
with regard to this representation. ese methods are subsumed under
the label cluster analysis (Everitt, 1974). Cluster analysis provides a
variety of algorithms to extract groups from the distances of objects in
a geometric space. e groups are called clusters and are constructed
such that objects within the same group are similar and objects that
belong to dierent groups are dissimilar. e similarity or dissimilarity
is measured by a distance-metric. A distance metric is a single number
that is constructed from the relative positions of two points in a
geometric space. Depending on the context, dierent distance metrics
may beappropriate. For example, the distance between two trees on
an open eld may bemeasured by their Euclidean distance (i.e., the
shortest straight line connecting the two trees). On the other hand, the
distance between two houses in downtown Manhattan may bebetter
captured by the city-block metric (i.e., the shortest path a car can take
from one block to the other). In the context of quantitative social
science, the objects that are compared with regard to their distances
are usually individuals. For example, two individuals may bemore
similar to one another than a third one with regard to their answers on
a numeric rating scale in a questionnaire. e squared dierences
between the individuals’ answers to all questionnaire items may then
beused to calculate the Euclidean distance of these individuals in an
abstract variable space that is spanned by the questionnaire items.
Based on the distances between all cases, the individuals are then
grouped into homogeneous clusters.
e results of a cluster analysis are actually very similar to the kind
of representation produced by qualitative type formation. Nevertheless,
the method of construction is completely dierent. Whereas the
qualitative strategy explicates the criterion of similarity by means of
extensive comparisons within and between cases, the quantitative
strategy starts with a set of variables as the basis of a geometric space
and constructs the clusters aerwards. In fact, the result of the analysis
is completely determined by the chosen variables, the used distance
metric, and the clustering algorithm. Consequently, there is no room
for interpretation with regard to the meaning of the clusters or their
essential properties. In other words, the dimensions of the geometric
space used in cluster analysis are chosen before the analysis is
performed and are thus not guaranteed to capture the essential
properties by which the objects under study dier. Sometimes, the
choice of variables is theoretically informed. In other cases, however,
variables are chosen without an explicit theoretical frame of reference.
Consequently, cluster analysis is less sensitive to new theoretical
discoveries than comparative analysis, because the criteria of similarity
are specied before the clusters are being constructed and may thus
bearbitrary with regard to the essential properties of the clusters.
Hence, in contrast to comparative analyses, cluster analysis does not
produce pure types in the above sense. Itmay provide a collection of
classes of objects that can be observed empirically, but are not
necessarily theoretically meaningful.
Despite these apparent shortcomings, cluster analysis comes with
some strong advantages. First, it is mathematically tractable, i.e., both,
the clusters and the dimensions of the geometric space, are explicit
mathematical objects. ey can beprecisely dened and communicated
in an unambiguous way. Second, the automated algorithms used in
cluster analysis ensure that the complete analysis is reproducible once
the variables and the distance metric are given. Finally, it is possible to
perform cluster analysis with an arbitrary number of objects and an
arbitrary number of variables. erefore, given a sucient data basis,
the results of cluster analysis more easily generalize to larger populations
than the results of qualitative type formation.

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5. Embedding typologies in geometric
spaces
Given the strengths and weaknesses of the qualitative and
quantitative approaches outlined above, it would beof great benet to
merge both strategies into an integrated research strategy. Such an
integrated approach would aim to construct theoretically rich
typologies alongside an explicit geometrical representation of their
horizons of comparison as the dimensions of an abstract geometrical
space. It is therefore important to not only perform both types of
analysis separately, but to ensure that all analogous concepts are
continuously translated and back-translated between the dierent
methodological approaches.
In the following, wewill outline how such an integrative strategy can
berealized. Weuse techniques borrowed from qualitative reconstructive
research such as comparative analyses and the construction of pure
types, as well as quantitative modeling techniques like k-means
clustering and linear discriminant analysis. e general rationale is to
start with an exploratory strategy using a qualitative reconstructive
approach, followed by modeling techniques that aim at nding a
quantitative representation of the qualitatively discovered pure types
that are embedded in a theoretically meaningful geometric space.
5.1. Construct theoretically meaningful
types
e rst step of our approach exploits qualitative reconstructive
techniques of type formation. e aim at this stage is to generate a
tentative typology that is both, theoretically rich and empirically
grounded. e results do not yet provide a formal, let alone
mathematical, description of the empirical structure under study. Nor
are they intended to generalize to larger populations. e main purpose
is to identify dierent qualia and to explore their essential features by
comparing them with respect to dierent characteristics. e result
should bea collection of pure types that can bedistinguished by means
of theoretically meaningful horizons of comparison.
e primary techniques used at this early stage of the research are
comparative analyses. From a representational perspective, comparative
analyses consist in identifying relevant empirical relations between the
objects under study. Mathematically speaking, a relation is nothing but
a subset of ordered tuples of objects. In the simplest case, the researchers
will compare every case to each of the remaining cases and judge them
to beeither similar enough to begrouped together or not. More complex
comparisons may include more than two cases at a time. For example,
two cases may besimilar to each other when compared to a third case,
but not when compared to yet another case, because, in the context of
additional cases, the criteria by which one compares the cases may
change. It is also possible to judge cases as being similar with respect to
one characteristic, but dissimilar with respect to another characteristic.
Regardless of the specic comparison procedures, comparative analyses
result in a more or less complex relational structure based on the
judgments of the researchers.
Since the comparative judgments in a qualitative study are based on
interpretative acts, rather than formalized procedures, the resulting
structure is, to some degree, subjective. Nevertheless, it is of course
possible to assess the degree of consensus between dierent researchers
and adjust the judgments such that they are intelligible across
individuals, as it is routinely done in qualitative interpretation groups.
Furthermore, comparative analyses naturally imply that researchers
reect on the kind of comparisons they perform as they proceed to
analyze the data. ereby, theoretically fruitful comparisons are
eventually identied whereas less useful comparisons are abandoned.
is selective component of comparative analyses eventually leads to an
abstract representation of the empirical structure by means of an
emergent typology, with the types being idealized contrasts (pure types)
with regard to theoretically meaningful characteristics (the horizons
of comparison).
5.2. Specify initial set of variables
e second step of the analysis takes the pure types and the horizons
of comparison as a starting point. e aim at this stage is to identify an
initial set of variables that are potentially meaningful with respect to the
theoretical typology constructed in the rst step.
e main technique used in this step is to extract the most relevant
and most specic characteristics of the pure types constructed in step
one and to transform them into a questionnaire. For some characteristics,
this may bestraightforward. Other, more abstract ones, may require
more attention. For example, an abstract concept like “relativism” is way
too vague to include it in a questionnaire. Consequently, researchers
have to partly de-construct the abstract characteristics of the pure types
to arrive at a set of unambiguous characteristics that can betransformed
into questionnaire items (e.g., “What is true for one person may
beuntrue for another person.”). Note that weare not dealing with some
kind of latent variable here–“relativism” is not something unobserved
underlying the concrete characteristics we want to include in the
questionnaire. It is a theoretical abstraction that results directly from
comparative analyses (Buntins et al., 2016). erefore, the relation
between the questionnaire items and the abstract construct is not one of
cause and eect, as implied by latent variable models like classical test
theory or item response theory. It is a logical relation that depends
strictly on the way the researchers use the abstract theoretical vocabulary
that emerged from comparative analyses (cf. Buntins et al., 2017;
Borgstede, 2019; Leising and Borgstede, 2019; Borgstede and Eggert,
2022). us, wecan only decide whether a question belongs in our
questionnaire on theoretical grounds. Since we are interested in
quantitative comparisons in the consecutive steps, it is reasonable to use
some kind of numeric answer type (e.g., a Likert scale). However, note
that the use of numerical scales does not necessarily imply psychological
measurement of an unobserved variable (Michell, 1999).
For the same reasons, standard psychometric criteria like convergent
and divergent validity or internal consistency are largely irrelevant with
regard to the questionnaire resulting from the characterization of the
pure types. In fact, the only relevant criterion for the questionnaire is
that the translation between the abstract theoretical constructs from the
qualitative analysis to concrete statements in everyday language is
successful (cf. Buntins etal., 2017). Like the qualitative analysis itself, the
construction of a questionnaire that is valid in this respect is subject to
interpretation and thus requires a corrective in the form of critical
discussion between researchers.
5.3. Ensure generalizability
Step three consists in assessing the theoretically derived
characteristics from step one in a large sample using the questionnaire

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developed in step two. e aim at this stage is to ensure generalizability
of the typology that was tentatively developed in step one.
e main issue to beaddressed when distributing the questionnaire
is the intended scope of the theory. Since our tentative typology was
constructed using qualitative methods, its empirical grounding will
most certainly belimited to a rather small number of cases. ese cases
are not a random sample but the result of a purposive sampling strategy
called theoretical sampling. eoretical sampling means that cases are
not selected before the data analysis, but as the theory evolves. Such a
strategy implies that data collection, data analysis and theory
development are parallel processes that inuence one another (Glaser
and Strauss, 1979). e intended scope of the theory is thus just as
much an emergent property of comparative analyses as the developed
typology and the horizons of comparison. Consequently, before
administering the questionnaire on a large scale, it is important to
analyze the results of the qualitative analysis with regard to the cases
that were sampled. Most importantly, one has to identify the common
characteristics of the cases (as opposed to the previous step, which
consisted in identifying the dierences between cases). ese
characteristics serve as a rst characterization of the population one
wishes to describe. In a way, they provide the boundary conditions of
the theory that is being developed.
Of course, there are many characteristics that are too general to
beuseful as boundary conditions. For example, the majority of human
subjects have two legs, two eyes, a nose etc. Unless one of these
characteristics is theoretically relevant (e.g., when the typology aims to
describe various forms of discrimination against people with
disabilities), they are rather uninformative and thus useless as boundary
conditions. On the other hand, when the common characteristics of the
cases are too specic, the resulting population may consist of very few
cases, and may even berestricted to only those cases that have actually
been sampled. erefore, an intermediate level of abstraction is required
to produce a workable best guess about the scope of the theory.
Based on these theoretical considerations, the target population for
the questionnaire assessment can eventually be specied. Once the
population is known, the best strategy is to adopt a random sampling
strategy and to sample as many cases as possible. Note that a priori
considerations about statistical power are not applicable, since weare
still in an exploratory stage. Consequently, the aim of the questionnaire
study is not to test statistical hypotheses, but to provide a representative
data source for a quantitative embedding of the qualitative typology
constructed earlier.
5.4. Formalize initial typology
We now enter the rst stage of quantitative modeling. Step four
consists in applying mathematical algorithms to the data obtained in the
previous step that produce a geometric representation of the
characteristics identied in step two alongside a set of empirically
derived clusters. e aim of step four is to produce an initial quantitative
approximation to the pure types constructed in step one.
As outlined in section 4, wepropose to use statistical clustering
methods to identify homogeneous groups of objects based on the
variables constructed from the numerical answers to the questionnaire
administered in step three. Since wealready have a tentative theory in
the form of pure types, we will use an algorithm that produces a
pre-determined number of clusters. e corresponding method is called
k-means cluster analysis (MacQueen, 1967).
e basic idea behind k-means clustering is that the variables
assessed in the questionnaire are interpreted as the dimensions of an
abstract geometric space. e notion of “space” is similar to the standard
use of the word for the three-dimensional space weuse to describe the
position and movement of physical objects. However, an abstract space
constructed from the numerical answers in a questionnaire can have an
arbitrary number of dimensions, each for every question. Moreover, this
abstract variable space does not correspond to a real physical object. It
is barely more than a quantitative representation of an implicit distance
structure. In other words, the abstract geometric space only serves the
purpose of providing a distance metric between objects allocated in the
space. is distance metric is then used to group objects according to
their similarity, where “similar” means that the distance is small and
“dissimilar” means that the distance is large.
e k-means clustering algorithm searches for a partition of the
objects into the specied number of clusters such that the average
deviation of the objects from the center of their assigned cluster is as
small as possible. e standard algorithm starts with an arbitrary initial
partition of objects into k clusters and calculates the mean values of the
objects in each cluster (hence the name k-means clustering). e
partition is then updated by re-assigning each object to the cluster that
is closest with regard to the distance metric. Aer the re-assignment, the
cluster means are re-calculated using the new members of the clusters.
e procedure is repeated until the clusters do no longer change
(Lloyd, 1982).
e result of this k-means cluster analysis is a mathematically
unambiguous partitioning of the cases that were sampled in step three
with regard to the characteristics identied in step two that were
themselves derived from the pure types constructed in step one. Wehave
thus a rst quantitative approximation to the pure types that made up
our tentative theory.
5.5. Formalize horizons of comparison
e steps one to four were concerned with a rst translation between
qualitative types and quantitative dimensions. However, the geometric
space constructed in the previous steps is not yet a faithful formalization
of the horizons of comparison that dierentiate between the pure types.
Neither are the clusters obtained in step four formal counterparts to the
pure types themselves. Due to the data-driven approach of the k-means
clustering algorithm, the clusters correspond to real types, rather than
pure types. ey do not abstract from the raw data on a theoretical level
but rather average over the objects that belong to the same cluster.
erefore, in the last two steps of the analysis, the quantitative
representation constructed so far will be transformed such that the
clusters will bepure types (in the sense that prototypical characteristics
are emphasized), and the corresponding geometric space is spanned by
a set of abstract dimensions that dierentiate maximally between these
pure types.
e method employed at this stage is linear discriminant analysis
(Klecka, 1980). Linear discriminant analysis is a statistical method
that transforms one geometric space into another geometric space,
such that the newly constructed dimensions dierentiate maximally
between groups of objects. In the simplest case, one starts with a
collection of objects that are divided into two groups (say, group A
and group B). Given a set of quantitative variables that describe the
individual objects, the method now calculates a weighted sum over
these variables to obtain a so-called discriminant variable. e

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weights in the summation are chosen such that members of group A
have, on average, low values on the discriminant variable, and
members of group B have, on average, high values on the discriminant
variable. e weights are adjusted until the discriminant variable
dierentiates maximally between the two groups. Linear discriminant
analysis can also beapplied if the objects are divided into more than
two groups. When the groups only dier along one continuum, the
method will still only yield one discriminant variable. Else, the
method will produce a set of several discriminant variables that form
a multidimensional discriminant space.
When applied to the results of a cluster analysis, linear
discriminant analysis yields an alternative, more abstract geometrical
embedding for the clusters. To emphasize the dierences between the
clusters, the discriminant space is routinely projected onto a space of
lower dimensionality (usually, two dimensions are sucient). In
contrast to the initial set of variables used to construct the clusters
(which were merely a best guess about the relevant horizons of
comparison), the discriminant space is constructed such that it
provides the most parsimonious and ecient way to distinguish the
clusters from one another. erefore, the result of step ve is a new
geometric embedding for the clusters obtained in step four, that
corresponds to a set of abstract horizons of comparison by which the
clusters can bedistinguished.
5.6. Formalize pure typology
e nal step of the analysis consists in a formal method to
construct pure types that are embedded in a quantitative geometric
space consisting of a small set of maximally dierentiating dimensions.
e corresponding geometric space has already been constructed in the
previous step. e aim of the last stage of analysis is to use this abstract
geometric space to update the initial clusters constructed in step four.
e linear discriminant space captures the abstract dimensions that
dierentiate maximally between the initial clusters. To construct new,
idealized, clusters (that correspond to the pure types tentatively
proposed in step one) weapply a second k-means cluster analysis—only
this time we use the discriminant variables (instead of the initial
variables) to dene the distance metric. Like before, weuse the same
number of clusters as in our initial typology. Since the discriminant
variables are determined from the initial variables, which in turn were
constructed from the description of the qualitative criteria of similarity
in step two, both, the geometric space and the clusters constructed in
this last step, are still grounded in the empirical structure wewish to
describe. However, in contrast to a simple cluster analysis (that yielded
real types), wenow have clusters in a discriminant space.
Since the discriminant space captures the abstract characteristics
by which the clusters dier, these new clusters are maximally dierent
with respect to the initial grouping. In other words, those
characteristics that strongly dierentiate between clusters are weighed
more strongly than those that dierentiate poorly. Consequently, the
updated clusters are idealizations of the original, empirically derived
clusters. ese idealized clusters capture the relevant horizons of
comparison by emphasizing those aspects in the data that are specic
to the clusters. erefore, the idealized clusters correspond to pure
types as characterized in section 3.
e result of the last step of analysis is a formal reconstruction of a
tentative qualitative typology based on a large sample of cases. is
formal reconstruction captures the implicit quantitative structure
underlying the act of grouping objects by similarity, as well as the
qualitative aspects of the empirical structure that is characterized by
pure types.
6. Exemplary application: Teachers’
epistemological beliefs
In the previous section, wepresented an integrated approach to
embed qualitative pure types in a quantitative geometric space to form
an overarching model of the underlying empirical structure. Our
approach builds on a representational integration of qualitative and
quantitative modeling strategies as outlined by Borgstede and
Scholz (2021).
We outlined how qualitative typologies that emerge from
comparative analyses can beinterpreted as an attempt to construe
abstract relational structures that are grounded in empirical
observations. Building on this abstract conception of qualitative
typologies, weproposed that the horizons of comparison from such a
typology betranslated into a collection of questionnaire items which are
then used as an initial variable space for a k-means cluster analysis.
Using linear discriminant analysis to construct an abstract geometric
space of maximally discriminating dimensions, wethen proposed to
revise the initial cluster solution based on these abstract dimensions.
is procedure yields a formal representation of pure types within a
quantitative space consisting of abstract dimensions.
In this section, wewill illustrate the feasibility of our method by
means of an exemplary application. e application is concerned with
the reconstruction of teachers’ epistemological beliefs and builds on a
published qualitative reconstructive analysis (Rau, 2020, 2021). Wewill
start with a brief review of the theoretical background of the study and
the results of the qualitative type formation. e main part of the
example focuses on the concrete procedure of formalizing the typology
proposed in Rau (2020) using the integration approach outlined above.
All statistical analyses were preformed using R version 4.0.3 (R Core
Team, 2020) with the additional packages janitor (Firke, 2021), MASS
(Venables and Ripley, 2002) and factoextra (Kassambara and
Mundt, 2020).
6.1. Background
Rau (2020) conducted a qualitative reconstructive study using the
documentary method (cf. section 3). e study examined
epistemological beliefs of teachers who teach a humanities subject. e
aim of the study was to describe how teachers generate knowledge
about cultural artifacts in the classroom. e study focused on the
following questions: (a) How do teachers deal with cultural artifacts
such as poems or images in their teaching practices? (b) How do
teachers interact with their students to esh out the meanings of these
cultural artifacts? e data were collected by group discussions (N = 19).
Cases were selected according to interviewees’ characteristics (e.g., level
of education, expert or novice in the teaching profession, subject
studied) following a theoretical sampling strategy (Rau, 2020, 2021).
e documentary method works by abstracting ndings and
nding a common theme that is common to all cases: the tertium
comparationis. e tertium comparationis identied in the study refers

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to the way that teachers justify their understanding of cultural artifacts
and was thus coined justication. Systematic comparisons within and
between cases revealed three basic ways of dealing with justication,
each of which constitutes a pure type in the sense of Weber (1904). e
three types were: (1) Contingency stops, (2) Orientation to application
(3) Appreciation of pluralism. ese three types comprise the basic
constituents of teachers’ epistemological beliefs. ey indicate how
teachers’ epistemological beliefs may guide instructional action in the
humanities and how justication of dierent meanings is ensured.
Teachers of type 1 expect pupils to interpret cultural artifacts within
their historical context of origin and to elaborate on authorial intention
by choosing a deductive method. Pupils have to learn epochal
knowledge and are not allowed to bring their own meanings for the
cultural artifact into the discourse. Teachers of type 2 ask, what personal
meaning cultural artifacts have for the individual pupils. e learners’
ability to articulate their personal meaning of the cultural artifact is a
characteristic of justication for teachers. ese teachers are eager to
learn about dierent meanings that pupils give to the same cultural
artifact. Teachers of type 3 choose dierent (theoretical) perspectives
to look at cultural artifacts. For example, they may interpret cultural
artifacts with a feminist reading. e empirical material shows that
these teachers engage in a discursive classroom conversation with their
pupils to agree on an interpretation of the cultural artifact. It is
important to these teachers that the pupils adopt a critical attitude and
learn to reect on their own about the meaning of cultural artifacts.
Justication is based on intersubjective validation between the teacher
and the pupils.
e within-cases and between-cases comparisons that generated the
tertium comparationis and the pure types also revealed the main
characteristics by which the three types may bedistinguished. ese
characteristics constitute the essential horizons of comparison by which
the identied, abductively formed, qualia dier. ese horizons of
comparison were: (a) Genesis of meaning: Which epistemological
approaches do teachers choose? (b) Certainty and limits of generated
knowledge: What do teachers expect from justication? (c)
Characteristics of the cultural artifacts: What are the ontological
characteristics of cultural artifacts in the humanities? Do teachers
perceive cultural artifacts as ambiguous and/or unambiguous? (d)
Relating teachers to pupils: How do teachers include pupils in the
genesis of meaning and knowledge? To what extent do teachers allow
pupils to discuss their own attributions of meaning in class? (e) Aims of
teaching in the humanities: What are the aims of teachers’ teaching in
relation to epistemic learning of the pupils? e three pure types, as well
as the ve horizons of comparison are summarized in Table1. e table
also shows how the dierent types can bedistinguished with respect to
these horizons of comparison.
6.2. Data
e qualitative typology put forward in Rau (2020) was used to
construct a questionnaire. e questionnaire contained 43 items that
were formulated such that they capture the ve horizons of comparison
that dierentiate between the three pure types. Since these horizons of
comparison were formulated on a high level of abstraction in the
original study, they were rst concretized and translated into everyday
language, such that respondents were able to understand them correctly.
For example, the rst pure type (“contingency stop”) can becharacterized
with respect to the rst horizon of comparison (“genesis of meaning”)
by the fact that teachers aim to convey the meaning that the originator
(supposedly) ascribed to the cultural artifact to their students. e
corresponding item generated to capture this aspect of the horizon of
comparison was: “When pupils interpret pieces of music, literature or
art, it is important that they carve out the author’s intention.” To ensure
item comprehensibility and content validity, the initial item set was
presented to two teachers in the eld of humanities and two experts in
the eld of teachers’ epistemological beliefs. Critical feedback from these
expert judgments was incorporated in a revised item set, which was then
used for a rst empirical study.
e sample consisted of 153 undergraduate students from a
Bavarian University with a focus on teachers’ education and humanities.
e students were recruited in university seminars and lectures. 83.1%
of the respondents identied as female, 15.5% as male, and 1.4% as
diverse. e median age of the participants was 19 years with an inter
quartile range of 4 years. 13.3% of the sample reported a migration
background. 54.9% of the participants were student teachers for primary
schools, 22.5% were student teachers for vocational schools, 18.3% were
student teachers for high schools, and 4.2% reported a dierent type of
school. 19.6% of the students reported at least some kind of teaching
experience, although most students (80.5%) were not teaching at the
time of the study.
e statistical analyses followed the steps described in section 5.
First, a k-means cluster analysis was conducted using the numerical
answers to the questionnaire items as a variable space and the Euclidean
distance as a distance metric. Since the qualitative analysis identied
three pure types, a three-cluster model was t to the data. All data were
TABLE1 Qualitative typology obtained from documentary method.
Type 1:
Contingency
stops
Type 2:
Orientation
to
application
Type 3:
Appreciation
of pluralism
Horizon of
comparison a:
Genesis of
meaning
Historicizing;
monoperspectival
Relativistic Multi-perspectival
Horizon of
comparison b:
Certainty and
limits of
generated
knowledge
Deductive method Extension of
decoding
performance
Advance of
knowledge;
intersubjective
validation
Horizon of
comparison c:
Characteristics
of the cultural
objectivation
Unique;
unambiguous
Ambiguous Ambiguous
Horizon of
comparison d:
Relating teachers
to pupils
Exclusion of
students from the
discourse
Supplementing
teacher
knowledge with
pupils’ views
Conversation about
decoding
Horizon of
comparison e:
Aims of teaching
in the
humanities
Imparting of
epochal knowledge
Search for further
knowledge;
surplus value for
pupils
Practicing a critical
attitude

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standardized within individuals before the analysis. Cases with missing
values were excluded. e so-obtained clusters were then used as
grouping variables in a linear discriminant analysis. Finally, a second
k-means cluster analysis was conducted with the discriminant variables
as a variable space.
6.3 Results and discussion
Figure 1 presents the results of the initial k-means cluster
analysis that was conducted using the questionnaire items as a
variable space. Since the actual variable space contains 43 dimensions
(one for each questionnaire item), the data was projected into a
two-dimensional space to enable a graphical depiction of the clusters.
e two dimensions in the gure were chosen as a reference
coordinate system such that they bind the maximum amount of
variance in the data.3
e three-cluster solution shows that there are two clusters
(cluster 1 and cluster 3) that are completely separate. ese two
clusters show no systematic dierences in the y-axis but are clearly
distinct with regard to the x-axis. Cluster 2 takes an intermediate
position on the x-axis and shows higher average values on the y-axis,
indicating that it diers from the other two clusters with regard to a
distinct dimension. However, cluster 2 has considerable overlap with
the other two clusters, especially with cluster 3. erefore, it remains
open at this stage of analysis, whether it actually captures a dierent
qualitative aspect of the data.
e three clusters from the original variable space were then used
as grouping variables in a linear discriminant analysis. Two discriminant
variables were constructed such that they are independent of one
another and dierentiate optimally between the three clusters. A second
k-means cluster analysis with three clusters was conducted using the
discriminant variables as a variable space (see Figure 2). Since the
principle components coincide with the discriminant variables, the x-
and y-axes of Figure2 can now beinterpreted as the two dimensions
that dierentiate maximally between the clusters and bind a maximal
amount of overall variance of the data.
Like in the initial cluster solution, clusters 1 and 3 dier mainly in
one dimension, which is indicated by the x-axis in Figure2. However,
in contrast to the initial solution, cluster 1 now has a higher overall
within-cluster variance than clusters 3 and 2, indicating that it is less
homogeneous than the other two. Like before, cluster 2 takes an
intermediate position on the x-axis and higher average values on the
y-axis, indicating that it mainly diers from the other two clusters with
regard to a dierent dimension.
In contrast to the initial solution, the updated clusters do no
longer overlap. is is a direct result of the newly constructed variable
space. Since the discriminant variables are specied such that they
maximally dierentiate between the clusters, existing dierences
between the original clusters are emphasized because those
questionnaire items that dierentiate more receive higher weights
when calculating the linear discriminant variables (which are, in fact,
just weighted sums of the original variables). e resulting clusters
are thus idealizations of the original clusters. Just like the pure types
3 Formally, these dimensions correspond to the first two variables obtained
from a principle component analysis (cf. Pearson, 1901).
in a qualitative typology, they are an abstract representation of the
observed qualia rather than a purely descriptive summary as it is
given by real types. Note that the data has not been changed to obtain
this idealized cluster solution. e individual item answers are the
same as before. e only dierence to the initial solution is that the
coordinate system has been changed by means of a linear
transformation (i.e., a weighted summation), such that the coordinates
dierentiate maximally between the clusters. us, the qualitative
dierences that were only partly visible in the initial solution are
revealed, alongside an abstract coordinate system that represents the
primary axes by which the clusters dier.
e last step of the analysis aimed to identify the semantic content
of the three idealized clusters. To get an impression about which
questionnaire items contribute most to the two discriminant variables,
the corresponding weighting factors were inspected. Comparing the
items with the highest and lowest relative weights for each of the
discriminant variables, it turned out that the rst dimension (the
x-axis in Figure2) corresponds to the distinction between a pluralistic
view (e.g., “Pupils should interpret pieces of music, literature or art
from dierent perspectives.”) and a dogmatic view on the meaning of
cultural artifacts (e.g., “It is important that pupils interpret pieces of
music, literature or art in line with established views from the
scientic community.”). e second dimension (y-axis in Figure2)
corresponds to the distinction between an orientation toward the past
FIGURE1
Initial cluster solution projected into two dimensions.
FIGURE2
Idealized cluster solution obtained from linear discriminant variables.

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(e.g., “I usually incorporate a history-dependent view on scientic
knowledge into my classes.”) and an orientation toward the present
with regard to the meaning of cultural artifacts (e.g., “e current
life-world of the pupils aects how they interpret pieces of music,
literature or art.”).
In light of these results, wecan contrast the three idealized clusters
as follows. Cluster 1 and 3 primarily dier on the pluralism-dogmatism
continuum, with cluster 3 leaning toward pluralism and cluster 1 toward
dogmatism. Cluster 2 can becharacterized as leaning toward pluralism,
as well, although less than cluster 3. e main dierence between
clusters 2 and 3, however, is not the degree of pluralism but that cluster
2 is characterized by an orientation toward the present, whereas cluster
3 leans toward a past orientation. is tendency toward the present
includes the current environment of the students and acknowledges the
relevance of the students’ own experiences and perspectives on the
meaning of music, art and literature.
ese characterizations capture essentially the same qualia as the
pure types from the original study. e fundamental horizons of
comparison were also reproduced almost exactly as in the qualitative
analysis. Wecan thus conclude that the qualitative pure types that were
identied by means of comparative analyses can in fact beformalized as
idealized clusters in a quantitative geometric space. e resulting
representation converges nicely with the qualitative analysis, thereby
sharpening the verbal descriptions within a mathematical model that
builds on data from a larger sample. e model captures both, the
qualitative and the quantitative aspects of teachers’ epistemological
beliefs, because it builds on a methodologically well-founded integration
of qualitative and quantitative research strategies within a
representational framework.
7. Conclusion
is article dealt with the question how qualitative and quantitative
research strategies can beintegrated such that qualitative types and
quantitative dimensions are represented within the same overarching
model. We argued that a true integration can only beachieved if
qualitative and quantitative modeling strategies are viewed in light of
a common methodological framework. e representational approach
put forward by Borgstede and Scholz (2021) provides such a
methodological background. In this paper, we rened the
representational approach and applied it to an empirical case study,
thereby demonstrating how qualitative and quantitative methods can
be merged to produce formal representations that capture both,
qualitative and quantitative, aspects in the data and integrate them
within a single model.
Our approach transcends the distinction between qualitative and
quantitative research by providing a common conceptual framework.
Within this framework, it is possible to translate between qualitative and
quantitative modeling approaches and to facilitate the simultaneous
discovery of both kinds of structures. For example, concepts like “pure
types” generally have no meaning in quantitative research. However,
from a representational perspective, pure types can beconceived as
idealized clusters in an abstract geometric space. Similarly, the concept
of a “distance metric” has no meaning in qualitative research. However,
in light of the representational view, a distance metric is just a formalized
version of the criterion of similarity or dissimilarity used to compare the
objects under study.
In this article, we focused on the question how qualitative
typologies can be embedded in a quantitative geometric space.
However, our approach provides a far more general rationale for the
construction of new research designs. e essential point is to realize
that qualitative and quantitative models are just dierent kinds of
abstract relational structures and that they both attempt to represent
empirical relational structures. Following this rationale, qualitative
comparisons within and between cases may be considered as the
empirical basis for various quantitative scaling techniques. Similarly,
quantitative representations may beexploited to extract qualitative
distinctions within and between cases. For example, the similarity
judgments of qualitative researchers may beused as primary data for
the construction of an abstract feature space by means of
multidimensional scaling (Borg etal., 1997). e abstract feature space
can then becompared to the horizons of comparison derived from
comparative analyses. Other applications might include psychometric
models with qualitative components (like multi-group Rasch models
or latent class analysis, cf. von Davier and Carstensen, 2007), the
embedding of qualitative contingencies in an abstract variable space by
means of correspondence analysis (Hirschfeld, 1935), or the application
of mathematical algorithms to identify specic similarities between
cases by means of formal concept analysis (Ganter and Wille, 1999).
e integration strategy outlined above requires profound
knowledge of both, qualitative and quantitative, modeling techniques.
In particular, the translation between dierent kinds of models depends
on an abstract understanding of the empirical and theoretical structures
involved in the analysis. Such an abstract understanding requires a level
of formalization that is rarely achieved in empirical research, let alone
in the context of theory building in educational science. Although
formal approaches to empirical research and theory formation may
bechallenging and sometimes seem cumbersome, wethink that they
are worth the eort. Our research example shows how the results from
qualitative and quantitative analyses to the same data may converge
using our representational approach. Moreover, the study shows that
the proposed strategy of integration enriches the theoretical scope of
the quantitative model components and scrutinizes the semantic
import of its qualitative aspects.
e representational approach emphasizes the similarities between
qualitative and quantitative research strategies and provides a
metatheoretical framework to identify relevant dierences at the same
time. Wehope that this overarching perspective will not only nd its
way into mixed methods research, but also facilitate communication
and foster mutual exchange between qualitative and quantitative
researchers in general.
Data availability statement
e raw data supporting the conclusions of this article will bemade
available by the authors, without undue reservation.
Ethics statement
Ethical review and approval was not required for the study on
human participants in accordance with the local legislation and
institutional requirements. e patients/participants provided their
written informed consent to participate in this study.

Borgstede and Rau 10.3389/feduc.2023.1087908
Frontiers in Education 11 frontiersin.org
Author contributions
MB conceived the conceptual background, developed the
methodology, conducted the statistical analyses, made the visualizations,
and wrote the rst dra of the manuscript. CR veried the conceptual
background and collected the data. All authors contributed to the article
and approved the submitted version.
Acknowledgments
We thank Jana Costa for her help with the formulation of the
questionnaire items.
Conflict of interest
e authors declare that the research was conducted in the absence
of any commercial or nancial relationships that could beconstrued as
a potential conict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and
do not necessarily represent those of their aliated organizations, or
those of the publisher, the editors and the reviewers. Any product that
may be evaluated in this article, or claim that may be made by its
manufacturer, is not guaranteed or endorsed by the publisher.
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