Modeling functions in motion: an interdisciplinary approach using Tracker and GeoGebra

Abstract

The teaching of motion often presents challenges in linking physical experimentation with mathematical modeling. This study examines how high school students construct and interpret models of uniformly accelerated rectilinear motion through a didactic sequence grounded in the modeling cycle, the Kinematics Working Space, and the theory of semiotic representation registers. Using Tracker and GeoGebra, students transitioned from empirical observations to algebraic representations. Analysis focused on seven explicitly implemented modeling phases showed that most students successfully constructed and validated models by coordinating graphical, algebraic, and verbal registers. While one group struggled with interpreting trajectory graphs and functional parameters, all engaged in meaningful conceptual work. The study confirms that theoretically grounded and technology-supported environments promote interdisciplinary articulation and foster a reconceptualization of the notion of function, which is not restricted to an algebraic expression but encompasses processes of covariation, the coordination of semiotic registers, and the modeling of relationships between physical quantities.

Keywords
Mathematical Modeling; Kinematics Working Space; Interdisciplinarity; function; Uniformly Accelerated; Rectilinear Motion

1. Introduction

The design of interdisciplinary educational activities supported by digital technologies has gained increasing relevance in recent years, particularly in STEM education and mathematical modeling contexts [¹, ², ³, ⁴]. However, the convergence of different disciplinary domains, such as Physics and Mathematics, continues to pose significant epistemological, methodological, and cognitive challenges, especially in high school and undergraduate settings. A paradigmatic case of this complexity is the teaching of kinematics, where the object of study – the motion of bodies – requires both experimental treatment and symbolic formalization. Although Physics and Mathematics have historically maintained a close relationship in the study of motion [⁵, ⁶], their epistemic frameworks differ substantially. Physics tends to privilege experimental validation and the use of physical magnitudes contextualized to real phenomena, while Mathematics constructs its models from abstract variables, operating in formalized and deductive frameworks [⁷, ⁸]. This epistemic gap generates learning difficulties for students, who must navigate between the two disciplines without always having tools that allow them to coordinate registers or recognize the meaning of mathematical objects in physical contexts [⁹, ¹⁰, ¹¹].

A particularly illustrative case is the modeling of uniformly accelerated rectilinear motion (UARM), in which the notion of function, understood as a functional relationship between physical quantities, plays a central role. Depending on the disciplinary lens, the same function can be interpreted with reference to experimental magnitudes (position, velocity, acceleration) or through abstract representations such as graphs, tables, and equations. However, the ability to move between these semiotic registers [¹², ¹³] and assign meaning to them depends on students’ experiences and the didactic design of the tasks involved. The study by Dutra et al. [³] investigated how an educational video game supports students’ understanding of graphs in kinematics through real-time visualization of motion. The sample consisted of 54 ninth-grade students (14–15 years old) from Uruguay. The results revealed high levels of motivation, engagement, and perceived learning, evidencing improvements in the interpretation of kinematics graphs and in the articulation between physical motion and mathematical representation. In a related study, Dutra, Suárez, and Marti [¹⁴] examined how computational thinking can be integrated into the teaching of kinematics through a teaching sequence based on programming and motion modeling. The sample consisted of three groups of ninth-grade students (14–15 years old) in Uruguay who already possessed basic knowledge of kinematics and Python programming. A learning situation was implemented in which students modeled accelerated motion, constructed functions, and analyzed graphs using computational programming. The results showed that students were able to apply computational thinking skills to analyze uniformly accelerated rectilinear motion (UARM), thereby strengthening conceptual understanding, graph interpretation, and the articulation between coding, mathematics, and physics.

The study conducted by Markus Hohenwarter, Zsolt Lavicza, and Wassong [²] analyzed the role of the dynamic mathematics software GeoGebra in the development of mathematical modeling and visualization processes across different educational levels. The work mainly consisted of a theoretical study and a review of previous teaching experiences; therefore, it did not focus on a single specific sample, but rather on various implementations of modeling activities supported by digital technologies. The authors emphasize that the use of dynamic software facilitates transitions between graphical, algebraic, and geometric representations, promoting processes of exploration, validation, and conceptual construction. The findings indicate that tools such as GeoGebra enhance functional understanding and the modeling of real-world phenomena, thereby strengthening the articulation between experimentation, representation, and mathematical formalization.

This study explores how students interpret and construct mathematical models of UARM through the combined use of two technological tools: Tracker, which allows the capture and quantification of real motion with physical units and variables, and GeoGebra as dynamic mathematics software. These environments not only operate in different representational registers but also activate distinct cognitive processes that reflect the underlying disciplinary logic of each domain.

To analyze the interplay between these tools, representations, and cognitive processes, we adopt the theoretical framework of the Kinematics Working Space (KWS) [¹⁵], based on the Mathematical Working Space (MWS) [¹⁶, ¹⁷], which allows us to characterize students’ epistemological positioning across three paradigms: Reality (KI), Quantification (KII), and Formalization (KIII). In parallel, we employ the Modeling Cycle of Blum and Leiss [¹⁸], which structures the modeling process into seven phases, making it possible to trace the students’ conceptual and representational progress from the empirical domain to formal abstraction. This dual theoretical framework enables a comprehensive analysis of how meaning is constructed and negotiated in interdisciplinary contexts.

The study involved 16 high-performing high school students from Universidad Nacional Autónoma de México (UNAM) enrolled in a differential and integral calculus course. This sample of students was selected because it was the only class to which access was available. Although this circumstance may introduce a potential bias in the results, implementing the learning situation was considered particularly valuable, given that the proposed methodology is innovative and largely unfamiliar to the students. Through a structured didactic activity, students recorded motion using Tracker, analyzed data, constructed algebraic models in GeoGebra, and justified their interpretations through written and graphical representations. The activity was designed to guide them progressively through all seven phases of the modeling cycle: understanding, simplifying and structuring, mathematizing, working mathematically, interpreting, validating, and presenting. Each phase was analyzed through the three dimensions of the KWS: semiotic, instrumental, and discursive genesis. The analysis focused particularly on how students construct new meanings for the concept of function in a physical context, how they coordinate semiotic registers, and how they transition between epistemological paradigms when modeling motion.

Section 2 presents the theoretical framework, Section 3 describes the methodology and didactic activity, Section 4 analyzes the results, and Section 5 presents the conclusions.

2. Theoretical Framework

2.1. Registers of semiotic representation

The study of phenomena related to mathematical knowledge requires a deep understanding of the notion of representation, as stated by Duval [¹⁹]. In this context, registers of semiotic representation play a fundamental role in the construction of knowledge, especially in the articulation of concepts coming from physics and mathematics. This research uses the theory of registers of semiotic representation and their conversion to analyze how students mobilize different representation systems when modeling UARM.

From a theoretical perspective, we start from two fundamental premises. First, mathematical understanding does not arise exclusively within a single register of representation, but requires the coordination of at least two different registers [²⁰]. This implies that the learning of a mathematical concept, such as function, cannot be considered complete if it is limited to a single representational system, such as the algebraic or the graphical. Second, semiotic representations are specific to a particular system of signs and, when converted to another system, may acquire different meanings for the user [²¹]. This phenomenon suggests that the transition from one register to another is not automatic and can sometimes generate difficulties in understanding the modeled phenomenon.

To analyze the production and conversion of semiotic representations in this study, we consider three fundamental aspects established by Duval [²²].

1. The structural aspect allows us to examine the nature of the signs, their rules of combination, and the dimensions in which they can be represented.

2. The phenomenological aspect focuses on the cognitive and perceptual limitations of students in producing and understanding signs.

3. The functional aspect is related to the cognitive operations that can be performed within a register, such as the formation of a representation, its processing through internal transformations, and the conversion from one register to another while preserving its meaning.

These three aspects will be used later when considering a register of semiotic representation within physics.

2.2. Kinematics Working Space

The Kinematics Working Space (KWS) arises as an extension of the Mathematical Working Space (MWS) [¹⁷] with the objective of analyzing and characterizing the teaching and learning of kinematics from an interdisciplinary perspective between Mathematics and Physics [¹⁵]. This theoretical framework allows us to study how students move from an intuitive and experimental understanding of motion to its mathematical formalization, addressing the differences in the representations and cognitive processes specific to each discipline.

Unlike traditional teaching, where Mathematics and Physics are usually presented as separate areas, KWS epistemologically articulates mathematical and physical knowledge, highlighting how each discipline models motion differently. In Physics, motion is described from measurable quantities, with emphasis on experimentation and interpretation of physical phenomena. Mathematics operates through symbolic systems and formalized structures in which functions may be treated as abstract objects independent of the physical phenomena from which they originally emerged. The interaction between these two approaches gives rise to distinct cognitive processes and therefore requires an analytical model capable of explaining how students coordinate and move between these domains of knowledge.

For this purpose, KWS introduces three paradigms that organize and structure the work in kinematics and are represented in diagrams [¹⁵], Figure 1:

• Reality paradigm (KI): Inspired by Aristotelian philosophy, this paradigm is based on the subject’s convictions, experiences, contexts and prior knowledge when approaching the study of movement. Here, students are expected to use mainly material artifacts, although technological tools may also appear with the aim of recreating and constructing the desired movement. The representations generated in this paradigm are mostly experimental and qualitative, including symbols, diagrams, and icons, which help to describe the phenomenon. The theoretical referential in KI is composed of basic knowledge about physical motion, without a structured mathematical formalization. The discourse in this paradigm arises from the perception of motion and is characterized by being descriptive, based on direct observations and intuitive arguments. In terms of validation, pragmatic evidence in the sense of Balacheff [²³] predominates, i.e., explanations that are supported by experience and common sense, without yet resorting to calculations or formal mathematical models. The schema for KI is shown in Figure 1a, where the representations are directly associated with the perceptions.

• Measurement and quantification paradigm (KII): Inspired by the historical process of the mathematization of motion developed between the Late Middle Ages and the Renaissance, and particularly by Galileo’s studies on kinematics, this paradigm marks the transition from qualitative analysis to the quantification of motion. Unlike KI, where explanations are built on sensory perception and common sense, in this paradigm measuring instruments are introduced that allow the quantification of variables such as time, position, velocity, and acceleration, thus establishing a first level of abstraction. In KII, artifacts can be material, technological, and/or symbolic, with the specific purpose of quantifying movement. Although mathematics begins to be present at this level, it does not yet appear in an axiomatic-deductive sense, but as a tool to represent and analyze the data obtained. In this sense, the forms of validation in KII are quantitative, derived from measurements and numerical calculations that validate the relationships between physical variables. Likewise, concepts and laws of physics begin to emerge, allowing a more structured modeling of motion and facilitating the transition to more advanced levels of abstraction. The schema for KII is shown in Figure 1b, where the representations are associated with the transition from perception of the physical motion phenomenon to visualization, considering visualization in the sense of Duval [²⁴], who defines it as the ability to relate different representations of the same mathematical object and thereby give it meaning.

• Formalization and idealization paradigm (KIII): At this level, motion is approached from the abstract, using mathematical models and functions to describe and predict it. This paradigm is associated with the historical development of the mathematization of motion, particularly through analytic geometry (Descartes and Fermat), infinitesimal calculus (Newton and Leibniz), and the subsequent analytical formalization of the concept of function (Euler). This paradigm also incorporates the vision of Einstein, who took abstraction even further through mental experimentation, where the conceptualization of motion no longer depends on direct physical experimentation, but on completely idealized theoretical models. In this paradigm, the instruments are symbolic and the proofs are intellectual in the sense of Balacheff [²³], characterized by reasoning based on chains of arguments and symbolic language. The mathematical formulation of motion seeks not only to describe specific cases, but to generate universal laws that can predict any idealized kinematic behavior. Mathematical thinking at this level is predominantly axiomatic-deductive, allowing the generalization of the phenomenon of motion from equations and theoretical principles. Here, the cognitive processes are visualization, construction and proving by means of mathematical demonstrations and formal argumentation in physics (Figure 1c).

Figure 1
KWS paradigm diagrams.

In addition, KWS structures the work in kinematics through three key dimensions, which allow analyzing the teaching and learning processes from an integral perspective:

• Semiotic Genesis: refers to representations in different registers of representation. In KI, representations are related to sensory perception (gestures, body movement). In KII, graphic and tabular registers emerge as quantification tools. In KIII, representations become symbolic and mathematically structured.

• Instrumental Genesis: Focuses on the use of artifacts to model movement. In KI, the artifacts are mainly material objects (physical objects in motion). In KII, technological and symbolic tools such as Tracker, GeoGebra, etc. are incorporated to analyze data. In KIII, the artifacts are mainly symbolic (equations, derivatives, theoretical models).

• Discursive genesis: Related to argumentation and proving of knowledge. In KI, discourse is based on sensory experience and intuition. In KII, justifications based on measurements and mathematical relations are introduced. In KIII, argumentation becomes deductive, based on mathematical demonstrations and physical laws.

In this study, the semiotic, instrumental, and discursive geneses of the KWS constitute not only theoretical categories but also analytical tools for interpreting the mathematical and physical work developed by the students. The semiotic genesis made it possible to analyze how students coordinated different registers of representation – physical, graphical, tabular, algebraic, and verbal – throughout the modeling process of uniformly accelerated rectilinear motion (UARM). The instrumental genesis enabled the examination of the mediating role of material and technological artifacts, particularly Tracker and GeoGebra, in the construction and transformation of representations. Finally, the discursive genesis allowed us to examine how students justified, interpreted, and validated their models through both physical and mathematical argumentation. These three dimensions were considered both in the design of the activity and in the analysis of students’ productions.

2.3. Modeling cycle

Several investigations in mathematics education have highlighted the role of mathematical modeling as a mechanism to promote the development of cognitive processes and nonmechanical ways of thinking, which can favor deeper and more meaningful learning. For example, Pollak [²⁵] emphasized the importance of connecting mathematics with other disciplines to enrich students’ conceptual understanding; Blum and Niss [²⁶] analyzed problem solving in applied contexts, highlighting modeling as an essential component in the teaching of mathematics; Blum [²⁷] extended this perspective by examining the teaching of mathematical modeling and its impact on the development of mathematical competencies, stressing the need to provide students with tools to construct and validate mathematical models. From this line of research, Blum and Leiss [¹⁸, ²⁸] studied the way in which students and teachers face modeling problems, identifying didactic obstacles and pedagogical strategies that can facilitate the transition between the real world and its mathematical representation. In a complementary manner, Borromeo-Ferri [²⁹] analyzed the cognitive processes involved in modeling, focusing on the difficulties that arise when switching between different registers of representation. From a broader perspective, Lesh and Doerr [³⁰] proposed an iterative approach to mathematical modeling, in which students generate, validate and refine models as part of a structured learning process; Kaiser et al. [³¹] provided an international overview of trends in the teaching of mathematical modeling, highlighting its growing relevance in different educational levels and disciplinary contexts. From these investigations and many others that have explored the impact of mathematical modeling in education, it has been shown that this approach is an effective strategy to guide students in the analysis of real-world phenomena and their translation into mathematical language, allowing them to give meaning to the mathematical objects involved. Mathematical modeling not only fosters the construction of knowledge, but also favors the articulation between diverse representations and the development of flexible mathematical thinking. For this reason, we have decided to employ this approach in this study not only for the design of the Uniformly Accelerated Rectilinear Motion (UARM) modeling activity, but also for the analysis of the results obtained, thus providing a more structured view of the learning processes involved.

The modeling cycle we adopted in this study is the one proposed by Blum et al. [³²], Figure 2, which allowed us to structure the activity progressively along seven phases. We opted for this model because its structure and organization of the phases facilitate the transition between the physical phenomenon observed in the real world and its mathematical representation, promoting a deep understanding of the concept of function in the context of Uniformly Accelerated Rectilinear Motion (UARM).

Figure 2
Modeling cycle diagram (Blum and Leiss, 2007, p. 225).

The 7 phases of the cycle are:

1. Understanding the task: In this first phase, a real-world phenomenon, problem, or situation is introduced that needs to be analyzed and modeled. Students explore the situation to generate a first qualitative model, based on their intuition and prior knowledge. This initial model is not yet mathematical, but incorporates physical and contextual elements from the real world that allow characterizing the phenomenon.

2. Simplifying/structuring: The information gathered in the previous phase is organized, including graphs, experimental measurements, simulations or data. The goal is to reduce the complexity of the problem and structure a realistic model that can be mathematized in the next phase. At this point, students must make decisions about which aspects of the phenomenon are essential and which can be ignored or idealized to facilitate the analysis.

3. Mathematizing: This phase marks the bridge between reality and mathematics. Students begin to formulate mathematical relationships that describe the phenomenon, identifying key variables and establishing equations that model their behavior. In the case of UARM, this involves recognizing the relationship between position, time, velocity and acceleration, laying the foundation for the construction of the mathematical function that describes the movement.

4. Working mathematically: In this phase, students operate with the mathematical model obtained in the previous phase, performing calculations, algebraic transformations and graphic analysis. Here, relevant mathematical concepts are applied, such as the quadratic position–time function associated with uniformly accelerated rectilinear motion (UARM), its derivatives, and its properties, in order to generate predictions and solve the proposed problem.

5. Interpretation: Once the mathematical results are obtained, they must be analyzed and contextualized in relation to the original phenomenon. Students are expected to interpret the values obtained in terms of the observed motion, establishing correspondences between the mathematical representation and the physical reality.

6. Validation: In this stage, students confront the results with the real situation to evaluate their relevance and accuracy. It is analyzed whether the mathematical model correctly describes the phenomenon and whether the simplifying assumptions adopted in previous phases are still valid. In case of inconsistencies, students can readjust the model and restart the cycle with modifications in the previous phases, thus ensuring an iterative process of refinement.

7. Presenting: The last phase consists of communicating the modeling process and its results. Students present their work, explain the decisions made in each phase, justify their model and discuss possible improvements or generalizations. This stage is crucial to consolidate learning, as it encourages reflection on the process and the meaning of the mathematical objects used in modeling.

3. Methodology

3.1. Characteristics of students and theacademic institution

The sample of students with whom we worked in this study corresponds to students of good academic level, enrolled in a campus of the National Preparatory School, belonging to the National Autonomous University of Mexico in the metropolitan area of Mexico. The participants were in their last year of high school and were between 17 and 18 years of age. At the time of the modeling activity, the students had already taken key subjects such as algebra, geometry and trigonometry and had begun to address differential and integral calculus content. Due to space limitations in the classroom, the activity was carried out with a reduced group of 16 students. The institution provided access to the necessary materials and equipment, and the study could take place in a suitable environment.

3.2. The modeling activity

The following didactic activity was designed to guide students in modeling Uniformly Accelerated Rectilinear Motion (UARM) by integrating experimental observation, digital tools, and mathematical representations. Through six structured tasks (Figure 3), students transition from observing real motion to constructing and interpreting mathematical models using Tracker and GeoGebra. A central aspect of this activity is the role of functions and equations in describing motion. Students analyze how position, velocity, and acceleration evolve over time by deriving and comparing equations. They establish correspondences between the position function, its first derivative (velocity equation), and its second derivative (acceleration), reinforcing the mathematical relationships that govern motion. Additionally, by working with parametric equations, they explore how trajectory can be represented algebraically, bridging the gap between graphical and algebraic representations.

Figure 3
Modeling activity

To design and implement the experimental activity on Uniformly Accelerated Rectilinear Motion (UARM), we integrated the Kinematics Working Space [¹⁵] and the Mathematical Modeling Cycle of Blum and Leiss [¹⁸]. These frameworks allowed us to structure an activity in which students not only explored acceleration from an experimental perspective, but also analyzed it mathematically through coordinated transitions between physical, graphical, tabular, and algebraic representations [²⁴]. In the design of the activity, the modeling cycle of Blum and Leiss provided the sequential organization of the tasks, guiding students from the observation of motion to the construction, interpretation, and validation of mathematical models. Simultaneously, the Kinematics Working Space framework informed both the design and the analysis through its epistemological and cognitive dimensions, particularly by examining how students transitioned across the paradigms of reality (KI), quantification (KII), and formalization (KIII), and how semiotic, instrumental, and discursive geneses were activated during the coordination of representations and the use of Tracker and GeoGebra. Our strategy combined experimentation with digital tools to guide students in the modeling process. We used Tracker to analyze the real movement, generating data in tables and graphs with variables from the physics domain, such as horizontal and vertical position, time, velocity and acceleration. We then used GeoGebra to transform this data into mathematical representations, including graphs with coordinates in terms of x and y, as well as algebraic expressions. This integration allowed students to understand the relationship between the concept of function and physical motion, promoting an interdisciplinary vision between physics and mathematics.

From the design of the activity, we established a bridge between empirical exploration and mathematical formalization, promoting the transition between different levels of abstraction in the study of UARM. To this end, (i) we defined the study problem in terms of the physical phenomenon of UARM and the difficulties faced by students in understanding this movement in relation to the mathematical function. Specifically, we addressed the lack of physical meaning attributed to the mathematical object of function in the context of UARM. (ii) We selected the appropriate technological tools to enable detailed observation of the phenomenon (Tracker) and subsequent mathematical modeling (GeoGebra); (iii) we structured the activity based on the Blum and Leiss [¹⁸] Modeling Cycle, ensuring that each phase would help students develop a progressive understanding of the phenomenon, from observation to model validation; and (iv) we incorporated the KWS as a theoretical framework to characterize and interpret how students transitioned between different paradigms (reality, measurement and quantification, and formalization).

3.3. Semiotic registers of representation in the modeling of UARM

In the UARM modeling activity, we consider five semiotic representation registers. One of them is specific to the domain of physics and the other four are traditional in mathematics.

• Physical Motion (PM): This register comes from kinematics and is generally not considered in mathematics. However, in this study we treat it as a register of semiotic representation due to its ability to fulfill the three cognitive operations described by Duval [²²].

  • –

    Formation: It is obtained by observing or recreating a UARM in a physical experiment, such as the movement of a rolling ball or the displacement of an object on an inclined track, etc.

  • –

    Processing: The experimental conditions can be modified, for example, by changing the velocity or comparing different motions.

  • –

    Conversion: The physical motion data can be transformed into other registers, such as graphs or algebraic equations.

  This register is key to introducing the notion of function through physical experimentation.

• Natural language (NL) plays an important role in modeling. In this study, we distinguish the written form, which appears in work guides and in the reports prepared by the students.

• Table of values (TV), which is constructed using the Tracker software. This change represents a first conversion of register of representation, since it makes it possible to transform the qualitative observation of the phenomenon into quantifiable information.

• Graphic

  1. –

     Graph Tracker (GT): Uses physics notation, contextualized to the movement.

  2. –

     Graph GeoGebra (GG): Uses standard mathematical notation with Cartesian axes.

• Algebraic

  1. –

     Specific Algebraic (SA): It is obtained by fitting experimental data in GeoGebra, providing a function that models the particular motion recorded.

  2. –

     General Algebraic (GA): It represents the general equation of UARM with its theoretical parameters.

3.4. Application of the Kinematics Working Space (KWS) in the UARM activity

The Kinematics Working Space allowed us to characterize the activity and analyze how students mobilized their knowledge by interacting with the phenomenon of Uniformly Accelerated Rectilinear Motion (UARM). From an epistemological perspective, the KWS helped us to structure the activity around the key concepts involved, such as the relationship between position and time, as well as the different registers of representation used in physics and mathematics. From a cognitive perspective, it allowed us to observe how students transitioned between different levels of abstraction and articulated their reasoning when interpreting experimental data and converting them into mathematical expressions.

To further analyze the learning process, we considered three fundamental dimensions within the KWS framework:

• Epistemological dimension: We established the conceptual framework of UARM, defining key physical variables such as position, velocity, and acceleration. In addition, we incorporated mathematical objects such as functions, derivatives, and parametric equations that are essential to describe and analyze uniformly accelerated motion. This allowed students to connect physical phenomena with their mathematical representations in a structured way.

• Cognitive dimension: We observed how students interpreted experimental data and connected them with algebraic and graphical representations.

• Semiotic, instrumental, and discursive genesis: We designed the activity to encourage students to transition between different registers of representation (graphical, tabular, algebraic, and verbal). In addition, we integrated technological tools such as Tracker and GeoGebra, which facilitated visualization and mathematical processing of the phenomenon.

During the activity, we used the KWS to analyze how students progressed through the paradigms that this model proposes:

• Reality Paradigm (KI): We sought to place students in this paradigm by recreating accelerated motion in a real environment. To do so, they used material artifacts and made video recordings that allowed them to capture and observe the phenomenon of motion experimentally.

• Transition from the reality paradigm to the measurement and quantification paradigm (KI–KII): Subsequently, we promoted a transition towards quantification of the phenomenon through the use of Tracker. By extracting numerical data and generating graphical representations of the movement, the students moved from a qualitative perception to a quantitative representation. In this process, Tracker played a key role, as it allowed them to measure and analyze the motion, facilitating the transition from the KI to the KII paradigm.

• Transition from the measurement and quantification paradigm to the formalization paradigm (KII–KIII): Finally, with GeoGebra, we asked students to fit mathematical models to the data obtained, transforming the empirical information into algebraic expressions that would describe the phenomenon of movement more accurately. This process allowed us to analyze up to what level of abstraction the students were able to reach, identifying whether they remained in the KII paradigm or whether they were able to advance to KIII, where mathematical modeling and generalization take on greater importance.

The KWS not only allowed us to structure the activity in a coherent way, but also functioned as a tool to analyze how students constructed meaning from experimentation. Throughout the process, we observed how semiotic, instrumental, and discursive registers emerged as students interpreted the data, formulated hypotheses, and validated their mathematical models.

3.5. Application of the Blum and Leiss Modeling Cycle

The Blum and Leiss Modeling Cycle served as a methodological structure to guide students in the construction and validation of UARM models. The activity went through the following phases:

1. Understanding the task: Selection of the material and planning of the experiment. In this phase, students identify the phenomenon to be analyzed and select the appropriate materials to record a UARM movement. They are asked to justify their choice considering observational and experimental aspects.

2. Simplification and structuring: Exploring variables and graphs in Tracker. Here, students analyze the data collected with Tracker, exploring the variables and graphs generated by default. They are asked to evaluate which of these representations may be relevant to their analysis, promoting the transition from the actual problem to a more structured representation.

3. Mathematization: Graphical analysis and mathematical representation in GeoGebra. Students relate the equations obtained in GeoGebra to the variables appropriate to the observed motion. They must also identify the coordinate axes and determine which of the three graphs x versus t, y versus t and y versus x represents the movement and why.

4. Mathematical work: Trajectory equation and parametric equations. In this phase, students generate parametric equations from the experimental data and analyze how these would represent the trajectory in an ideal case. Here, mathematical calculations begin to take center stage, linking the equations to the kinematics of the UARM.

5. Interpretation: Comparison of equations of motion. Students compare the obtained equation with the theoretical equation of UARM, determining the term-to-term correspondence. In addition, they derive the position equation and relate it to the velocity equation $v_f=v_0+\text{at}$. Finally, they find the second derivative and analyze its physical meaning, establishing a link between the mathematical models and their interpretation in the context of motion.

6. Validation: Analysis of results from the graphs, velocity and acceleration. Here, students evaluate whether the obtained models adequately represent the physical phenomenon. They compare their experimental results with the mathematical predictions and explain what they can conclude from the velocity and acceleration graphs, validating or adjusting their model according to the observed discrepancies.

7. Presentation: This phase was not explicitly developed in the activity questions; however, the students presented their results at the end of the sessions in order to disseminate their results and obtain feedback from their classmates and teacher

4. Analysis of Results

In this study, the modeling cycle proposed by Blum and Leiss [¹⁸] and the Kinematics Working Space (KWS) were articulated in a complementary manner both in the design of the activity and in the analysis of students’ productions. The modeling cycle provided the temporal and epistemological structure of the process, organizing students’ work into successive phases – understanding, simplification, mathematization, mathematical work, interpretation, and validation – which made it possible to identify the progression from empirical observation to the mathematical formalization of uniformly accelerated rectilinear motion (UARM). In turn, the KWS operated as an analytical tool to characterize the nature of the cognitive work developed at each phase of the cycle, identifying the mobilized paradigms (KI: reality, KII: quantification, and KIII: formalization), as well as the semiotic, instrumental, and discursive geneses activated during task resolution. Thus, while the Blum and Leiss cycle enabled us to describe “at what stage” of the modeling process the students were situated, the KWS allowed us to analyze “how” they constructed meaning, coordinated registers of representation, and transitioned across different levels of abstraction. This methodological articulation enabled a fine-grained analysis of the transitions between physical experimentation, quantification through Tracker, and algebraic formalization in GeoGebra, revealing how students resignified the notion of function within an interdisciplinary context connecting mathematics and physics.

4.1. Understanding and simplifying the motion phenomenon, phases one and two

The initial phase of the modeling cycle, corresponding to the understanding and construction of the movement phenomenon, represents a fundamental moment in which students begin to construct the physical situation from their previous knowledge and experience and their direct experimentation. At this stage, the cognitive processes related to the reality paradigm (KI) of the Kinematics Working Space (KWS) are mainly activated, in which knowledge emerges from sensory perception, intuition, and previous convictions of the students. From the epistemological approach of the KWS, this phase is characterized by a representation of the phenomenon centered on the recording of Physical Motion (PM), in which students mobilize material artifacts – such as everyday objects – to generate a movement representative of the UARM. In this study, the four participating groups activated, in the reality paradigm (KI), the three geneses: the instrumental, by selecting and manipulating the artifacts to construct the movement; the semiotic, by generating the PM representation of the phenomenon and modifying the variables involved, which were perceived with their senses; and the discursive, by contrasting their perceptions of the PM with their previous theoretical referential, which allowed them to argue whether the variables corresponded to a UARM or whether it was necessary to repeat the experiment. The written and oral responses in natural language (NL) in the guides reflect an argumentative discourse of a descriptive nature, linked to a practical and intuitive knowledge of the movement. This type of argumentation is consistent with the type of pragmatic validation described by Balacheff [²³], in which judgments are based on immediate experience rather than on theoretical structures or formal demonstrations. It should be noted that the students freely chose both the artifacts and the context of the movement, which made it possible to identify the richness and diversity of their initial conceptions, as well as the way in which these guided the construction of the physical model. The movement phenomena recorded were as follows: group 1 dropped headphones on the inclined leg of a table (Figure 4a); group 2 filmed one of its members running in an accelerated manner from rest (Figure 4b); group 3 attempted a vertical shot with a soccer ball, which resulted in a parabolic shot, which they decided to analyze upon recognizing that, although it was not a rectilinear motion, it was uniformly accelerated (Figure 4c); finally, group 4 dropped a ball in free fall (Figure 4d).

Figure 4
Movements made by students.

It is also noteworthy that, although there was not yet a symbolic representation of the phenomenon, the students began to establish causal and temporal relationships in terms of variations in position or changes in state (rest-motion), which is an indicator of the emergence of cognitive structures conducive to modeling. This observation is in line with Duval’s approach, who maintains that the construction of meanings begins even in non-formal registers, as long as they are articulated with cognitive activities of formation and conversion of representations. From the perspective of semiotic registers, the use of natural language to describe and justify the experiment allowed us to observe the way in which students construct meanings before accessing graphic, tabular, or algebraic registers, as we can observe in Group 1’s response to the first question (the other groups gave similar responses), Figure 5.

Figure 5
First answer of Group 1.

This register becomes a key bridge between the perception of the physical phenomenon and its eventual translation into mathematical representations. Additionally, the KWS allows us to analyze how the articulation between material artifacts and semiotic meanings is established, highlighting that in this phase the artifact is part of a process of instrumentalization since its use is intentional and conscious to model the phenomenon of movement. This stage is necessary so that later, with the incorporation of technologies such as Tracker and GeoGebra, the instrumental genesis is activated at more abstract levels.

Finally, in this phase, the understanding of the phenomenon is not limited to a phenomenological recognition, but implies an epistemic positioning from which students attribute meaning to the movement in terms of observable physical variables. This initial appropriation, still situated at the level of empirical experience, establishes the basis for the progressive development of an interdisciplinary discourse between Physics and Mathematics, in which the notion of function can be re-signified.

4.2. Mathematizing, phase 3

The mathematization phase of Blum and Leiss’ modeling cycle represents the key transition between observed empirical reality and formal mathematical formulation.

In this stage, students begin to establish correspondences between the variables of the physical phenomenon and the mathematical objects that allow its description. From the framework of the KWS, this transition is mainly situated in the passage from the paradigm of reality (KI) to the paradigm of measurement and quantification (KII), where a new cognitive configuration oriented towards abstraction through symbolic and quantitative registers is activated.

In the study, this transition was mediated by the combined use of two digital tools: Tracker, associated with the physics domain, and GeoGebra, corresponding to the mathematics domain. The implementation of both software tools configured a dual instrumental trajectory that allowed a progressive cognitive reorganization in the students. Tracker provided an experimental measurement environment in which students could access data in tabular (TV) and the graphical Tracker (GT) register, using physics notation and units. GeoGebra, in contrast, allowed them to operate with formal mathematical representations in the graphical GeoGebra (GG) and specific algebraic (SA) registers.

This change of registers implied a double challenge: on the one hand, the need to convert the physical representations obtained with Tracker into abstract mathematical representations; on the other hand, the need to preserve the physical meaning of the phenomenon during this process. This moment required students to perform register conversion operations between disciplinary domains, which implies a high degree of cognitive complexity.

In question 2, students were asked to decide if it was relevant to add other variables and graphs in the analysis of the movement. Three of the four groups – groups 1, 2 and 3 – answered affirmatively, identifying the importance of including the velocity and acceleration graphs to better understand the phenomenon. This decision evidences a first step towards mathematization, as students begin to articulate dimensions of motion that are not directly observable but measurable, which corresponds to an activation of the instrumental and semiotic genesis of the KII paradigm. In terms of the KWS, this moment represents the emergence of visualization in a technical sense, in which graphical and tabular representations in Tracker begin to acquire analytical value. To illustrate the students’ work in Tracker, Figure 6 shows the Tracker screen for Group 3.

Figure 6
Work in Tracker by Group 3.

In question 3, students worked with GeoGebra, copying tables of values from Tracker to obtain functions and graphs. They were asked to change the $x$ and $y$ variables to those that physically corresponded to the motion, to name the axes appropriately, and to analyze the meaning of each of the graphs.

They were also asked to identify which of the three graphs – $x$ versus $t$, $y$ versus $t$, or $y$ versus $x$ – represented the motion of the object. Some important difficulties emerged here:

• Group 1 correctly identified that the $x\left(t\right)$ graph represented horizontal motion. However, from both mathematical and physical perspectives, the graph $y\left(t\right)$ represents the temporal variation of the vertical position rather than the spatial trajectory of the object, which instead corresponds to a relationship between spatial variables, $y\left(x\right)$ (Figure 7a).

• Group 3 noted that their motion was two-dimensional, but did not clearly distinguish the $x$ versus $t$ graph nor did they properly identify the trajectory graph. Even so, they claimed that the $y$ versus $t$ graph represented the motion, but placed it where $y$ versus $x$ appeared, showing confusion in the correspondence between graph and phenomenon (Figure 7b).

• Groups 2 and 4 did not answer the question, which may be interpreted as a cognitive obstacle associated with the coordination of semiotic registers and the construction of the physical meaning of graphical representations. From Duval’s perspective [¹², ¹³], conversion between registers constitutes one of the most complex cognitive activities in mathematical learning, since it requires preserving the meaning of the object while transitioning across different systems of representation. In this case, although the students had access to graphs generated in Tracker and GeoGebra, they were unable to establish stable semantic correspondences between these representations and the observed physical phenomenon. Similar findings have been reported in research on kinematics learning, where students frequently confuse the spatial trajectory with position–time graphs or interpret graphical representations in a pictorial rather than functional manner [⁴, ¹¹]. Likewise, the absence of a response may be interpreted as evidence of difficulties in covariational reasoning, particularly in the simultaneous coordination between physical variation and graphical representation, as highlighted by Thompson and Carlson [³³].

Figure 7
Answers to task 3.

These findings indicate difficulties in coordinating registers and constructing coherent meanings across disciplinary representations. Although students had access to graphic representations, they did not always manage to establish solid semantic links between them and the observed phenomenon.

In question 5, students were asked to compare the equation modeling the motion obtained experimentally (functional fit with GeoGebra) with the position equation associated with the position–time function of uniformly accelerated rectilinear motion (UARM) $d=d_0+v_{0}t+\frac{1}{\text{2}}{\text{at}}^2$ and to identify the physical meaning of each of its terms. This task required recognizing the formal structure of the quadratic function and establishing semantic correspondences between the coefficients of the fitted function and the physical parameters of the motion, activating the conversion between specific (SA) and general (GA) algebraic registers, as well as the use of natural language (NL) for argumentation.

From the framework of Blum and Leiss’ modeling cycle, this phase is located within mathematization, since it involves the passage from the empirical to the theoretical model. In terms of the KWS, the semiotic genesis is mainly mobilized – by the transit between SA and GA – and the discursive genesis – by the justification of the physical meaning of each parameter, both typical of the quantification paradigm (KII).

The four groups managed to perform this task adequately, establishing correspondences between the terms of the fitted quadratic function and the parameters of the physical equation for UARM. This result indicates that, in this phase, all students were able to activate the coordination between algebraic registers, as well as to endow the mathematical representations involved with physical meaning. Figure 8 presents the most representative responses, those of Groups 1 and 2. They established precisely the correspondence between the terms of the fitted function and those of the theoretical UARM equation. The $y_i$ dentified d0 as the initi $a_l$ position, v0 as the initial velocity, and the quadratic coefficient as half the acceleration. These responses show a correct articulation between the experimental setting and the physical-theoretical model, as well as a meaningful appropriation of the mathematical object function within a physical context.

Figure 8
Answers to task 5.

The ability demonstrated by the students in this task evidences an advance in the interdisciplinary resignification¹ of mathematical objects, consolidating the transition towards forms of reasoning characteristic of the KII paradigm, in which symbolic representation acquires meaning in relation to modeled physical phenomena.

From the KWS, this phase mobilized the semiotic genesis – through the production and reinterpretation of graphs and equations –, the instrumental genesis – through the use of GeoGebra as a symbolic mediator – and, to a lesser extent, the discursive genesis, as the arguments were oriented towards the identification of variables but not always towards the justification of their choices.

4.3. Working mathematically and interpretation, phases 4 and 5

The fourth phase in the modeling cycle corresponds to the moment in which students develop formal mathematical operations on the generated models, either to solve the problem posed, validate hypotheses, or deepen their understanding of the functional relationships involved. In this study, this phase was materialized through the analysis of the fitted functions and the deduction of the trajectory equation by means of symbolic operations, as well as obtaining the first and second derivatives with their corresponding physical interpretation.

From the KWS, this phase clearly activates the quantification paradigm (KII) and establishes the transition to the formalization and idealization paradigm (KIII), especially when students operate algebraically with parametric functions and execute symbolic derivations. The most clearly mobilized geneses are the semiotic – by the transit between different algebraic and graphic registers –, the instrumental – by using symbolic artifacts such as derivatives –, and the discursive – not only when justifying the physical sense of each transformation performed, but also when mathematically deriving the equations of motion and trajectory by means of a symbolic discourse.

The main task consisted of obtaining the equation of the trajectory of motion using the parametric position equations $x\left(t\right)$ and $y\left(t\right)$, and expressing this trajectory in a rectangular equation, as $y\left(x\right)$. This procedure involved isolating the time variable in one of the functions and substituting it in the other, a fundamental operation to consolidate the notion of function and its link with the observed physical phenomenon. The results show relevant differences between the groups:

• Groups 1 and 2 correctly performed the expected algebraic procedure. They isolated the time variable from the vertical position function and substituted it into the horizontal function, obtaining the equation of the trajectory in rectangular coordinates. Moreover, their expressions retained formal coherence and were interpreted within the physical context of the observed motion. This performance evidences an activation of the [Sem-Dis] plane of the KII paradigm and an initial transition towards KIII, since the students used symbolic artifacts as formal mathematical tools and justified their relevance with disciplinarily valid arguments, Figure 9. Group 3, although they did not develop a manual algebraic procedure, used GeoGebra as a technological tool to perform the substitution of variables. They obtained the equation of the trajectory directly in the software, avoiding symbolic development, but showing functional understanding of the operation. In terms of the KWS, this group activated the [Ins-Dis] plane of KII, with a predominance of technologically mediated instrumental genesis. The response evidences a partial understanding, supported by the use of digital tools rather than symbolic procedures, Figure 10.

Figure 9
Answers to task 4, group 1 and 2.

Figure 10
Answers to task 4, group 3.

Group 4, on the other hand, presented a limited response. They recognized that the trajectory corresponded to a quadratic equation and wrote its general form $a{x}^2+bx+c$, but they did not perform any operation linking such expression with the data obtained, nor did they justify its construction. This performance indicates a weak activation of semiotic and discursive genesis, remaining within the KII paradigm, without transiting to KIII.

Complementarily, in subquestions 5a and 5b, students were asked to calculate the first and second derivatives of the position equation with respect to time, and compare the results with the physical equation of the final velocity $vf=v_0+at$. This activity made it possible to observe students’ understanding of the concept of derivative, which was used not only as an algebraic procedure, but also as a tool for interpreting relationships of variation between physical quantities.

• Groups 1, 2 and 3 correctly performed the algebraic treatments. They identified the first derivative as the velocity and the second as the acceleration, and established correspondences between the results obtained and the theoretical equations of motion. These groups activated the [Ins-Dis] plane of the KWS within the KII paradigm, demonstrating that the derivative was used as a meaningful symbolic tool. Moreover, the verbal justifications presented evidence a progressive consolidation of the discursive genesis associated with functional reasoning, Figure 11.

• Group 4, on the other hand, did not perform the derivatives nor did they present an algebraic development. Their response was limited to an intuitive comparison with the physical equation, without procedures to validate their assertions, which suggests that they failed to use differentiation as a symbolic tool to integrate the notion of derivative in the context of the analyzed movement, Figure 11.

Figure 11
Answers to task 5 a) and b), groups 1–4.

Overall, the results of this phase show that the students were able, for the most part, to use mathematical operations to transform, derive and reinterpret the functions obtained from the observed motion. This ability is indicative of a significant appropriation of the function object and an increasingly abstract understanding of the physical relationships involved. From the KWS point of view, this progress represents the beginning of a transition towards the formalization paradigm (KIII), which should be consolidated through activities that deepen the formal validation and generalization of the models obtained.

4.4. Validation, Phase 6

The validation phase within the Blum and Leiss modeling cycle is the moment when students evaluate whether the mathematical model constructed adequately represents the observed physical phenomenon. At this stage, the judgment on the adequacy of the model is based on the coherence between the results obtained (table of values, graphs, functions, derived equations) and the real characteristics of the analyzed movement. From the KWS, this phase activates mainly the quantification paradigm (KII), with elements of the formalization paradigm (KIII), especially when students integrate physical, algebraic and graphical arguments in their validation discourse. At the semiotic register level, an effective articulation between algebraic (SA, GA), graphical (GT and GG), tabular (TV) and natural language (NL) representations is expected.

In this study, validation was addressed through question 6, in which students were asked to interpret the results obtained (graphs, velocity and acceleration) to explain the analyzed movement. This exercise made it possible to identify the degree of appropriation of the function object as a modeling tool, as well as the students’ abilities to move between registers and construct solid disciplinary arguments, Figure 12.

Figure 12
Answers to task 6), groups 1–4.

• Group 1 presented a response that evidenced a structural appropriation of the mathematical model. They correctly identified the quadratic term in the UARM equation and explained its physical meaning: “each time the time is larger the position at $x$ would increase even more”. They recognized the difference between URM and UARM from the effect of squaring time on position, which shows a functional understanding of the model. Their discourse activates semiotic genesis by linking algebraic structure to physical behavior, and discursive genesis by justifying the model in natural language. This response reflects a validation that articulates the general algebraic (GA) and natural language (NL) registers, and sits solidly on the [Sem-Dis] plane of the KII paradigm.

• Group 2 offered an even deeper response on functional relationships between quantities. They pointed out that “if you derive position versus time, you get velocity, and if you derive velocity, you get acceleration,” explicitly stating the relationship between function and successive derivatives. They recognized the parabolic form of the position-versus-time graph and constructed a coherent discourse that integrates notions of differential calculus applied to the UARM context. This response evidences an activation of the discursive genesis that approaches the formalization paradigm (KIII) by the degree of understanding of the mathematical objects and their significance.

• Group 3, with a more synthetic discourse, also managed to validate their model. They indicated that the movement presented a constant acceleration congruent with gravity, and that “the $v$-versus-$t$ graph helped us to understand this more”, evidencing a functional association between the graphical behavior and the magnitude represented. Although they did not develop explicit algebraic operations, they activated the [Ins-Dis] plane of the KWS, and their validation relied mainly on the interpretation of the graphical representation (GT, GG). Their functional understanding was more empirical than structural, but sufficient to validate the model in terms of the observed phenomenon.

• In contrast, group 4 presented an incomplete and disjointed validation. They mentioned terms such as “acceleration is not constant” and “mechanical energy”, but without linking these notions to the representations generated during the activity. The discourse lacks explicit references to graphs, equations or functional relationships between variables. They remain in the KI paradigm of the KWS, with no clear activation of semiotic or discursive genesis. The use of technical terms does not translate into an effective validation of the model, which evidences a rupture between the constructed model and its interpretation.

Overall, the results of this phase show that three of the four groups were able to validate their models based on functional, structural and phenomenological arguments. The notion of function was treated as a mathematical object linked to the UARM phenomenon, either through its algebraic form, its derivatives or its graphical behavior. This process evidences a resignification of the function object within an interdisciplinary context, supported by the coordination of registers, the use of technological tools, and the development of argumentative discourse. However, the difficulties observed in group 4 highlight the need to reinforce strategies that promote representational and discursive articulation as a means of conceptual validation.

5. Conclusions

The results obtained in this study show the value of a robust theoretical-methodological construct, consisting of Blum and Leiss’ modeling cycle, the Kinematics Working Space (KWS) and Duval’s theory of registers of semiotic representation, for the design of interdisciplinary activities and for the rigorous analysis of the students’ work. These three frameworks made it possible to structure a didactic situation that progressively guided students from the initial understanding of a physical phenomenon to the formalization of a mathematical model, while enabling a fine reading of student productions at each stage of the process. The modeling cycle provided a coherent sequence for organizing the tasks and delimiting epistemic expectations at each stage; the KWS, for its part, made it possible to identify transitions among the KWS paradigms, from empirical experience to mathematical formalization and to analyze the mobilization of semiotic, instrumental, and discursive genesis as indicators of cognitive work; and the theory of registers of semiotic representation provided a powerful tool for examining students’ coordination, conversion and processing of representations at different levels.

In this context, the incorporation of Tracker and GeoGebra as disciplinary artifacts played a fundamental role in the transition between the physical and mathematical domains, not only as technological tools, but also as epistemic instruments that enable the construction of meaningful models. Tracker allowed students to capture real motion, quantify it and visualize it through graphs and tables that maintain the variables and units of physical discourse. GeoGebra, on the other hand, offered a symbolic space where students could graphically represent data, adjust functions, and then operate with derivatives and link these processes with the formal structures of mathematical language. The complementarity of both environments made it possible to articulate heterogeneous disciplinary registers, enabling the reconstruction of the phenomenon from multiple perspectives and favoring the development of an integrated vision of modeling. Thus, Tracker and GeoGebra functioned as bridges between registers, as means to activate different genesis within the KWS, and as catalysts of interdisciplinary thinking around function.

The analysis of the students’ responses shows diverse but, in general, highly significant trajectories. Group 1 displayed a consistent discourse throughout all phases, characterized by a strong articulation between the formal aspects of the model and its physical interpretation, managing to validate the fitted equation with structural and phenomenological arguments. Group 2, on the other hand, performed an excellent integration between the notions of derivative, function, and physical magnitude, using a clear functional discourse that evidences appropriation of concepts from a contextualized perspective. Group 3, although using less technical language, was able to recognize relationships between graphs and magnitudes, validate the model in terms of physical parameters such as gravity, and link technological tools with their interpretation of movement. Group 4, although it presented greater difficulties, especially in the articulation of registers and in the validation of the model, showed previous knowledge about the phenomenon that, although disorganized, constitutes a base on which to build more solid processes of resignification. Overall, the students made significant progress in the understanding of the function as a mathematical object applied to the analysis of UARM. Most of the students not only managed to construct coherent functional models, but also to establish correspondences between algebraic terms, graphs, and physical magnitudes. Even so, tensions persist in the semantic interpretation of the graphs, in the distinction between types of representations (trajectory, movement), and in the precise identification of variables, which indicates that the notion of function, although worked out and understood in many of its representations, still requires systematic didactic work to be consolidated. The achievements attained by the students – especially in terms of articulation of registers, validation with their own arguments and use of disciplinary tools – allow us to affirm that the methodological approach used constitutes a powerful way to promote interdisciplinary, meaningful and conceptually integrated learning.

These results are consistent with previous research on kinematics learning, which has identified persistent difficulties in the interpretation of motion graphs, particularly in coordinating trajectory, position–time, velocity–time, and acceleration–time representations. Similar to studies on kinematics graphs and multiple representations, our findings show that students do not always distinguish between the spatial trajectory $y\left(x\right)$ and the temporal evolution of a coordinate; however, they make significant progress when working with dynamic tools and experimental data [³]. Nevertheless, unlike approaches primarily focused on visualization, simulation, or gamification, this study demonstrates that the articulation between Tracker and GeoGebra not only improves graph interpretation, but also promotes algebraic construction, the interpretation of physical parameters, and the resignification of the position–time function in uniformly accelerated rectilinear motion (UARM). This finding is consistent with recent studies highlighting the potential of GeoGebra, Tracker, and computational environments to connect experimentation, modeling, and conceptual understanding in kinematics [³⁴]. In this sense, the distinctive contribution of the present study lies in showing how transitions across the KI, KII, and KIII paradigms of the KWS enable a more precise characterization of the progression from physical experience to the mathematical formalization of motion.

Data Availability

The entire dataset supporting the results of this study is available upon request from the corresponding author, Claudia Gabriela Reyes-Avendaño. Reason for restriction: The dataset contains student-generated materials and educational research data involving human participants; therefore, public access is restricted to protect participant privacy and confidentiality.

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¹

We understand the resignification of the notion of function as the process through which students reinterpret a mathematical object previously studied in purely school-based contexts, attributing to it new meanings associated with relationships of variation between physical quantities.

*Correspondence email address:

[e-mail]

Editor-in-Chief:

Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  16 Oct 2026
• Date of issue
  2026

History

• Received
  16 Dec 2025
• Reviewed
  06 Aug 2026
• Accepted
  08 Aug 2026