Geometric distance and semantic structure: A methodological innovation for mixed methods research with HUDAP and ATLAS.ti

Abstract

This article presents a rigorous and replicable methodological alternative for articulating quantitative and qualitative analyses within mixed-methods research, contributing to interpretive coherence by relating quantitative proximity structures to inductively derived qualitative codes. The proposed approach integrates minimum-distance and mean-value analysis (employing WSSA1 and POSAC via HUDAP software) with qualitative content analysis in ATLAS.ti to construct cohesive interpretive categories. It enables the alignment and meaningful links of Euclidean clusters of quantitative variables with inductively derived qualitative codes, enhancing the interpretive coherence of complex datasets. Through a case study in Technology and Informatics teacher education, the article demonstrates how a didactic unit based on scientific instrument construction fosters professional competencies in pre-service educators. Grounded in abductive reasoning, this integrated strategy offers a methodological contribution to mixed-methods research by bridging geometric and semantic analyses within a transferable educational framework.

• Provides a systematic, replicable model linking Euclidean clusters (HUDAP: WSSA1/POSAC) with inductively coded qualitative categories (ATLAS.ti).

• Contributes to interpretive depth and internal validity through methodological triangulation.

• Offers a transferable application for STEM teacher education by fostering competencies through hands-on instrument design.

Introduction

Within the framework of mixed-methods research, one of the enduring challenges (stemming from fundamental epistemological differences) is the coherent articulation between quantitative and qualitative analyses. While the quantitative approach prioritizes the objective measurement of variables, qualitative analysis is oriented toward the contextualized interpretation of meaning. This dissociation has prompted the development of methodological strategies aimed at fostering a deeper integration between both paradigms, particularly in educational research where it is essential to grasp both the structure and significance of complex phenomena (Leech and Onwuegbuzie, 2009; Mauceri, 2016).

Contemporary mixed-methods scholarship frames integration not as the mere coexistence of qualitative and quantitative strands, but as the deliberate and explicit relating of components so that they become mutually illuminating and yield findings that are greater than the sum of their parts (Bazeley, 2017). From this perspective, integration is strengthened when it is enacted through the analysis process (i.e., when data structures, analytic steps, and inferential moves are intentionally connected) rather than being postponed to a narrative synthesis at the level of conclusions. The contribution of the present study is situated within this analytic view of integration; specifically, geometric proximity patterns derived from quantitative variables (HUDAP) are used as a structural scaffold to be iteratively examined alongside inductively developed qualitative codes and categories (ATLAS.ti), so that the relationship between variable-oriented regularities and process-oriented meanings is made transparent and auditable. In doing so, the paper aims to advance an integration mechanism that goes beyond treating qualitative findings as post hoc explanations of correlations, by specifying how cross-strand linkage is constructed and how meta-inferences are warranted (Bazeley, 2017, 2024).

This paper proposes to contribute to this ongoing dialogue by proposing an innovative methodological articulation strategy that combines the use of HUDAP software (specifically its Weighted Similarity Structure Analysis (WSSA1) and Partial Order Scalogram Analysis with base Coordinates (POSAC)) with qualitative content analysis supported by ATLAS.ti. The proposal is grounded in the principle that quantitative variables can be represented as points in Euclidean space, enabling the application of essential geometric criteria to identify latent structures and minimum distances between items (Amar and Toledano, 2001). These structures can be organized into meaningful clusters, which are subsequently linked to emergent categories from qualitative analysis, thereby facilitating interpretive convergence (Finfgeld-Connett, 2014; Franzosi et al., 2013; Oleinik et al., 2014; Ramírez Cano, 2019).

The study adopts an exploratory-sequential design guided by abductive logic, allowing for interpretive inferences that oscillate between empirical induction and theoretical deduction (Richardson and Kramer, 2006; Verd and Lozares, 2016). This logic has proven particularly effective in contexts requiring the integration of heterogeneous data. The case study is situated within a teacher training process in the field of Technology and Informatics, implemented through a didactic unit (DU) focused on the construction of scientific instruments, which enabled the collection of both quantitative and qualitative data (Oleinik et al., 2014).

Accordingly, the purpose is not only to present the empirical outcomes of this experience but also to test a methodological model that offers a rigorous, replicable, and epistemologically grounded alternative for linking qualitative interpretations with quantitative data (Saunders et al., 2018). The combined use of HUDAP and ATLAS.ti not only facilitates this articulation but also opens up new possibilities for analyzing complex data through simultaneous structural and semantic representations. As such, this proposal positions itself as a methodological contribution to the field of educational research, especially in settings that demand the integration of diverse forms of evidence.

The methodological proposal presented in this study offers a substantial contribution to educational research conducted under mixed-methods paradigms, particularly for those investigations that face the challenge of articulating qualitative and quantitative data without sacrificing interpretive depth or structural rigor. Although applied in a teacher education context, its utility extends to evaluative studies, research in pedagogical innovation, analyses of professional practices, and explorations of formative processes across various educational levels. The abductive approach that underpins this integration enables the construction of interpretive relationships between data structures and emergent meanings, rendering it a viable alternative for complex studies requiring multiple, articulated levels of analysis.

Theoretical framework

This study employs key methodological concepts that are essential for understanding the proposed approach. Abduction, understood as an inferential logic that enables the generation of interpretive hypotheses from empirical data, guides the articulation between qualitative and quantitative methods. In turn, methodological articulation refers to the process of operational convergence between emergent semantic structures and metric patterns derived from the applied instruments. Within this context, essential geometry is employed (via the HUDAP software and its tools WSSA1 and POSAC) as a strategy for representing variables in Euclidean space, facilitating the identification of structural cores within the data. These concepts are revisited consistently throughout the article to support the methodological proposal and its empirical application (Ramírez Cano, 2019; Saunders et al., 2018).

Contemporary educational research faces the challenge of integrating diverse ways of knowing. In this regard, mixed-methods approaches offer an opportunity to transcend the limitations of exclusively quantitative or qualitative paradigms by combining measurement tools with interpretive strategies (Leech and Onwuegbuzie, 2009; Mauceri, 2016). However, such integration is not without tensions, particularly given that each approach responds to distinct epistemological paradigms: positivism/post-positivism in the case of quantitative analysis, and constructivism/interpretivism for qualitative inquiry (Denzin and Lincoln, 2012). This fundamental condition demands the development of strategies that enable genuine articulation between methods, fostering dialogue grounded in the logics of each paradigm.

One of the most effective tools for achieving such articulation is the use of abductive logic. According to Richardson and Kramer (2006), this strategy refers to the process of examining facts and formulating a theory to explain them. Through this process, useful explanations are identified, which essentially constitute inferences from observed phenomena. As such, abduction occupies an intermediate space between deduction and induction.

In studies involving complex data, abduction functions as a bridge between empirical induction and theoretical deduction, giving rise to interpretations that emerge not solely from the data nor exclusively from pre-existing frameworks, but rather from their dynamic intersection (Santiago-Delefosse et al., 2015; Verd and Lozares, 2016). This logic proves especially valuable when combining quantitative structural procedures with emergent qualitative categories, as proposed in this study. As illustrated in Figure 1(a), the research findings are situated midway between the ascending path of empirical discovery and the descending trajectory of theoretical constructs.

Figure 1.

(a) Procedure of abductive research and (b) dynamics of abductive inquiry.

Moreover, as depicted in Figure 1(b), the starting point of this approach is the research question that prompted the formulation of the problem and the definition of intended outcomes. Based on these objectives, a methodological design is developed to determine the instruments to be employed. Once the instruments are defined, fieldwork is conducted to gather empirical data for subsequent analysis, interpretation, and presentation through data triangulation (Campos, 2009; Ramírez Cano, 2019). It is worth noting that this process includes essential feedback loops that enhance the analysis and interpretation of results. Rather than claiming a holistic capture of the phenomenon, the procedure is presented as an explicitly traceable integration pathway that links quantitative structures to qualitatively evidenced meanings, while acknowledging that any integrated account remains partial and contingent on the operationalizations used. This positioning is also compatible with the view that methods can be performative, insofar as analytic choices shape what becomes visible as an integrated account rather than revealing a single totality.

From this perspective, the analysis of geometric distances using HUDAP-WSSA1 becomes particularly relevant. Amar and Toledano (2001), in the HUDAP user manual, explain that this technique is based on the principle that each variable can be represented as a point within a Euclidean space, and that the relationships among these variables are expressed through relative distances, rather than merely linear correlations. The WSSA1 tool enables the identification of structural similarity patterns among items, generating groupings that can be visualized as spatial configurations or semantic networks, thereby facilitating the detection of thematic clusters based on internal affinities.

The choice of the WSSA1 tool responds to the need to move beyond traditional statistical analyses that are limited to linear correlations among variables. Unlike more conventional techniques such as factor analysis or linear regression, WSSA1 allows variables to be represented as points in Euclidean space, thereby enabling the analysis of their relationships in terms of geometric distance. This feature is particularly valuable for identifying latent structures and conceptual groupings within the data, allowing for the detection of similarities among items that may not be evident through classical correlational methods. Its capacity to identify structural cores, particularly within Likert-type scales, provided a solid foundation for its subsequent articulation with emergent qualitative categories derived from content analysis (Amar and Toledano, 2001; Sayago, 2015).

In the quantitative strand, “geometric distances” are calculated between questionnaire items (aspects), not between abstract themes. The dataset is a rectangular matrix in which rows correspond to participants (n = 12) and columns to aspects/items (n = 24) scored on Likert-type scales. HUDAP first computes Pearson correlations for all item pairs, producing a symmetric intercorrelation matrix R_if. WSSA1 then represents each item V_i as a point in a Euclidean space and estimates inter-item distances such that higher positive association corresponds to shorter distance (and weaker association to larger distance), while preserving the monotonic ordering between input associations and output distances as closely as possible. In practical terms, two items are “close” when participants tend to score them similarly across cases, and “far” when their response patterns diverge.

In plain terms, the “thematic clusters based on internal affinities” are groups of questionnaire items that tend to behave similarly across participants. After WSSA1 positions each item as a point in a two-dimensional Euclidean space, items that appear close to one another form local regions in the map because their response patterns are highly correlated. These regions are interpreted as clusters “with internal affinities” because the items within the same region share stronger mutual associations than with items outside the region. The “latent structures” referred to in this paper are therefore latent regularities in the quantitative response patterns among items (i.e., the underlying grouping implied by the inter-item correlation structure), not latent qualitative meanings. These quantitative clusters are then interpreted alongside the qualitative categories to explain what the observed response-pattern groupings may represent in substantive terms. Figure 2 presents an example of the tool’s output.

Figure 2.

Two-dimensional space diagram. HUDAP-WSSA1.

The analysis conducted using WSSA1 can be complemented with the application of the POSAC tool, also available in the same software suite. To organize the correlation of the items, POSAC generates groups based on the mean rating across participants for each aspect (n=24) and presents them from highest to lowest according to the results. Since the averages of some aspects may coincide, the program places them in the same group; this organization is referred to as “profiles.” It is important to clarify that these profiles are a POSAC-derived ordering and grouping of the same questionnaire aspects (items), intended as an analytic reference rather than as a new set of constructs. To distribute these profiles in a two-dimensional diagram, POSAC calculates new coefficients according to the monotonicity of the values obtained in the profiles. These values are placed along a positive diagonal, with the weakest profile at the bottom and the strongest at the top. The remaining profiles are then distributed from lowest to highest under this condition (Amar and Toledano, 2001; Ramírez Cano, 2019). Figure 3 displays an example of the output generated by the POSAC tool.

Figure 3.

Example of POSAC output.

To enhance the analysis, qualitative content analysis, supported by software such as ATLAS.ti, enables the identification, coding, and interpretation of emergent units of meaning derived from the discourses and texts under examination. This type of analysis is grounded in a categorical logic that does not rely on pre-established structures; rather, it constructs inductive analytical frameworks based on the density and recurrence of codes, which, for methodological purposes, are referred to as categories (Ramírez Cano, 2019; Saunders et al., 2018).

The strength of this technique lies in its ability to capture the symbolic, contextual, and experiential dimensions of the phenomenon under study, providing a deep interpretive layer that complements the structural perspective offered by quantitative data. Figure 4 illustrates an example of qualitative data analysis using the software.

Figure 4.

Example of analysis using ATLAS.ti.

Ultimately, the true innovation of this proposal lies in its ability to articulate both strategies within an abductive logic, allowing for the comparison and interconnection of quantitative structural groupings with emergent qualitative categories. This methodological intersection not only enhances triangulation, but also enables new ways of constructing meaning from complex data.

The key lies in recognizing that both types of analysis (geometric and semantic) operate on relational structures. In other words, while one works through distances between items, the other engages with densities among meanings. Here, references to “density” are used only as a descriptive indicator of where interpretive attention concentrates in the corpus; they are not treated as a measure of importance or validity. Low-frequency codes may still be analytically significant, especially when they index boundary conditions or discrepant cases. Together, they provide a more transparent and analytically warranted representation (within the limits of the available evidence) of the phenomenon, validated both empirically and interpretively (Ramírez Cano, 2019).

Methodology

To demonstrate the innovative potential of the methodological proposal presented herein, this article draws upon a recently completed doctoral research project in the field of technology education. The study focused on analyzing the development of professional competencies in pre-service teachers through an interdisciplinary didactic unit centered on the construction of low-cost scientific instruments. The methodology employed, along with the results obtained, provides the empirical framework from which the strategy for articulating quantitative and qualitative analysis is outlined and exemplified. Accordingly, the methodological design implemented in this research is presented, highlighting the elements that enable operational convergence between structural and semantic techniques as part of a mixed-methods approach guided by abductive logic (Ramírez Cano, 2019; Ramírez-Montoya and Lugo-Ocando, 2020).

The study is framed within a mixed-methods design. The combination of qualitative and quantitative approaches in such a framework can be analyzed from various perspectives regarding their organization and integration, such as: the timing of method implementation (sequential, parallel, or iterative), the methodological weighting (equal status or dominant status), and the moment of integration (during data collection, analysis, and interpretation) (Leech and Onwuegbuzie, 2009; Ramírez-Montoya and Lugo-Ocando, 2020). In this research, integration was enacted during analysis and consolidated during interpretation through an explicit linking procedure that aligns HUDAP-derived region membership with ATLAS.ti coding profiles, under a qualitative-priority orientation guiding interpretive decisions. Finally, the concluding stage involved a joint interpretation of qualitative and quantitative evidence to support warranted meta-inferences.

Moreover, the study employed an abductive approach, as it supports the structured integration of quantitative and qualitative data, allowing for the contrast between empirical patterns and emergent categories, as previously discussed. This strategy is justified by the need to overcome the traditional methodological fragmentation between paradigms, thereby facilitating a holistic understanding of the educational phenomenon under investigation: the development of teachers’ professional competencies through a constructivist-oriented didactic experience (Ramírez Cano, 2019). Table 1 summarizes the methodological proposal.

Table 1.

Summary of the methodology employed in the study.

Methodological framework	Research strategy	Research method
Mixed methods	Abductive	Case study
Data collection instruments and processing techniques	Tools	Validity and rigor criteria
Questionnaires, specific tests, and interviews. (SLEI and Likert scale—Cronbach’s Alpha)	ATLAS.ti, SPSS, HUDAP (WSSA1 and POSAC)	Onwuegbuzie and Leech Model (2007) and Guba and Lincoln

Source: Author’s own elaboration.

The target population of this study consisted of pre-service teachers in the area of Technology and Informatics, enrolled in the Bachelor of Electronics Education program at the Universidad Pedagógica Nacional de Colombia. The intervention was carried out through the design and implementation of a didactic unit (DU) centered on the construction of a scientific instrument focused on the generation and detection of electromagnetic waves at a frequency of 27 MHz. The purpose of the DU was to create an active and interdisciplinary learning environment that fostered the integration of science and technology education. This DU enabled the application of a set of data collection instruments, which included, on one hand, Likert-type scales to record perceptions and achievement levels across various competencies (quantitative analysis), and on the other, observation techniques, field journals, and semi-structured interviews to capture participants’ experiences, meanings, and reflections (qualitative analysis) (Ramírez Cano, 2019).

The quantitative inputs were Likert-type questionnaire items (1–5) designed to capture students’ self-reported perceptions related to the didactic unit (e.g., perceived relevance of specific aspects of the unit, confidence in using particular tools/procedures, and perceived development of targeted competencies). Each item was a brief statement scored on the same response scale, and the resulting dataset formed an 5 × 5 matrix (participants × items). When the paper refers to “scores by domain,” this denotes item sets grouped a priori by construct (e.g., competence-related vs. instrument-related aspects), where a domain score is computed as the mean of its constituent items (with consistent coding direction), while WSSA1 primarily operates on the inter-item correlation structure derived from the full item matrix.

For the quantitative analysis, HUDAP software was employed, specifically its WSSA1 and POSAC tools. This technique allows for the representation of a visual and mathematical structure that identifies latent groupings among variables, going beyond simple linear correlation. These groupings reflect semantic proximities among the evaluated dimensions, enabling the recognition of underlying conceptual cores that emerge from participants’ response patterns.

The qualitative analysis was conducted in ATLAS.ti through an inductive and iterative open-coding procedure applied to textual meaning units derived from the qualitative records, consistent with contemporary guidance that conceptualizes coding as an analytic act of systematically assigning concise labels to data segments in order to capture salient meaning (Saldana, 2021). In the first-cycle phase, clause-to-paragraph segments were coded line-by-line using descriptive codes to represent topical content, process codes to capture actions and practices, and in vivo codes when participants’ wording carried specific analytic significance; concurrently, analytic memos documented coding rationales as well as boundary decisions. In the second-cycle phase, the initial code set was refined through focused consolidation and patterning to reduce overlap, stabilize code meanings, and generate higher-order emergent categories, which were subsequently organized into thematic families; the density and frequency of these families supported the identification of interpretive axes. Accordingly, the resulting code system was not treated as a software-generated list of terms, but was formalized in a codebook through operational definitions, explicit inclusion/exclusion criteria, and anchor quotations.

Building on this qualitative foundation, the analytic workflow operationalizes integration in line with established mixed-methods strategies (Bazeley, 2017, 2024). Quantitative outputs (minimum-distance and mean-value patterns; WSSA1/POSAC) were first generated to identify proximity structures among variables, producing interpretable regions and clusters. Qualitative data were then inductively coded and organized into categories within ATLAS.ti, with explicit code definitions and decision rules documented in a codebook. Integration was enacted through a structured linking step in which (a) cases (analytic units) were aligned across strands, (b) code-occurrence summaries and category profiles were compared across quantitatively defined regions, and (c) discrepant or extreme cases were inspected through return-to-text verification. Supplementary Materials include the initial and refined codebooks, along with ATLAS.ti outputs that support transparency and auditability. This sequence corresponds to integration strategies such as comparing coded qualitative patterns across subgroups defined by variables, typology development, and iterative multi-phase analysis, while keeping the underlying textual evidence available for audit and verification (Bazeley, 2017), thereby supporting warranted meta-inferences rather than a purely illustrative juxtaposition of findings.

To avoid the risk of treating complex computations as a rhetorical device and to provide analytic warrants and safeguards, the quantitative and qualitative components were specified as an auditable chain of analytic decisions. On the quantitative side, the unit of analysis was the questionnaire item (aspect) scored on a Likert-type scale, represented as a vector of responses across participants (i.e., a participants × items matrix). All preprocessing steps (scaling and the distance metric used to compute Euclidean proximity) were established prior to interpretation; WSSA1 was used to identify candidate structural cores by minimizing within-region dispersion, and POSAC was used to represent the resulting partial order and regional structure in a way that preserves relational constraints. On the qualitative side, codes were operationally defined and stabilized through rule-based refinement (definitions, inclusion/exclusion criteria, and anchor quotations), as documented in the codebook. Integration was conducted at the case (analytic-unit) level: each case’s HUDAP region membership was aligned with its ATLAS.ti coding profile, enabling comparisons of code distributions and category patterns across regions. Interpretations were treated as abductive and hypothesis-generating, and were subjected to robustness checks, including (i) inspection of discrepant/extreme cases, (ii) return-to-text verification for each proposed structural–semantic correspondence, and (iii) sensitivity checks against alternative plausible regional partitions or coding consolidations when interpretations appeared unstable. These safeguards ensure that the proposed linkage functions as a methodological integration mechanism rather than an impressionistic alignment of outputs.

The methodological contribution of this study is operationalized through an explicit articulation process. Specifically, each category in ATLAS.ti was interpreted not only based on its internal content, but also in terms of its potential connection to the structural cores identified by HUDAP, seeking correspondences, contrasts, or indirect connections between quantitative patterns and qualitative meanings. In other words, the geometric groupings obtained via WSSA1 were not interpreted in isolation, but were compared with the categories constructed in ATLAS.ti. This strategy is embedded within an abductive logic that enables the formulation of explanatory hypotheses about the links between quantifiable dimensions and constructed meanings, generating grounded interpretive inferences (Ramírez Cano, 2019).

Figure 5 provides a synthesis of the articulation strategy employed. As illustrated, the integration was conducted in two successive phases, corresponding to two evaluation moments within the research process. The first phase involved an exhaustive qualitative analysis through open coding and the generation of emergent categories using ATLAS.ti. These categories provided a solid and context-sensitive interpretive foundation for the phenomenon under study.

Figure 5.

Synthesis of the process of combining the qualitative and quantitative information obtained in the example.

Subsequently, a structural analysis of the quantitative data was conducted using the WSSA1 and POSAC tools from the HUDAP software. This enabled the identification of geometric groupings among items during the initial phase. In the second phase, data from the second stage of evaluation were analyzed. Here, the same qualitative analysis was performed; however, it incorporated axial and nodal categories, which served to cluster the previously identified open categories. In this paper, axial categories refer to second-cycle groupings that consolidate multiple open codes around a shared analytic dimension, whereas nodal categories denote the highest-order integrative nodes that organize the axial categories into a small framework for mixed-methods integration.

Thereafter, the quantitative analysis was carried out as described earlier, followed by the construction of theoretical insights. This methodological sequence allowed for a contrastive and integrative articulation between qualitative categories and quantitative structures, generating an abductive linkage grounded in both the semantics of discourse and the metric relationships among variables. In the following Results section, an example of this articulation is presented through graphs and specific analyses.

To initiate the articulation process, we report the qualitative coding conducted in ATLAS.ti on interviews, field journals, and observational notes. Following an inductive content-analytic procedure, meaning units were coded line-by-line to generate operationally defined open codes, which were then consolidated through constant comparison and organized into higher-order categories. Figure 6 summarizes the most frequent open codes (labels), while their methodological status as codes is established through explicit definitions, inclusion/exclusion criteria, and anchor quotations provided in the codebook (Supplementary Material). The most prominent analytic areas include Design, Knowledge, Competencies, Communication, and Commitment.

Figure 6.

Most frequent open codes from initial inductive coding.

The labels displayed in Figure 6 represent operationally defined codes rather than software-derived terms; their definitions, boundary rules, and anchor quotations are documented in the initial and refined codebook (Supplementary Material), in accordance with established coding cycles (Saldana, 2021) and transparency requirements in content analysis (Krippendorff, 2019). To avoid conflating software outputs with analytic constructs, the items shown in Figure 6 were treated as provisional labels during initial coding and were subsequently refined into methodologically warranted codes through rule-based consolidation: each code was assigned an operational definition, delimited by explicit inclusion and exclusion criteria, and supported by at least one anchor quotation. Refinement procedures included merging near-synonyms, resolving overlap, and checking code boundaries against the original text to reduce category drift. Finally, frequency and quotation density are reported only as descriptive indicators of salience (not as validity criteria) since interpretive claims are grounded in the documented codebook and the underlying textual evidence (Bingham, 2023; Ramírez Cano, 2019).

Following Figure 5, the next step involved a descriptive analysis of the data collected. For this purpose, SPSS software was used. A box plot was employed to present the results, as this type of visualization is particularly useful for displaying data distribution, detecting outliers, comparing groups, and observing variability and dispersion. Figure 7 illustrates the representation of the data after conducting a statistical analysis of quantitative information.

Figure 7.

Box graph for the Importance behavior in the initial phase.

Subsequently, an analysis was conducted on the Likert-type scales, processed using HUDAP and its WSSA1 and POSAC tools. This analysis revealed clustering patterns within the evaluated dimensions, allowing the identification of conceptual cores characterized by high internal coherence. Figure 8(a) shows three clusters obtained from the geometric distance analysis. As depicted, it represents a spatial configuration of items organized based on their structural similarities. The figure illustrates how certain items cluster within Euclidean space, highlighting thematic cores.

Figure 8.

(a) Regions and categories delimited in the correlation matrix for Importance—initial phase (left) and (b) regions delimited in the POSAC scalogram for Importance—initial phase (right).

Upon reviewing the 24 evaluated aspects and correlating them with the WSSA1 output, a category was derived to encompass them specifically: Competencies and Instruments; Laboratory and Technology Preparation. In line with this analysis, the POSAC tool provided insight into the aspects that, according to respondents, were considered most to least important. Figure 8(b) illustrates the delimitation of these regions.

In the second phase of the research, the same procedure was followed. However, in the ATLAS.ti analysis, the 128 open categories identified were grouped into higher-order categories, referred to as axial categories (13 in total), and ultimately into nodal categories: Knowledge, Problem Solving, and Competencies. Figure 9 illustrates the relationship between the axial categories and the nodal category Knowledge.

Figure 9.

Relationships of the nodal category Knowledge with the axial coding categories.

Note that categories such as Competencies, Teaching, Evaluation, Content Integration, and Self-training are linked to this nodal category. Additionally, observe that quotations (identified with numbers) are associated with these categories, providing evidentiary support and thereby fulfilling the Dependability criterion.

To meet the Credibility criterion, a thorough review of both the results and the analytic process was conducted, including a peer review by experts familiar with the topic and methodology. This ensures compliance with the rigor and quality criteria for qualitative research as suggested by the Onwuegbuzie and Leech Model (2007).

The outputs from the quantitative analysis in the second evaluation phase are similar to Figures 7 and 8, but based on a different dataset corresponding to this subsequent stage.

In this context, it is important to emphasize that the innovative nature of the proposed methodological approach becomes particularly evident during the analysis and articulation of the graphical results. In other words, the structural cores derived from WSSA1 and POSAC were systematically contrasted with the qualitative categories. This process, guided by abductive logic, enabled the establishment of interpretive correspondence between quantitative structures and emergent qualitative meanings. For example, the quantitative cluster grouping items related to competencies and instruments correlated with the nodal categories of Competencies, Teaching, and Evaluation. Within this framework, axial categories such as Knowledge, Content Integration, Teaching, Evaluation, Science, Technology, Mathematics, and Laboratory were linked and correlated in the analysis.

Thus, the categories generated using HUDAP, along with the associated items representing aspects evaluated in the quantitative instruments, related to the hermeneutic unit constructed in ATLAS.ti and, by extension, with the quotations from various documents and the theoretical insights developed during the process. This integration enables what can be considered cross-validation of the analyzed phenomenon across multiple analytical layers. Such a strategy results in a more robust theoretical construction, with a level of argumentation that spans from the deep interpretive analysis typical of qualitative methods to the comprehensive scope offered by quantitative approaches.

Another innovative element of the proposed methodology was the comparative analysis strategy used to review the results and descriptive analysis across both phases. This comparison sought to correlate aspects that students identified as relevant to their training as teachers in AT&I (Automation, Technology, and Informatics), with the level of satisfaction reported for those same aspects, each of which was evaluated during both phases.

Building on the structural–semantic articulation established in the preceding analyses, the results are synthesized by emphasizing how quantitative proximity structures and inductively constructed qualitative categories converge to support coherent meta-inferences. At this stage, the integration is sufficiently demonstrated through the region-based interpretation of HUDAP outputs and the corresponding coding profiles documented in the codebook. An additional option for presenting the Importance–Satisfaction results was explored by crossing the data matrices and calculating, for each participant and evaluated aspect, the difference between Importance and Satisfaction scores; a corresponding graph was generated for each phase, and these values were subsequently visualized through a heat map. This strategy (drawn from telecommunications engineering and commonly used in security analysis to detect network weaknesses through color-coded representations of attacks, vulnerabilities, and strength variables) introduces an additional analytic layer that may distract from the core contribution. For this reason, the present manuscript concentrates on consolidating the interpretive implications of the structural–semantic linkage and on delineating the methodological contribution of the proposed workflow, while a separate methodological paper is projected to develop the heat-map approach with the required technical detail.

Ultimately, after reviewing the various analytical processes, it can be concluded that this methodological approach not only allowed for the confirmation of findings from multiple perspectives, but also facilitated the generation of new interpretive hypotheses regarding teacher professional development in innovative educational contexts.

This methodological triangulation significantly strengthens the analysis, promoting a richer and more nuanced understanding of the phenomenon under investigation.

Discussion

The findings obtained allow for a critical reflection on the potential and challenges of methodological integration between quantitative structural analyses and emergent qualitative categories through an abductive logic. Unlike traditional approaches in mixed-methods research (which often apply triangulation strategies in parallel or sequentially without achieving genuine epistemic convergence (Campos, 2009; Creswell and Inoue, 2025; Denzin and Lincoln, 2012; Oleinik, 2011)) the methodological proposal developed here succeeds in articulating both dimensions in a dialogical, structural, and semantic manner. This represents a significant advancement compared to the linear models of methodological combination that still predominate in educational research.

Importantly, the quantitative geometry does not “validate” qualitative meanings, nor do qualitative codes “explain” statistical structure by default. The contribution lies in making the integration step explicit; specifically, HUDAP outputs are used to propose candidate relational structures, while the qualitative evidence is used to warrant (or reject) those candidates through documented coding rules and return-to-text checks. Accordingly, the reported correspondences are presented as abductive meta-inferences with clearly stated analytic conditions, rather than as deterministic mappings.

In comparison with previous works that have explored methodological articulation (Fetters et al., 2013; Teddlie and Tashakkori, 2012), this study introduces an innovation by employing specific tools such as HUDAP (WSSA1 and POSAC) to identify geometric patterns in quantitative data, and to interpretively link them with inductive categories obtained in ATLAS.ti. This articulation enables the detection of latent relationships between meanings and structures that do not easily emerge through conventional analysis techniques. In this way, the study contributes not only to the internal validity of the results, but also to the generation of new interpretive hypotheses about the educational phenomena under investigation.

Moreover, the abductive logic employed goes beyond mere methodological complementarity. It constitutes an alternative form of reasoning that enhances theoretical construction through the dialogue between empirical experience and conceptual frameworks. This aligns with contemporary proposals that advocate for the heuristic value of abduction in social research (Richardson and Kramer, 2006; Verd and Lozares, 2016) and offers a promising path for studies aiming to move beyond linear explanation or simple correlation.

Content analysis cautions against reifying “content” as something objectively contained in texts and warns that coding can devolve into shallow counting if analytic procedures and inferential steps are not made explicit and contestable (Krippendorff, 2019). In this study, codes are treated as interpretive instruments rather than measurements of inherent textual properties; consequently, frequency and quotation density are reported only as indicators of analytic salience. To mitigate typical risks (such as decontextualization, category drift, and overconfident claims) three safeguards were implemented: (i) explicit unitizing decisions and context specifications for each coded segment; (ii) rule-governed code definitions (operational meaning, inclusion/exclusion criteria, anchor quotations) documented in the codebook; and (iii) return-to-text verification whenever structural–semantic linkages were proposed. These steps are intended to ensure that the analysis supports replicable, critically evaluable inferences from texts to the contexts of their use, rather than relying on opaque transformations or rhetorical appeals to computation (Krippendorff, 2019).

However, several limitations of the study must be acknowledged. First, the effectiveness of this articulation largely depends on the conceptual rigor with which instruments are designed and data interpreted. Additionally, the use of specialized software such as HUDAP and ATLAS.ti requires advanced technical training, which could limit its applicability in contexts where such resources are unavailable. Furthermore, the case study nature of the research imposes constraints on generalizability although the depth of analysis achieved justifies this methodological choice (Ramírez-Montoya and Lugo-Ocando, 2020).

Future research could explore the replication of this methodology in other educational settings, disciplines, or levels of instruction, as well as the integration of artificial intelligence tools to automate the identification of structural-semantic correspondences. It would also be relevant to advance the development of formal criteria for validating the hybrid categories that emerge from this abductive convergence.

Conclusions

The research conducted validated a methodological strategy that coherently and rigorously integrates qualitative and quantitative approaches within a mixed-methods design guided by abductive logic. Through the combined use of ATLAS.ti and HUDAP (specifically the WSSA1 and POSAC tools) it was possible to link emergent categories from qualitative content analysis with structural cores derived from the analysis of geometric distances among items. This articulation offers an alternative to integrating heterogeneous data, particularly those rooted in the epistemological divide between the two paradigms.

Furthermore, the study demonstrated the feasibility of establishing relationships between constructed meanings and structural patterns, facilitating the generation of interpretive inferences with a high degree of internal validity. In particular, the methodology enabled the contrast between nodal, axial, and open categories obtained through qualitative analysis and quantitative clusters defined by the internal coherence of the evaluated items. This convergence provided a more comprehensive understanding of the phenomenon under investigation (namely, the development of professional competencies in teacher education) and contributed to the construction of a theory grounded in both empirical evidence and interpretive frameworks.

Finally, this proposal represents a methodological contribution to the field of mixed-methods research, as it offers an operational alternative for the abductive integration of qualitative and quantitative data, using complementary geometric and semantic resources. Its application may be extended to other educational or social contexts where it is necessary to construct knowledge from diverse yet convergent structures.

The supplementary material includes the initial and refined codebook (definitions, inclusion/exclusion criteria, and anchor quotations), exported as standard ATLAS.ti reports to ensure transparency. Additionally, the analysis files used in HUDAP (WSSA1 and POSAC) are provided to facilitate methodological replication and auditability.

Supplemental Material

sj-docx-1-mio-10.1177_20597991261443800 – Supplemental material for Geometric distance and semantic structure: A methodological innovation for mixed methods research with HUDAP and ATLAS.ti

Supplemental material, sj-docx-1-mio-10.1177_20597991261443800 for Geometric distance and semantic structure: A methodological innovation for mixed methods research with HUDAP and ATLAS.ti by Jimmy William Ramírez Cano, William Manuel Mora Penagos and Diego Mauricio Rivera Pinzón in Methodological Innovations