An Experience with Pre-Service Teachers, Using GeoGebra Discovery Automated Reasoning Tools for Outdoor Mathematics

Abstract

This paper presents an initial output of the project “Augmented Intelligence in Mathematics Education through Modeling, Automatic Reasoning and Artificial Intelligence (IAxEM-CM/PHS-2024/PH-HUM-383)”. The starting hypothesis of this project is that the use of technological tools, such as mathematical modeling, visualization, automatic reasoning and artificial intelligence, significantly improves the teaching and learning of mathematics, in addition to fostering positive attitudes in students. With this hypothesis in mind, in this article, we describe an investigation that has been developed in initial training courses for mathematics teachers in several universities in Madrid, where students used GeoGebra Discovery automated reasoning tools to explore geometric properties in real objects through mathematical paths. Through these activities, future teachers modeled, conjectured and validated geometric relationships directly on photographs of their environment, with the essential concourse of the automated discovery and verification of geometric properties provided by GeoGebra Discovery. The feedback provided by the students’ answers to a questionnaire concerning this novel approach shows a positive evaluation of the experience, especially in terms of content learning and the practical use of technology. Although technological, pedagogical and disciplinary knowledge is well represented, the full integration of these components (according to the TPACK model) is still incipient. Finally, the formative potential of the approach behind this experience is highlighted in a context where Artificial Intelligence tools have an increasing presence in education, as well as the need to deepen these three kinds of knowledge in similar experiences that articulate them in a more integrated way.

1. Introduction

Effective mathematics teaching involves understanding how students interact with mathematical concepts in various learning environments, including those outside of school. The benefits of expanding mathematical learning beyond the classroom and exploring the real world to discover and apply mathematical concepts have been well recognized for a long time (Hattie et al., 1997; Çengelci, 2023).

In this context, one powerful resource for connecting mathematics to the real world is Math Trails. These are routes with marked stops “where walkers formulate, discuss, and solve interesting mathematical problems” inspired by their surroundings (Shoaf et al., 2004, p. 4). By incorporating urban and natural elements into the learning experience, Math Trails help students recognize and apply mathematical ideas in everyday situations (Ludwig et al., 2020). In the case of pre-service teachers, Math Trails participants are engaged in a mathematical problem-solving activity in meaningful and real settings, which may play a significant role in shaping students’ future experiences with mathematics and their perception of what mathematical practice is (Martínez-Jiménez et al., 2022).

On the other hand, mobile technologies have been successfully implemented in several outdoor learning proposals in recent years and have gained importance not only to study motivational and mathematical beliefs perspectives but also from aspects related to mathematics learning and performance (Benito et al., 2023; Ariño-Morera et al., 2024). One example is the MathCityMap application (Ludwig & Jesberg, 2015), developed at the Goethe University of Frankfurt in 2012, which aims to facilitate and improve—both in the educational context and in public use—the creation and development of outdoor Math Trails. In the educational setting, the use of MathCityMap brings significant benefits, such as student motivation and meaningful learning of the mathematical content addressed in the context of this application.

In addition, numerous studies have explored the teaching and learning of mathematics using dynamic geometry systems (DGSs), with many teachers incorporating these tools, especially GeoGebra, into their instructional practices (Hohenwarter & Preiner, 2007). The use of these applications offers several benefits in the mathematics learning process, including fostering a creative and dynamic understanding of concepts, enhancing critical and analytical thinking, promoting logical–mathematical reasoning, enabling dynamic demonstrations, verifying conjectures, sparking interest in mathematics and developing collaborative work skills, among others (Recio et al., 2019; Weinhandl et al., 2024).

Despite its growing but slow inclusion, both in classrooms and in training programs, it has long been known that the development of problem-solving skills within DGS environments is a relevant aspect of teacher training (Hohenwarter & Lavicza, 2007). If teachers perceive that the technology is user-friendly and effective in teaching math concepts, this perception can foster positive attitudes towards using it (Hartsell et al., 2009). One approach aimed at achieving successful integration in teacher training is the TPACK model (Koehler & Mishra, 2009), which highlights the three types of knowledge that should be considered when planning learning activities with ICT: technology, pedagogy and content. These elements must be interconnected, not simply added together, to design activities that create meaningful links among them. In initial teacher training, this framework helps pre-service teachers reflect on their attitudes and the role of ICT in mathematics classrooms (Marbán & Sintema, 2021).

For extending the possibilities of dynamic geometry systems, the incorporation of algebraic capabilities, as well as automatic discovery and demonstration of geometric constructions, are key. In this context, it is worth mentioning GeoGebra Discovery, an extended version of GeoGebra that includes a collection of automated reasoning tools for elementary geometry. These tools and features enhance the visual exploration and understanding of Euclidean geometry constructions built on the graphics window of GeoGebra Discovery by answering—with mathematical rigor—different questions posed by the user concerning the verification or discovery of geometric properties on the construction, having the potential to improve the teaching and learning of geometric reasoning (Kovács et al., 2022).

Nowadays, with the increasing role of Artificial Intelligence tools in daily life, the inclusion of experiences with automated reasoning tools, such as GeoGebra Discovery, in the initial training of mathematics teachers could be of greater relevance. Accordingly, in this paper, we describe and evaluate an experience that attempts to put into practice a novel proposal that merges the different issues we have described so far: Math Trails, Dynamic Geometry and Automated Reasoning, within the TPACK framework. Thus, during the initial training courses for future mathematics teachers, taking place in several universities of Madrid, we have asked students to model, explore, conjecture, confirm and interpret by using the automated reasoning tools available in the popular dynamic mathematics program GeoGebra and its fork version GeoGebra Discovery, geometric features of diverse real objects that appeared in a math trail along the classroom and surrounding areas. Let us remark that although the proposal for this activity has already been announced by some of the current authors of this manuscript (see Ariño-Morera et al., 2024; Botana et al., 2020), this is the first time it has been put into practice. In summary, the goal of this paper is to present the context and the development of the experience and to assess its potential educational relevance by evaluating the different data collected during the experience.

After summarily describing in the next Section the main ingredients of this experience (Mathematics Trails, GeoGebra Discovery dynamic geometry and automated reasoning tools and the TPACK pedagogical framework), we proceed, in the following Section, to detail the elements (design, research questions, sample, procedure, data analysis) involved in the description of experience. Finally, we analyze, according to the TPACK model, the evaluations expressed by the participants in relation to aspects such as satisfaction with the experience, perceived usefulness, difficulties encountered, learning acquired, elements of the software considered most interesting, perceptions about its future applicability in teaching practice and suggestions for improvement, concluding with a positive overall assessment of its relevance, both for current students and for their future performance as mathematics teachers.

2. Theoretical Framework

2.1. Mathematical Trails

The concept of the Math Trails emerged as a proposal aimed to popularize mathematics during the 1980s and 1990s (Blane & Clarke, 1984). Although the idea of a math trail can be considered from other perspectives, such as recreational, informative and touristic, the importance of mathematical walks as a school activity has been recognized early on by the educational community (Cahyono et al., 2025).

Math Trails support the STEAM educational approach by contextualizing mathematics with real-world objects, often drawn from architecture, engineering, or the arts; by promoting the use of technology, including mobile devices, GPS and measuring tools; encouraging collaboration through group work and enhancing, as well as enhancing learners’ mathematical literacy (Cahyono et al., 2025; Haas et al., 2021). This pedagogical approach takes advantage of the richness of the environment to explore and apply mathematical concepts, thus fostering meaningful learning and a contextualized understanding of mathematics.

An example of a successful mobile application and web portal for Math Trails is MathCityMap (Ludwig & Jesberg, 2015). It aims to facilitate and enhance, both in the educational context and for public use, the creation and development of outdoor mathematical walks. Some of its key features are the possibility to create digital classrooms for conducting trails with students, GPS geolocation for tracking the trail, chat functionality for interaction among students and teachers, gamification elements, hints for problem-solving and immediate feedback upon task completion (Jablonski et al., 2020).

MathCityMap has been used in several proposals in Initial Teacher Training, some of them introducing the creation of trails by pre-service students as a rich problem-posing activity, as well as experiences complementing MathCityMap with other technologies like Augmented Reality (Barbosa & Vale, 2020; Benito et al., 2023; Gurjanow et al., 2017; Haas et al., 2021; Martínez-Jiménez et al., 2022). All of them report positive valuations from pre-service teachers, high levels of engagement and the view of Math Trails as a meaningful activity for their professional future.

2.2. GeoGebra Discovery and Real Objects

The last decade has seen the development of GeoGebra Discovery (Kovács et al., 2022), an extension of GeoGebra. Although GeoGebra already includes automatic reasoning tools, GeoGebra Discovery extends some of its features related to the automatic discovery and proof of geometric properties. It has different versions, working on and offline, over diverse operating systems, that can be freely downloaded1.

For example, GeoGebra Discovery, using the Relation command, is able to automatically discover ratios and inequalities holding between two algebraic expressions involving segments, while in GeoGebra, the same command is operative only for testing equalities. This is particularly useful for the exploration of real objects, as shown in Figure 1. Indeed, for obtaining the displayed Relation (k = (1/2 (√5 − 1)) × r or k = (1/2 × (√5 + 1)) × r, where k and r are the lengths of the two sides of the pink door, we have, first, to place a photo of the door in GeoGebra Discovery’s graphics view. Then, we should place over the image some points, like K, P (that look like the extremes of the longer side), L (the extreme of the shorter side, perpendicular to KP), i.e., three of the corners of the door. Next, we make some geometric construction starting from these points (in Figure 1: the circle centered at K passing through L, intersecting the line defined by the long side at M). Since this point M does not coincide with P, we might consider the midpoint O of KM and construct the circle centered at O and going through L. This new circle intersects the long side at a point P’ very close to P, so we assume that our construction reasonably represents the two sides of the door. Finally, we ask GeoGebra Discovery to find the precise ratio holding between the segments k = KL and r = KP’, and the answer is the golden ratio.

Another example of the singular features of GeoGebra Discovery is displayed in Figure 2, describing the output of the Discover command, which automatically displays all results, subject to some programmed limitations concerning the different tests—collinearity of three points, cocyclicity of four points, equality of segments, perpendicularity of lines, etc.—involving a chosen point such as point H in Figure 2 and the midpoint of F and G, where F and G are obtained from three fixed points D, C, E, so that DC is perpendicular to CF and of equal length, and DE is perpendicular to EG and of equal length. The obtained information allows the user of this command to discover that the position of H does not depend on the initial position of D, only on those of C and E, since the obtained results show that HC and HE have equal lengths and are perpendicular.

On the other hand, Figure 3 shows another feature of GeoGebra, the LocusEquation command, plotting the curve that a certain point J must verify if it is subject to some restrictions.

We refer the reader to Kovács et al. (2022) for more details about the algorithms, implementation and performance of the automated reasoning commands in GeoGebra Discovery.

Let us remark that the use of GeoGebra Discovery for outdoor mathematics seeks to expand the possibilities offered by the technology (such as MathCityMap) in Math Trails. Thus, as illustrated in the previous Figures, GeoGebra Discovery could be conceived as a kind of digital tape measure, providing math trail walkers with a reasoning tool that allows them to perceive, on the photos of real objects placed on the GeoGebra Discovery window, a kind of automatically augmented reality that could help them to enhance and support their problem-solving skills in real-life environments, something already suggested in Ariño-Morera et al. (2024) and Botana et al. (2020).

Yet, the starting point for the project is the perception of the scarcity of experiences carried out in classrooms with GeoGebra Discovery. The need to urgently respond to this lack has been pointed out, specifically mentioning GeoGebra Discovery’s automatic reasoning tools by researchers in mathematics education (Carvalho, 2024; Hanna & Yan, 2021; Kovács et al., 2022; Russo, 2023), with multiple references to the interest and need to experiment with its incorporation into training and, particularly, into teacher training.

2.3. TPACK

For an effective integration of technology in a mathematics classroom, teachers need a wide variety of knowledge, not only for choosing the appropriate technology for a specific mathematics content but also to adapt pedagogical strategies to the context (Niess et al., 2009). This knowledge was described by Mishra and Koehler (2006), combining the pedagogical content knowledge (PCK) of Shulman (1986) with the knowledge of technology for learning, creating the technological–pedagogical content knowledge (TPACK). The model thus considers three main domains: mathematical content knowledge (CK), pedagogical knowledge (PK) and technological knowledge (TK), but it crucially stresses the manner in which these domains interrelate so teachers can construct effective solutions in their classrooms. See Section 3.6 for an extension on each of the model’s domains. The interactions among the three knowledge groups are expressed as intersections of the domains of the TPACK model (see Figure 4). With the ongoing evolution of technology, teaching practices and classroom contexts, the TPACK “provides a dynamic framework for viewing teachers’ knowledge necessary for the design of curriculum and instruction focused on the preparation of their students for thinking and learning mathematics with digital technologies” (Niess et al., 2009, p. 7).

Koehler and Mishra (2009) mention that expert teachers can simultaneously integrate technology, pedagogy and content knowledge, bringing TPACK every time they teach, although the path to achieve it is not straightforward. According to Niess et al. (2009), the concurrence of technological knowledge (TK) into pedagogical content knowledge (PCK) requires several steps until a teacher can create knowledge from the intersection of pedagogy, content and technology. These authors propose a model for the development of mathematics teachers’ TPACK, which shows how teachers’ understanding of mathematics TPACK grows as they integrate technology into their teaching (see Figure 5). The model consists of the following five stages (Niess et al., 2009, p. 9):

• Stage 1. Recognizing, where teachers demonstrate the knowledge to use technology and understand its alignment with mathematics content, but they still do not use it in their teaching.
• Stage 2. Accepting, where teachers develop either a positive or negative attitude toward using appropriate technology in teaching and learning mathematics.
• Stage 3. Adapting, where teachers participate in activities that lead them to make a decision on whether or not to adopt the use of appropriate technology for teaching and learning mathematics.
• Stage 4. Exploring, where teachers actively integrate technology with teaching and learning mathematics, searching for implementations throughout the curriculum.
• Stage 5. Advancing, where teachers incorporate and evaluate the integration of technology as an essential and regular part of their teaching.

Figure 5. Teacher stages of understanding the integration of TK into PCK (Niess et al., 2009).

Figure 5. Teacher stages of understanding the integration of TK into PCK (Niess et al., 2009).

Rakes et al. (2022) argue that a true intersection of technological knowledge (TK) and pedagogical content knowledge (PCK) can only be achieved at Stage 5. Indeed, the progression from an instrumental use of technology to a pedagogically grounded and disciplinarily coherent approach does not occur spontaneously but requires intentional curricular planning and formative experiences designed for that purpose (Rakes et al., 2022). This highlights the importance of teacher training that helps both in-service and pre-service teachers recognize and understand the three dimensions of knowledge. It is crucial not to overlook the complexity of each knowledge area and the interrelationships among them, as failing to do so may result in oversimplified approaches or potential failure (Koehler & Mishra, 2009).

With respect to Initial Teacher Training, existing reports on the development of pre-service teachers’ TPACK within mathematics teaching and learning contexts remain limited, particularly regarding the evolution of their levels of understanding of the process and their perceptions of using technology in their professional future (Marbán & Sintema, 2021). It is known that pre-service teachers’ learning experiences in their training programs play a critical role in enhancing their TPACK and that there is a need for more exemplary TPACK experiences to enable future teachers to comprehend the links among content knowledge, specific pedagogical approaches and appropriate technologies (Tondeur et al., 2019).

As (Niess et al., 2009) points out, the emergence of a new technology requires a process of rethinking its acceptance for teaching and learning mathematics, as well as the content and the pedagogies involved. The developing stages shown in Figure 5 are, therefore, not linear or behave as an increasing regular pattern. In this context, this paper presents the case of GeoGebra Discovery’s novel Automatic Reasoning tools in Initial Teacher Training, which offers future teachers a learning experience and reflection on its connections with mathematical content and its pedagogical possibilities.

3. Research Design and Method

3.1. Research Questions

As stated in the Introduction, and given the novelty of the proposal, mixing for the first time, Math Trails, Dynamic Geometry and Automated Reasoning, the question we have addressed in this research experience is the evaluation of the training potential of this type of experience for the development of professional teaching knowledge in initial training in mathematics.

In fact, we will show how participation in the experience carried out in this study based on the use of GeoGebra Discovery automatic reasoning tools for teaching and learning geometry, aimed at future mathematics teachers in outdoor learning contexts, makes it possible to identify relevant manifestations of technological, pedagogical and disciplinary knowledge, as well as their interrelationships in line with the TPACK model. Therefore, from the evaluations expressed by the participants in relation to aspects such as satisfaction with the experience, perceived usefulness, difficulties encountered, learning acquired, elements of the software considered most interesting, perceptions about its future applicability in the teaching practice and suggestions for improvement, it is expected that evidence will emerge of the training potential of this type of proposals for the development of professional teaching knowledge in initial training in mathematics.

More specifically, this study addresses the following research questions:

• What manifestations of technological, pedagogical and content knowledge (according to the TPACK model) can be identified in the participants’ responses after engaging in an outdoor learning experience using GeoGebra Discovery?
• What are the participants’ perceptions regarding the usefulness, applicability and challenges of this experience for their future teaching practice?

3.2. Design

This study is framed within experimental designs, specifically of the pre-experimental type. The design was operationalized through the implementation of an intervention applied to different independent groups belonging to different universities and the subsequent collection of data through a single post-measurement. The data were analyzed through a methodological approach that integrates qualitative categorical content analysis with descriptive quantitative support, allowing a richer and more detailed understanding of the results obtained. This type of design is particularly appropriate in exploratory studies such as this one.

3.3. Sample

An incidental sampling technique was used to carry out the study due to the accessibility and availability of participants during the data collection period. The participants (140) were students from three universities in the Community of Madrid: Universidad Complutense de Madrid, Centro Universitario La Salle and Universidad Rey Juan Carlos. The participating students were currently taking the Double Degree in Early Childhood Education and Primary Education (Doble Grado en Maestro en Educación Infantil y Maestro en Educación Primaria, Universidad Complutense de Madrid), the Degree in Primary Education (Grado en Educación Primaria, Centro Universitario La Salle and Universidad Rey Juan Carlos), the Master’s Degree in Research and Innovation for the Teaching and Learning of Experimental, Social and Mathematical Sciences (Máster en Investigación e Innovación para la Enseñanza y el Aprendizaje de las Ciencias Experimentales, Sociales y Matemáticas, Universidad Complutense de Madrid) and the Master’s Degree in Teacher Training for Secondary and High School, Vocational Training and Language Teaching (specializing in mathematics) (Máster en Formación del Profesorado de ESO y Bachillerato, FP y Enseñanza de Idiomas, Universidad Complutense de Madrid and Universidad Rey Juan Carlos) (see

Table 1. Sample description.

Variable	Category	Frequency	Percentage
Gender	Male	40	28.57
	Female	100	71.43
Age	20–21 years	26	18.57
	22 years	53	37.86
	23–27 years	47	33.57
	28 years and beyond	14	10.00
University	Universidad Complutense de Madrid	66	47.14
	Centro Universitario La Salle	59	42.14
	Universidad Rey Juan Carlos	15	10.71
Degree	Degree in Primary Education	64	45.71
	Double Degree in Early Childhood Education and Primary Education	36	25.71
	Master’s Degree in Research and Innovation for the Teaching and Learning of Experimental, Social and Mathematical Sciences	8	5.71
	Master’s Degree in Teacher Training for Secondary and High School, Vocational Training and Language Teaching (specializing in mathematics)	32	22.86

). The sample size represents the total number of accessible and eligible participants who met the inclusion criteria of current enrollment in these programs and voluntary consent to participate. While incidental sampling limits the generalizability of the findings, it was considered appropriate given the exploratory nature of this study.

3.4. Instrument

To evaluate the results of the experience with the students, part of an ad hoc questionnaire consisting of 15 open-ended questions was used. These questions address issues related to satisfaction with the experience, usefulness, perceived difficulty, learning of concepts acquired, aspects of the software used considered most interesting, perceptions about the possible implementation of experiences of this type in their future as teachers and issues that stand out from the experience and suggestions about it (see Appendix A).

Regarding the reliability analysis of the questions, two independent coders performed the categorization of a random subset of 20% of the responses, applying intercoder reliability criteria. A Cohen’s Kappa coefficient of 0.82 was obtained, indicating a high level of inter-coder agreement. Likewise, Krippendorff’s Alpha was calculated, where a value of 0.85 suggests high stability in categorization.

For the validity analysis, a mixed inductive and deductive coding process was used, in which the responses were organized into thematic categories aligned with the research objectives. A theoretical saturation analysis was applied, considering that saturation was reached when, after analyzing 90% of the responses, no new relevant categories emerged, which guarantees the solidity of the identified themes.

3.5. Procedure

The experience was similar for each of the participating Degrees, taking place in a single 90 min session. The session began with an introduction to GeoGebra Discovery, with special emphasis on the automatic reasoning tool known as Relation and its potential use in Math Trails and MathCityMap. Subsequently, different types of geometric proportionality ratios (such as the golden ratio2, the root of two ratio3 and the silver ratio4) were addressed, illustrating them through practical examples (e.g., paper size according to International Organization for Standardization (2007), historical artwork and monuments, etc.). Thus, students made different constructions according to their level of mathematical knowledge and analyzed them using the specific capabilities of GeoGebra Discovery. After that, the students left the classroom with the objective of identifying and photographing real, rectangular-shaped objects with their cell phones. The task was to select elements that could be linked to possible mathematical activities in the context of a future mathematical walk, perhaps with the concourse of MathCityMap. They were then asked to work with these images in GeoGebra Discovery, formulating conjectures about the proportion between the sides of the chosen objects (the golden ratio, the root of two or the silver ratio), also considering the possibility that none of these might be fulfilled, which opened the door to the discovery of other possible proportionality relationships. To test their hypotheses, the students made geometric constructions directly on the photographs uploaded to the GeoGebra Discovery graphics window and analyzed them using GeoGebra Discovery’s Relation tool, which automatically determines the relationship between the selected segments (see Figure 1).

At the end of the session, the ad hoc questionnaire mentioned above was applied to all students to evaluate different aspects of the experience.

3.6. Data Analysis

A categorical content analysis with a mixed deductive–inductive approach was conducted based on the TPACK model. Open-ended responses were coded according to their domains. This was complemented by descriptive quantitative analyses to support and enrich the qualitative findings. The process was developed with qualitative rigor criteria, such as double coding and validation by consensus. The coding was binary (1 if the domain in question is present in the response and 0 if not).

It should be noted that the same response may have been coded in multiple domains of the TPACK model. This depends on its association with different types of knowledge. Specifically, the different dimensions of the TPACK model have been adapted, as follows, to the experience carried out with GeoGebra Discovery automatic reasoning tools. These tools support the teaching and learning of geometric concepts in the training of future teachers:

• CK (content knowledge): responses from future teachers mentioning geometric concepts worked on during the GeoGebra Discovery experience, such as geometric ratios (the golden ratio, the root of two and the ratio of silver), proportionality, similarity, etc., were coded in this domain.
• PK (pedagogical knowledge): the responses that allude to reflections on how students learn geometric concepts and what difficulties they may have were associated with this domain.
• TK (technological knowledge): all answers referring to the use and/or mastery of the GeoGebra Discovery automatic reasoning tools used throughout the experience were coded in this domain. Example response, “Learning to prove things using GeoGebra Discovery”.
• PCK (pedagogical knowledge of the content): the answers of the future teacher were associated with this domain in which they allude to how geometric content seen during the experience can be taught, alluding to specific pedagogical methods but without mentioning technology.
• TCK (technological knowledge of the content): the answers in which the future teacher identifies geometric content that can benefit from the use of GeoGebra Discovery automatic reasoning tools used in the experience were coded in this domain and how teaching strategies can change due to the availability of these type of resources, showing how technology allows representing, exploring or teaching such content.
• TPK (technological–pedagogical knowledge): responses in which the future teacher alludes to technology as a facilitator of the construction of geometric knowledge were coded in this domain, showing the importance of guiding students in the use of automatic reasoning to explore and validate geometric properties.
• TPACK (technological, pedagogical and integrated content knowledge): the responses of prospective teachers integrating geometric content knowledge, pedagogical strategies and technological tools based on dynamic exploration, conjecture, reasoning and automated validation learned in the GeoGebra Discovery experience were associated with the TPACK model.

With respect to the TPACK mathematics teachers’ development framework of Niess et al. (2009), since the participants were all pre-service teachers with no professional activity, the stages considered in this study are the first two: recognizing and accepting. The analysis of those stages has been made from the questionnaire in Appendix A with a qualitative analysis of items 3, 4 and 10 for the recognizing stage and items 11, 12 and 13 for the accepting stage.

In addition, comparative analyses were made according to the university, degree and gender of the prospective teachers participating in the experience.

The inclusion of these variables responds to the need to understand how contextual and personal factors can influence the development and articulation of technological, pedagogical and disciplinary knowledge during initial teacher training. Previous research has pointed out that institutional culture, the academic level of the student body and gender can significantly condition the conceptions and practices of future teachers in relation to the educational use of technology (Koehler & Mishra, 2009; Li et al., 2015; Tondeur et al., 2017). These variables allow, therefore, the identification of differentiated profiles in the TPACK model and the guiding of more contextualized formative strategies.

From this perspective, the data analysis was developed using a qualitative strategy based on the thematic categorization of the open-ended responses, coded according to the dimensions of the TPACK model. The presentation of results by means of percentages has a descriptive purpose, aimed at identifying patterns and contrasts between the different groups of participants without inferential pretensions. This methodological decision responds to the nature of the instrument used—open-ended responses with non-exclusive multiple coding—and to the intention of preserving the interpretative and contextual richness of the discourses.

4. Results

4.1. Global Results by Domain and Development of the TPACK Model

The results of the analyses show that the domains of technological knowledge (TK), content knowledge (CK) and pedagogical knowledge (PK) are the most represented in the set of responses of prospective teachers on the evaluation of the experience. In global terms, TK appears in 33.63% of the codings, CK in 27.80% and PK in 19.59%. This presence of the three domains of the TPACK model shows that the participants have focused their attention mainly on the aspects related to the mastery of technological tools (especially GeoGebra Discovery), the geometric content addressed and the basic pedagogical implications.

On the other hand, dimensions involving the integration among types of knowledge such as TCK (technological content knowledge), TPK (technological–pedagogical knowledge), PCK (pedagogical content knowledge) and TPACK as a whole appear less frequently. TCK is present in 6.98% of the cases, TPK in 4.75%, PCK in 5.76% and TPACK in only 1.49% of the responses, in line with the fact that the more integrative levels of the TPACK model require greater professional maturity and teaching experience, and that they tend to develop in a more progressive manner (see

Table 2. Percentage of presence of each TPACK model dimension.

Dimension	Percentage
TK	33.63
CK	27.80
PK	19.59
TCK	6.98
TPK	4.75
PCK	5.76
TPACK	1.49

). This result is in line with the TPACK development model of Niess et al. (2009) since the final stages of adapting, exploring and advancing are not expected for pre-service teachers.

The results of the analysis according to each of the questions of the questionnaire applied (see

Table 3. Percentage of presence of each TPACK dimension in the answers to each questionnaire item.

Items	TK	CK	PK	TCK	TPK	PCK	TPACK
1	33.57	25	15	9.29	5.71	4.29	1.43
2	22.14	11.43	7.14	2.86	0.71	1.43	0.71
3	37.86	30	17.14	10	4.29	5.71	1.43
4	22.14	10	5	5.71	2.86	0.71	0.71
5	33.57	13.57	4.29	6.43	1.43	0.71	0.71
6	9.29	34.29	2.86	3.57	0.71	0.71	0.71
7	5.71	28.57	1.43	2.14	0	0.71	0
8	37.14	10	2.86	2.86	1.43	0.71	0.71
9	30	7.14	0.71	2.14	0	0.71	0
10	18.57	14.29	16.43	4.29	2.86	3.57	1.43
11	20.71	27.14	34.29	6.43	5	13.57	1.43
12	16.43	35.71	30	5.71	7.86	13.57	2.14
13	22.86	9.29	28.57	2.14	5	4.29	0
14	27.86	21.43	20.71	7.86	7.14	7.86	4.29
15	16.43	15	20	2.14	5	2.14	0

) show that the presence of TK is found especially in those questions that refer to the use of GeoGebra, such as the following question, “8. Indicate what aspects about GeoGebra have you reinforced during the experience” (with 37.14%) or the question, “3. Indicate what aspect/s have been most useful to you in the experience” (37.86%). A representative example of this type of response is “Learning to check things using GeoGebra”. These types of responses denote a focus on the technological tool within the recognizing developing stage without necessarily implying a reflection on how that tool integrates with teaching or content.

The responses on CK are especially concentrated in questions such as “12. Indicate the advantages of carrying out a similar experience with your future students” (35.71%), with a direct connection with the accepting developing stage, or in the question “7. If you have learned a new mathematical concept during the experience, indicate what concept/s it is” (28.57%). Here, we have identified expressions such as “See mathematical concepts in your daily life” that reflect the connection between the disciplinary content and its applicability, although without necessarily alluding to how to teach it.

Regarding the PK dimension, this stands out in questions 11 and 12 about reasons why they would or would not carry out a similar experience with their future students (34.29%) and the advantages of carrying out a similar experience with their future students (30.0%). In this context, we found responses such as “It is a very visual application that can help many people who have difficulty imagining certain mathematical aspects and need to try and see them”. These types of statements are contained within the accepting developing stage, driving attention to diversity, student learning experience and motivation, which are key aspects of pedagogical knowledge.

The integration among domains of the model (TCK, TPK, PCK, TPACK) appears less frequently, which could be interpreted as a reflection of the level of professional development of the participants. Since these are students in initial training, it is logical that the integrated dimensions appear to a lesser extent since they require a holistic view of teaching practice. However, some answers point to a more complex reflection on teaching with technology, for example, “The possibility of materializing certain concepts that are complex, which would facilitate their understanding. In addition, students participate actively in an experiential and playful way where they must face challenges and overcome themselves in order to continue advancing.” In fact, this fragment could be coded as TPACK for its simultaneous reference to technology, disciplinary content and didactics, which are articulated in the same meaningful experience.

In relation to the accepting stage in the TPACK development of the participants in the study, 70% showed positive attitudes towards the use of MathCityMap and GeoGebra Discovery with a similar experience in their future classrooms, 25% had negative opinions and only 5% did not show a clear decision. These results showed extensive development at this stage. The main reasons included in the positive responses were the visual capacities of the technologies used, the meaningful and real-life learning experience and possibility of increasing students’ motivation. On the contrary, the main major concerns and barriers to implementing the activity in the future include the lack of technological resources and time, the diversity of student levels, lack of confidence and teacher training and aspects related to the use of technology such as students’ distraction or excessive use of screens.

4.2. Results by TPACK Domain According to University and Degree

The results of the analyses by university of origin and degree allow us to establish certain trends in the way in which the different domains of the TPACK model are manifested. The differences observed in the percentages of the presence of each domain, as well as in the quality of the responses, suggest that the institutional and educational context directly influences the way in which future teachers conceive the relationship among technology, pedagogy and content.

Specifically, in this analysis, the qualitative responses of students were grouped according to their university and degree, making it possible to identify differentiated patterns depending on the level of education (see

Table 4. Percentage of each dimension by university and degree.

University	Degree	TK	CK	PK	TCK	TPK	PCK	TPACK
Centro Universitario La Salle	Degree Primary Ed.	38.82	20.94	24.94	4.47	5.41	4.24	1.18
Universidad Complutense de Madrid	Double Degree Early Childhood Ed./ Primary Ed.	35.38	33.49	12.97	9.67	3.54	3.77	1.18
	Master’s Degree Teacher Training for Secondary	30.56	27.08	19.10	6.94	4.86	8.68	2.78
	Master’s Degree Research/ Innovation	29.13	32.04	19.42	7.77	4.85	5.83	0.97
Universidad Rey Juan Carlos	Degree Primary Ed.	22.45	42.86	24.49	2.04	0.00	8.16	0.00
	Master’s Degree Teacher Training for Secondary	27.54	25.75	22.75	7.78	7.19	7.78	1.20

).

In the group of undergraduate degrees, relevant differences are observed among institutions, especially in the emphasis that each one places on the different domains of TPACK.

In the first place, the responses of the students of the Bachelor’s Degree in Primary Education taught at the Centro Universitario La Salle participating in the experience show a marked orientation towards technological knowledge (TK), which reaches 38.82% of the total number of codifications. This percentage, the highest in TK among all the degrees analyzed, reveals a strong presence of instrumental and technical aspects in the students’ perceptions. To this are added relevant percentages in pedagogical knowledge (PK, 24.94%) and content (CK, 20.94%), which configures a relatively balanced profile, although with a clear predominance of the technological component. The presence of technological–pedagogical knowledge (TPK, 5.41%) and pedagogical knowledge of content (PCK, 4.24%) is moderate, while the TPACK domain—which implies a deep integration of the three elements—is limited (1.18%). This pattern is reflected in responses such as the following, “GeoGebra allows classes to be more visual and fun, I would use it to capture the students’ attention”, where an awareness of the motivational and visual value of technology can be appreciated, as well as showing a favorable attitude towards the inclusion of this technology in their professional future (accepting stage).

On the other hand, the responses of the future teachers of Primary Education who are studying for a degree in Primary Education at the Universidad Rey Juan Carlos show a different trend. Here, content knowledge (CK) represents 42.86% of the coding, constituting the highest value recorded in this domain. To this is added a significant presence of pedagogical knowledge (PK, 24.49%), but with a lower weight of the technological component (TK, 22.45%). Even more relevant is the total absence of the TPACK domain, which suggests a conceptual separation between technology and educational practices. This is illustrated in responses such as, “It helped me to better understand the concept of similarity, although the tool itself did not seem so useful”, where a more passive use of technology is appreciated without an integrative reflection on its role in teaching, showing less progress through the development stages.

The results of the analysis of the responses of the students of the Dual Degree in Early Childhood Education and Primary Education taught at the Universidad Complutense de Madrid show a balance between the content (CK, 33.49%) and technological (TK, 35.38%) domains. However, pedagogical knowledge (PK) drops considerably (12.97%), suggesting a training focused on the technical and disciplinary, but with less attention to didactic strategies. In contrast, technological content knowledge (TCK) has one of the highest presences in the analysis (9.67%), indicating an outstanding ability to represent mathematical concepts using digital tools and evidence of accomplishing the developing stage of recognizing. A response that illustrates this trend is: “With GeoGebra I have been able to see how mediatrices are constructed, which helps me to understand it better”. Despite this, TPACK remains low (1.18%), showing that, although there are efforts to integrate technology with content, these are not completed with deep and experienced pedagogical reflections.

Regarding master’s degrees, the results of the analysis of the responses of the students of the Master’s Degree in Teacher Training for Secondary and Baccalaureate, Vocational Training and Language Teaching (specialization in mathematics) taught at the Universidad Complutense de Madrid show a complete and mature approach from the point of view of the TPACK model. Although technological knowledge (30.56%) and content knowledge (27.08%) are still the most representative, pedagogical knowledge reaches 19.10%, and the integrative domains also have a significant presence: PCK (8.68%), TCK (6.94%) and TPK (4.86%). This group stands out for having the highest presence of TPACK of all the groups analyzed (2.78%), which suggests a real capacity to articulate the three knowledge domains in didactic proposals with technology and compliance with the recognizing and accepting developing stages. This integrative competence is reflected in answers such as “Thanks to GeoGebra, I was able to show proportionality dynamically so that the students could discover it”, where both the use of the resource and a pedagogical intention oriented to meaningful learning can be appreciated.

The results for the Master’s Degree in Teacher Training for ESO and Bachillerato, FP and Language Teaching (specialization in mathematics) taught at the Universidad Rey Juan Carlos are also balanced from the point of view of the TPACK model, although with a somewhat lower presence in each domain. Technological (27.54%), content (25.75%) and pedagogical (22.75%) knowledge are evenly distributed, and the mixed domains of TCK (7.78%), TPK (7.19%) and PCK (7.78%) reflect a more systematic integration than in other degrees. TPACK, with 1.20%, although lower than the results of the master’s degree taught at the Universidad Complutense de Madrid, is in the upper middle range. This balance is manifested in recognizing answers such as “The activity allows students to visualize the content while interacting with the tool”, where the intention to link the conceptual, didactic and technological is evident.

Finally, the results regarding the students of the Master’s Degree in Research and Innovation for the Teaching and Learning of Experimental, Social and Mathematical Sciences taught at the Universidad Complutense de Madrid, in spite of its interdisciplinary approach, show a similar profile to that of the Double Degree in Early Childhood Education and Primary Education taught at the same university, with a strong presence of content knowledge (CK, 32.04%) and technological knowledge (CK, 32.04%), presents a similar profile to that of the Double Degree in Early Childhood Education Teacher and Primary Education Teacher taught at the same university, with a strong presence of content knowledge (CK, 32.04%) and technological knowledge (TK, 29.13%), accompanied by intermediate pedagogical knowledge (PK, 19.42%). Mixed domains are moderately represented (TCK: 7.77%; PCK: 5.83%; TPK: 4.85%) and TPACK appears with a small percentage (0.97%). This is striking, given that the STEAM approach should, in principle, encourage greater integration among these domains. Some answers, such as “It has been interesting to use technology to teach mathematics from a more practical perspective”, show an incipient integrative orientation, but there is still a lack of depth in the articulation of the three components of the model.

4.3. Results by TPACK Domain According to the Gender of the Participants

The results of the comparative analysis between men and women (see

Table 5. Percentage of each dimension by gender.

Gender	TK	CK	PK	TCK	TPK	PCK	TPACK
Male	35.90	27.51	17.48	7.23	5.36	5.13	1.40
Female	32.70	27.92	20.46	6.88	4.49	6.02	1.53

) reveal that technological knowledge (TK) is the domain with the highest presence in both groups, although there is a slight difference in favor of men, with 35.90% compared to 32.70% in women. This can be interpreted as a greater focus by males on the use and operation of tools such as GeoGebra Discovery. This difference does not necessarily imply a greater technical competence but perhaps a more explicit discursive attention to technology. For example, many of the male’s recognizing responses include phrases such as, “I liked seeing how to use GeoGebra to represent dynamic figures”. While women tend to insert these recognizing evaluations within more pedagogical or comprehension contexts, “I have found it useful to check concepts with GeoGebra, especially thinking about how to explain them to my students”.

In terms of mathematical content knowledge (CK), both groups show a very similar level, with 27.51% in males and 27.92% in females. This convergence suggests that the disciplinary dimension has been internalized in a similar way, regardless of gender.

In the PK (pedagogical knowledge) domain, a notable difference is observed, with women registering 20.46%, while men remain at 17.48%. This difference could be interpreted as a greater sensitivity or attention to the pedagogical aspects of the teaching–learning process on the part of women. Female responses tend to allude more frequently to student diversity, motivation and didactic design. An example of this is the recognizing response, “It is a very visual and dynamic activity, ideal for students with spatial difficulties”. Although these types of pedagogical reflections also appear among men, they do so less frequently.

The dimensions involving integration among components (TCK, TPK, PCK, TPACK) are less frequent in general, although here too, small differences are detected (see Table 5). This corroborates that female responses present slightly stronger integration in the domains related to pedagogy (PCK, PK) and in the complete view of the TPACK model, although in absolute terms, these differences are small. Moreover, the fact that the TPACK dimension has a low overall presence (1.40% and 1.53%) in both genders confirms that this type of integration requires a more advanced level of reflection and experience, which is to be expected in a population in initial training.

5. Discussion and Conclusions

The results of the experience show a notable presence of technological knowledge (TK) in the responses of the participating students, followed by disciplinary content knowledge (CK) and, in third place, pedagogical knowledge (PK). The integrated dimensions of the TPACK model appear to a lesser extent, suggesting that these types of experiences that introduce innovative digital tools, such as GeoGebra Discovery, are beginning to generate awareness about the educational use of technology and its didactic applicability, although they do not always promote a deep and interrelated integration of the three types of knowledge that make up the TPACK model (Koehler & Mishra, 2009; Rakes et al., 2022). Thus, the low occurrence of the model’s integrated dimensions (TCK, PCK, TPK, TPACK) in the ratings suggests that prospective teachers are still in the early stages of developing strong didactic–technological competence.

This phenomenon is consistent with previous research, which highlights that the effective integration of technology in teaching requires time, sustained experiences and reflective accompaniment (Niess et al., 2009; Marbán & Sintema, 2021). In this direction, the results obtained in all groups of the study reflect the extent of the recognizing and acceptance stages in the development of TPACK in Figure 5. The participants were able to align the technology used (MathCityMap and GeoGebra Discovery) with the mathematical content, and most of them formed an opinion (favorable or unfavorable) about its use in their future as teachers.

The results obtained by university and degree confirm that the full integration of technological, pedagogical and content knowledge (i.e., the presence of the TPACK domain) is still scarce in initial teacher education. However, there are significant differences in the way in which the different types of model knowledge manifest themselves. Master’s programs, especially the Master’s Degree in Teacher Training for Secondary and High School, Vocational Training and Language Teaching (specializing in mathematics) offered at the Universidad Complutense de Madrid, show a greater maturity and balance in the integration of the model’s domains, which can be attributed both to the students’ previous training and to the professionalizing approach of this type of study (Martínez-Jiménez et al., 2022). In contrast, undergraduate degrees tend to polarize between a technological (Centro Universitario La Salle) or disciplinary (Universidad Rey Juan Carlos) orientation, while the Double Degree in Early Childhood Education and Primary Education and the Master’s Degree in Research and Innovation for the Teaching and Learning of Experimental, Social and Mathematical Sciences, despite their potential, do not consolidate a full integration of the TPACK model.

Likewise, the results of the experience also allow us to draw conclusions according to the gender of the participants, which provides a nuance of great pedagogical interest. Female students tend to focus their evaluations on pedagogical and attentional aspects and on the student experience, with a greater representation of domains such as PK, PCK and TPACK. Males, on the other hand, manifest a more technical focus, highlighting the use and management of software (TK, TCK). This finding is in line with recent studies that show different training sensitivities according to gender, which invites consideration of differentiated and equitable strategies in teacher training processes (Hartsell et al., 2009).

Taken together, these results reinforce the need to explicitly incorporate in teacher training curricula proposals that promote the profound articulation among technology, content and pedagogy. TPACK competence, understood not only as a sum of knowledge but as a complex and situated integration, should be addressed as a key transversal competence for teacher professional development in the current context, characterized by digitization, mobility and active learning in real scenarios (Cahyono et al., 2025; Ariño-Morera et al., 2024).

In this sense, experiences such as the one described here (focused on the use of tools such as GeoGebra Discovery in mathematical modeling activities in real environments) allow the opening of authentic spaces of didactic and technological exploration, favoring not only the understanding of geometric content but also the reflection on how to teach it and with what resources. To maximize their impact, it is essential that these proposals are developed systematically, with didactic planning that contemplates the progressive integration of the domains of the TPACK model and that provides future teachers with opportunities to reflect, experiment and build knowledge from practice.