Introduction

Today, most educational dynamic geometry environments (DGEs) allow the symbolic manipulation of geometric constructions and graphic representations. GeoGebra stands out as a DGE that fully integrates the traditional features of a DGE with the algebraic features of computer algebra systems, which is why GeoGebra can be used as a dynamic geometry and algebra environment (DGAE) with dynamic multi-representations (Hohenwarter & Jones, 2007). Dynamic properties increase the ability to examine mathematical concepts and relationships (e.g. Baccaglini-Frank et al., 2013; Nagle & Moore-Russo, 2013; Olive et al., 2010) and improve ‘the reasoning, understanding, and conceptualization of mathematical objects’ (Villa-Ochoa & Suárez-Téllez, 2021, p. 5). Dynamic behaviour presents a dilemma, as, on the one hand, it allows students to create and transform graphic representation through algebraic notation, which can otherwise be a challenging task, making algebraic manipulation and transformation more accessible to students (Hohenwarter & Jones, 2007), and, on the other hand, the outsourcing of translation between representations can be problematic.

It is widely acknowledged in mathematics education research that handling the various representations of a concept, including its associated processes and objects, plays a significant role in mathematical reasoning and concept development. For example, Sfard (1991) argues that shifting between representations of the same object is a necessary step toward reification, and Duval (2006) believes that being able to access and translate between different mathematical representations is crucial to all mathematical understanding and activity. Furthermore, Duval worries that the outsourcing of the translation of representations deprives students of an awareness of the one-to-one mapping between graphic visual values and algebraic terms.

Pedersen et al. (2021) suggest that, to address such issues, task designers could require students to predict changes in these representations when using digital technology. Moreover, students’ justifications tend to rely on empirical knowledge (Harel & Sowder, 2007) or phenomenological evidence (Baccaglini-Frank, 2019). This tendency is enhanced by the dynamic properties of environments, which allow students to interact and observe representations that appear as real virtual objects that can be experienced phenomenologically (Baccaglini-Frank, 2019; Leung & Chan, 2006). In general, predicting results and strategies supports the development of students’ reasoning abilities (Kasmer & Kim, 2011; Miragliotta & Baccaglini-Frank, 2021), which is why prediction tasks may address both the translation of representations and students’ phenomenological tendencies. Additionally, in this study, I hypothesise that predicting the dynamic behaviour of objects in a DGAE allows students to reason about algebraic properties based on their mathematical conceptual knowledge while still capitalising on the realness of virtual objects and their dynamic properties.

In Denmark’s education system, the mastery of mathematics is considered mathematical competence including mathematical reasoning competency as described in the KOM framework (Niss & Højgaard, 2019) (KOM abbreviates ‘Competencies and the Learning of Mathematics’). Research on the use of digital tools in mathematics education’s interplay with the development of mathematical competencies are becoming increasingly common (e.g. Bach, 2022; Geraniou & Jankvist, 2019; Geraniou & Misfeldt, 2022; Højsted, 2021; Jankvist & Geraniou, 2022; Thomsen, 2022). Mathematical reasoning competency includes the spectrum of forms of mathematical reasoning across the scope of mathematical mastery, from early mathematics education to expert mathematicians. This study focuses on justification as a particular aspect of reasoning competency, as justification is predominant in everyday teaching in mathematics classes in lower secondary education (age 13–16). Moreover, justification has been given little attention in the research on reasoning in general (Stylianides & Stylianides, 2022).

To address how students use tools in conjunction with their mathematical competencies, prior studies (Bach, 2022; Geraniou & Jankvist, 2019; Thomsen, 2022) have drawn on the instrumental approach to mathematics education (IAME) (Drijvers et al., 2013; Trouche, 2003, 2004, 2005). This approach highlights how tools become instruments used to solve mathematical tasks, but it does not delve into the processes of justification. In a recent study by Gregersen and Baccaglini-Frank (2022), we examined how the processes described in the IAME approach can be analysed as a componence of justification processes by using an adapted version of Toulmin’s (2003) argumentation model. However, we did not extensively elaborate on the significance of students’ conceptual understanding in justification processes.

Additionally, the KOM framework (Niss & Højgaard, 2019) has no concepts with which to analyse students’ knowledge and conception, but it does recognise them as ingredients in mathematical competencies as the exercise of any mathematical competency involves some subject matter. In order to expand upon the conceptual aspect of students’ justification processes in conjunction with digital tools, Geraniou and Jankvist (2019) have taken initial steps by suggesting that schemes (Vergnaud, 1998) as the cognitive component of the IAME may be useful in articulating ‘the role of conceptual knowledge in relation to the mathematical competency’ (Geraniou & Jankvist, 2019, p. 41). If so, it might be possible to link students’ conceptual knowledge and development, reasoning competency and tool use by analysing students’ schemes. Accordingly, this study aims to explore how students use tools in a DGAE in justification processes when predicting dynamic behaviour from both a reasoning competency and a conceptual perspective.

The framework and context of the study are further elaborated below, after which the aim will be concretised into two research questions. The explorations are then conducted as a case study (Thomas, 2011b) of the justification process of a pair of students solving a prediction task embedded in a restricted GeoGebra environment. The case is analysed in three steps. First, the potentials and constraints (Trouche, 2005) of the relevant tools are considered, followed by an analysis of the students’ justification process and tool use using an adapted Toulmin’s model (Gregersen & Baccaglini-Frank, 2022). Finally, the students’ process is analysed with regard to the components of the scheme (Vergnaud, 1998).

Theoretical Framework

Reasoning Competency, Arguments and Justification

The KOM framework defines a mathematical competency as ‘someone’s insightful readiness to act appropriately in response to a specific sort of mathematical challenge in given situations’ (Niss & Højgaard, 2019, p. 14; italics in original). Out of eight distinct competencies, this study is confined to the reasoning competency. Students exercise reasoning competency when they analyse or produce mathematical arguments (Niss & Højgaard, 2019). This can consist of oral or written arguments in various forms, in this case, justification. An argument is a chain of statements linked by inference in support of mathematical claims or solutions to mathematical problems (Niss & Højgaard, 2011, 2019).

A person’s competency is an evolving situated entity that is developed over time through active participation in mathematical situations. Competency development involves expanding the degree of coverage, radius of action and technical level. Coverage pertains to the different aspects of a competency, such as active participation in various forms of reasoning. The radius of action considers the diverse contexts in which the competency can be applied, spanning various domains and social situations. The technical level addresses the sophistication of concepts, theories and methods.

Justification is the process of supporting mathematical claims and choices when solving problems when students are asked to explain and warrant their answers concerning a given problem (Stylianides & Stylianides, 2022). In all mathematical reasoning, arguments are put forward to change the epistemic value (the degree of certainty) of a claim (Duval, 2007). The epistemic value can be considered from the perspective of the reasoner or the general mathematics community (Duval, 2007; Harel & Sowder, 2007; Jeannotte & Kieran, 2017; Knuth et al., 2019).

Considering an argument from a structural standpoint, Toulmin (2003) suggests a geometric structured model (see Fig. 1), considering what constitutes a valid argument from epistemological and psychological perspectives. Toulmin’s argumentation model structures the fundamental components of an argument, including the claim, qualifier, data and warrant. A claim is a statement along with its epistemic value (i.e. qualifier). The qualifier expresses the probability of the claim (e.g. false, possible, more possible or true) and is established based on evidence (i.e. data that supports the claim and the warrant, which connects the data to the claim). Finally, the rebuttal limits or counters the claim.

Fig. 1
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Geometric structure of the elements of an argument (Toulmin, 2003)

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The Instrumental Approach to Mathematics Education

The IAME conceptualises how a tool becomes an instrument for solving mathematical tasks through the process of instrumental genesis (Artigue & Trouche, 2021; Trouche, 2003). The process comprises a subject (from here on, a student) and a material or non-material artefact. The student knows objects or concepts particular to the situation and the use of artefacts. The artefact mediates the students’ actions on objects, which are influenced by the potentials and constraints of the artefact. As the student uses the artefact, it becomes a tool for a particular situation or task. This could be using a polygon tool (the artefact) in a DGE to construct a triangle (a task concerning an object). Instrumental genesis has a dual nature: instrumentalisation and instrumentation. Instrumentation is the constraints and possibilities imposed on the student’s actions, while instrumentalisation is the student imposing a personal use. Over time, as the student uses the tool for similar situations, the process of instrumental genesis unfolds to develop an instrument. An instrument is a cognitive unit that consists of both scheme and artefact.

Schemes concern perceptual and gestural goal-oriented activities in ‘the invariant organization of behaviour for a certain class of situations’ (Vergnaud, 1997, p. 12). Schemes include a generative component: rules-of-action, which shape behaviour based on situational variables. The purpose of rules-of-action is not to be true, but to be effective. In addition, schemes include the conceptual components of the operational invariants: theorems-in-action about concepts-in-action. Theorems-in-action are often not explicitly stated, but rather held-to-be-true statements that, according to mathematical theory, can be true or false. Theorems-in-action provide insight into the world of objects: the concepts-in-action that can be relevant or irrelevant to the situation or task.

Moreover, invariant behaviour is relative as schemes are adapted by inference pertaining to contexts and circumstances (Pittalis & Drijvers, 2023; Vergnaud, 2009). This relativism reflects the instrumental genesis as an instrument develops over time. In the same manner, the stability in a scheme for a certain class of situation is reached over time. Consequently, the schemes of students using an unfamiliar artefact or solving an unfamiliar task will be less stable, rules-of-action may be ineffective and theorems-in-action may be wrong (Ahl & Helenius, 2018). Finally, Vergnaud (1998) emphasises the significance of possibilities of inference within schemes, acknowledging that inference and computation are inherent in any activity.

In the IAME, the conceptual aspect of the epistemic use of tools is prevalent (Shvarts et al., 2021). For example, Drijvers et al. (2013) consider the dualistic process of activity and conceptual knowledge as a technique-scheme duality in the instrumental genesis process. Epistemic use is most often explored through the identification of the invariant behaviour across users of an artefact and the conceptual understanding underlying different usage schemes. Alternatively, through the analysis of the operational invariants of students’ developing schemes, Rezat (2021) explicates the students’ rationales as expressions of knowledge. Such insight is indeed relevant in justification processes. Thus, by adopting a similar approach to that of Rezat, the scheme-technique duality can provide insight into the co-evolution both of conceptual development and of justification in the instrumental genesis process.

Rabardel argues that, ‘it is necessary to analyze and understand what these activities are from the perspective of the users themselves’ (2002, p. 31). Indeed, this is a prominent concern. Therefore, I take an inclusive approach to technique, one encompassing all gestures involved in student tool use, including hand movements, direct interactions with the artefact and verbal expressions of imagined activity. In addition, oral explanations of action can provide insight into concepts- and theorems-in-action (Rezat, 2021), since ‘enunciation plays an essential part in the conceptualization process’ (Vergnaud, 2009, p. 89).

Instrumented Justification

Traditionally, the IAME has been applied to analyse students’ learning techniques when solving particular mathematical problems utilising a digital tool, such as determining the solutions to an equation utilising CAS (e.g. Artigue, 2002; Jupri et al., 2016). In this case, students utilise GeoGebra to predict the translation of symbols. Furthermore, students may present arguments in support of or against certain claims arising from their solution process. In order to capture such processes, Gregersen and Baccaglini-Frank (2022) introduced an analytical tool by reinterpreting Toulmin’s model, in light of the scheme-technique duality, and termed the process instrumented justification (IJ). Based on this analytical tool, IJ is described as ‘a process through which a student modifies the qualifier of one (or more related) claim(s) using techniques in a digital environment to generate and search for data and warrants constituting evidence for such claim(s)’ (Gregersen & Baccaglini-Frank, 2022, p. 135; italics in original). An elaboration of the analytical tool is provided below. Please refer to Fig. 2.

Fig. 2
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Adaptation of Toulmin’s model into an analytical tool for students’ instrumented justification

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Toulmin’s model is most often applied in mathematics education research to analyse a finalised argument or chains of sub-arguments. However, in the IJ analytical tool, the unit of analysis is the process from a claim to a restatement of that claim, along with a change in the qualifier. The students generate data through techniques as evidence to support or refute the initial claim, and change the qualifier from ‘possible’ to ‘more possible’, ‘less possible’, ‘true’ or ‘false’. The close connection between data and techniques appears in the analytical tool as connected frames correlating a technique to the data it produces. The schemes (Vergnaud, 1998) that direct and organise techniques generating data contain conceptual elements and rules that regulate actions that are seen as warrants that connect the data to the claim and can be inferred from students’ techniques and verbal expressions (Rezat, 2021).

Figure 2 shows a generic diagram of the IJ analytical tool as an adaption of Toulmin’s model. In continuous sub-processes, the first uttered claim, along with its qualifier, is noted in the top right corner in grey, so below is the re-claim with a new qualifier. Finally, the rebuttal consists of the limitations of the claim or counterarguments as in Toulmin’s (2003) original model.

A Prediction Task to Situate an Instrumented Justification Process

The task presented in Fig. 3 originates from a sequence of tasks developed during my Ph.D. study. Collectively, the sequence explores a microworld of variable points in GeoGebra, which are ordered pairs containing a variable in more or less complicated algebraic expressions. The current task is the first prediction task in the sequence.

Fig. 3
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Above is the restricted interface of a GeoGebra app for predicting the movements of points A and B, presented in the orange box: the available tools are ‘move’, ‘point’, ‘pen’ and ‘erase’, and below are the questions posed

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In question 1, the students are required to predict how the variable points will move in the co-ordinate plane, which the students must visualise and explain using a highly restricted interface in a GeoGebra app. The restrictions are enforced to prevent students from constructing the points by typing them into the algebra view. The tools available are ‘move’, ‘point’, ‘pen’ and ‘erase’. The student can turn on the trace of constructed points to trace any dragging of the point. In question 2, the students must justify their predictions in writing in a Word document. Research has demonstrated that prompting students to predict outcomes can encourage mathematical reasoning using previous knowledge (Kasmer & Kim, 2011; Lim et al., 2010). Some research studies students’ prediction as a product or the processes by which predictions emerge (Miragliotta & Baccaglini-Frank, 2021). I follow the latter approach, viewing student predictions as an instrumented justification process and treating predictions as claims about assumed dynamic behaviour in GeoGebra. Unlike traditional positions of prediction in mathematics education as a statement or conjecture anticipating either the solution to the problem or the strategy used to reach a solution (e.g. Boero, 2002; Kasmer & Kim, 2012; Palatnik & Dreyfus, 2019), the intention in this case is to leverage predictions and thus give ‘students the opportunity to defend or refute ideas’ (Kim & Kasmer, 2007, p. 298). Consequently, I consider the prediction task as a problem in itself, one that requires students to engage in IJ and operationalise their knowledge about variables and dynamic behaviour in GeoGebra.

Inspired by physics education, the prediction task involves anticipating outcomes in a way that is akin to experimental testing (Højsted & Mariotti, 2021; White & Gunstone, 1992). The dynamic behaviour of objects in GeoGebra creates the impression of real virtual objects, simulating movement and behaviour comparable to physical objects that can be experienced phenomenologically (Baccaglini-Frank, 2019; Leung & Chan, 2006). The prediction of such dynamic behaviour can be tested in the environment. In fact, although not part of the case presented, following the prediction task, the students are asked to test their predictions and consider the outcomes.

Student Knowledge of Variables and Dynamic Properties

In the prediction task, the concept of the variable is central. I will briefly elaborate on the concept of variables from a conceptual perspective and in relation to dynamic behaviour in DG(A)Es. The dual nature of concept formation and development in mathematics education research, which involves processes and objects, is widely recognised (Douady, 1991; Dubinsky, 1991; Noss et al., 2009; Sfard, 1991). For young students, concepts are initially tied to processes within specific numeric situations. Ideally, these concepts evolve into abstract objects, enabling the exploration of structures and relationships (Douady, 1991).

Concerning variables, Noss et al. emphasise that generalisation involves moving beyond the specific, recognising the structural properties, relationships and patterns that variables (and constant) represent. Introducing variables often marks students’ first step into objectification, requiring them to perceive a letter as representing all values subject to the same computational manipulation as numeric values. In addition, Noss et al. (2012) problematise the static representations of paper-and-pencil tasks, arguing that such representations hinder students’ progression in conceiving variables as ‘the inevitably static (and therefore specific) figure that can be presented on paper is often problematic for students as it lacks a rationale for thinking generally’ (p. 64).

The dynamic behaviour of objects in a DG(A)E reflects the process–object nature of concept formation as either a discrete collection of examples or continuous movement. Indeed, Miragliotta and Baccaglini-Frank (2021) describe that, in predicting dynamic objects, students may pin-point specific positions or envision, enact or imitate continuous movements. This also holds true for variable points, which can shift between positions in a co-ordinate plane or move along a trajectory. In a fully generalised conception, a variable point transcends dynamic properties, taking on the form of a line. The structural properties are then defined by the position of the trajectory or a line in relation to the co-ordinate system and other variable points.

Research Questions

After explaining the theoretical frameworks and laying out the task details, the research aim can be formulated as specific research questions:

In predicting the dynamic behaviour of variable points in a restricted GeoGebra environment, how can students’ use of the point tool, trace function and pen tool interplay with their justification processes?

To what extent can an analysis of the components of the student’s scheme provide links between the process of instrumented justification, the student’s mathematical reasoning competency and conceptual knowledge?

Research Design and Method

Design of a Case Study

This study aims to conduct a fine-grained analysis of the conceptual evolution of students engaged in IJ to link reasoning competency, tool use and students’ conceptual knowledge. To achieve this objective, the study is designed as a case study of a singular key-case (Thomas, 2011b) that will follow the IJ processes and the evolution of schemes. The case is presented as a temporal account (Thomas, 2011b), based on transcripts of students’ utterances, descriptions and pictures of gestures, and screenshots of their computer screen that capture specific moments. The case provides the reader with contextual insights into the development of the use of particular tools for the prediction of dynamic behaviour, by exemplifying the intricate development of such a process (Thomas, 2011a). Furthermore, the case demonstrates the potential of the described task to provide a context for students’ IJ justification processes.

The case consists of the IJ processes of the two students, Lev and Rio, who were collaborating on solving the prediction task. This pair was chosen because the students engaged in an IJ process characterised by the development of their tool use and their prediction. In particular, Lev was verbal about his assumptions throughout the process, making it possible to infer his warrants. Lev and Rio regularly use GeoGebra in mathematics class, though they mainly use the graphic view and geometric tools, including points and tracing. They have only used the algebra view to provide information on constructed objects. In the introductory part of the task sequence, students were introduced to constructing static points through the algebra view. In class, they have been introduced to the definition of a variable as an expression of all values, and procedures concerning variables in equations, functions and formulas. The pair was acquainted with plotting points on the co-ordinate system but had no experience with variable points before the experiment.

Regarding reasoning competency, the prediction task requires the students to expand their radius of action, as variable points are a new task. Concerning coverage, the students have experience with justification processes and prediction from their regular mathematics classes, mostly in the form of estimating the results of a computation. Concerning the technical level, Rio and Lev have no experience justifying variables as generalised numbers.

Data Collection

Data were collected from a class of 7th-grade students aged 13 to 14 during a classroom experiment. To encourage the students to express their assumptions and justifications, they were paired up and shared a computer to solve the task sequence. Additionally, the students were instructed to verbalise their thoughts and arguments while solving the tasks. OBS studio was used to capture the students’ screen, voices, faces and upper bodies on video recordings during the experiment. In the classroom, the mathematics teacher and I were present to assist students with any questions or issues they may have had while completing the tasks. The video recordings were transcribed. The gestures were described and, if necessary, accompanied by images.

Data Analysis

The theoretical framework examines the case from three perspectives: the artefact, IJ and conceptual understanding. Each perspective also divides the analysis into three steps.

In step 1, following the IAME, I describe each of the three artefacts to have a clear understanding of the limitations and constraints of the available artefacts in terms of solving the task (Drijvers et al., 2013).

In step 2, I analyse the case using the analytical tool for IJ processes. The IJ model’s components are identified using a theory-guided structured coding approach (Mayring, 2015). The categories were developed and revised in a cyclic process until applicable across students’ IJ processes. Claims and re-claims are identified and constitute the analysis unit. A claim is an uttered tentative or final solution to the task, and a reclaim is an uttered statement similar to or referencing the claim along with an implied change in epistemic value. Any rebuttal is then identified. Techniques are then identified in the unit together with the corresponding data produced. Techniques can be performed or imagined in verbal expression. The data is the products of students’ interactions with an artefact and their verbal interpretations of the data produced as evidence for or against the claim. The change in the qualifier of a claim is inferred from the students’ actions and utterances, such as hesitation or continued search for data, which can indicate a lack of conviction in the claim’s truth. Inferring warrants is a demanding process that requires interpreting how students’ techniques and the data produced are relevant to the claim. Warrants are typically implicit, so formulating them is an explicating procedure and requires a narrow qualitative content analysis (Mayring, 2015). The formulation of warrants is revisited to ensure consistency with the source data in videos and transcripts.

Step 3 categorises warrants as either rules-in-action or theorems-in-action. Then, the order of appearance of theorems-in-action is used to infer the ‘possibilities of inference’ drawn between theorems-of-action concerning different concepts-in-action. The analysis method is further addressed and discussed in the step 3 of the ‘Analysis’ section.

Analysis

Step 1: Analysis of the Potentials and Constraints of Tools Specifically Regarding the Tasks

For the prediction task, students must anticipate the dynamic behaviour of points A = (1, s) and B = (s, 1) in the restricted interface. This requires translation from symbolic to graphic representation, which is typically outsourced. To aid in this process, students have access to the move tool, the point tool, the trace function and the pen tool. The point tool enables the placement of free points on the coordinate plane, allowing subsequent movement using the move tool. When using the point tool, students must assign numerical values to each point they place. For example, A = (1, s) is expressed as a singular case. Multiple values of the variable (s) can be depicted by plotting several points, shifting a single point or activating the trace function, which leaves a track of points where the point is dragged across the screen. However, tracing can be challenging when moving a free object as it is susceptible to cursor movements. The point tool and trace function are specifically designed for mathematical objects and properties. In contrast, the pen tool allows free drawing, requiring students to apply mathematical properties or functionality, such as the notation of values, sketching, plotting, tracing points or drawing lines.

Step 2: Analysis of the Instrumented Justification Processes

Here, I provide an analysis of Rio and Lev’s IJ process. It is presented as two sub-processes: the first proces is captured in 1a (Figs. 4 and 5) and 1b (Figs. 6 and 7), and the second in subproces 2 (Figs. 8 and 9). The first sub-process is lengthy and is, thus, divided into excerpts 1a and 1b to make the analysis accessible. Each excerpt includes a transcript, screenshots of the students’ work in GeoGebra and an analytical IJ model. Between sub-processes 1b and 2, a brief intermission is described, which is not considered significant for the overall process.

In the IJ analysis model, warrants are labelled according to concepts (WV for variable and WP for ordered pairs or points) and numbered according to appearance. As warrants reappear in the process, they are referred to by these abbreviations. If warrants are challenged of the two students, the warrant is assigned to the student expressing the specific warrant. In the transcripts, I am referred to as Me, gestures are described in square brackets and author notes are in italics. Rio is in control of the shared computer and mouse in all excerpts.

Instrumented Justification Sub-processes 1a

Fig. 4
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(a) The state of GeoGebra after excerpt 1; (b) illustrates the movement done with curser; (c) Rio’s hand gesture

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Fig. 5
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Instrumented justification sub-process 1a through the lens of the analytical tool

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Instrumented Justification Sub-process 1b

Fig. 6
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(a) Traces of points A and B ‘starting’ in (1, 1); (b) messing up the trace; (c) drawing trajectories of A and B limited by the ‘starting point’ in (1, 1)

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Fig. 7
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Instrumented justification sub-process 1b through the lens of the analytical tool

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Between Instrumented Justification Sub-processes 1b and 2

The students assert the claim by expressing that they have now shown how the points move and go on to the next task, justifying their prediction, which requires a written answer in a Word document. They begin by referencing WV2 and restating the claim, but then Rio seems to realise that their claim is faulty and returns to GeoGebra to reconsider their answer to question 1. Then, Rio’s following ‘monologue’ occurs, leading to a new claim (claim 2). As Lev is only observing in this excerpt, it is impossible to infer whether he shares the listed warrants. The written answer is not revisited.

Instrumented Justification Sub-process 2

Fig. 8
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(a) Extending trajectories to 0 on axes; (b) extending trajectories into negative numbers

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Fig. 9
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Instrumented justification sub-process 2 through the lens of the analytical tool (as the student instrumented justification process continues into the testing step, the epistemic value of the claim is only possible)

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Evolution of the Elements in the IJ Process

Rio and Lev put forward two different claims:

  • Claim 1: Every time one (point) shifts one place, the other one also shifts one place, so they keep moving further and further away from each other at a 90-degree angle (with the rebuttal ‘the starting point is (1, 1) which is also a limit’).

  • Claim 2: They will always be one away (from the axes), and then they can just change infinitely because of s.

When claim 2 is presented, it causes claim 1 to become less credible and ultimately be proven false. The two claims are not contradictory; instead, the second claim extends the first. The patterns of the trajectories are consistent in being perpendicular along the axes. However, the description evolves from only having positive directions in the first claim to having both positive and negative directions. Consequently, the limit is refuted. In sub-process 1a, claim 1 is based solely on data with s = 4 and the corresponding positions of points A and B. Rio imagines how the points will move from the starting point of (1, 1). Although Rio does not explicitly state why this is the starting point, it is later labelled as such in 1b and can be inferred to WP2. In sub-process 2, line 42, Rio explains that the constant within the ordered pairs defines the starting point. This understanding may have already existed in 1a. Therefore, WP2 can be specified as WP2’.

I now return to sub-process 1a. The techniques encompass plotting points with the point tool corresponding to a chosen value of the variable and imagining the position of points. In sub-process 1a, the warrants WV1, 2, 3 and WP1 are inferred. In sub-process 1b, Rio produces data to support claim 1 by moving the points to different positions corresponding to other variable values. He then changes his technique and traces through the dragging of points to generate sets for each point. The shift in technique also evolves WV1 into WV4 and WP1 into WP3. The tracing spurs a discussion between the two students about whether the initially chosen values of the variable (four and one) are also the limits of the trace. Though struggling with the trace function, Rio realises that the limit of four is unjustified, as he can continue the trace for higher values of s, evolving the warrant WV4 into WV5. At this point, Rio changes the technique again as he continuously struggles with tracing. Instead, he draws the trajectories with the pen tool from the ‘starting point’ in positive directions to the edge of the graphic view. The limit of one is maintained, which can be explained by WP2’, though this has not yet been expressed, and WP3 evolves into WP5. I will address Rio’s thinking further in the coming analysis of the development of the components of schemes.

In sub-process 2, Rio recalls that the variable can both increase and decrease in value, which is inferred to WV6. The realisation seems ultimately to unravel the issues regarding the starting point, as the limit is moved to 0 and then infinitely into negative numbers. This process is expressed in warrants WP6, WV7, WV8 and WP7. Each relates to generating new data as the trajectories are elongated. This process resolves the issues concerning the limit of the variable, and claim 2 is conceived of, in which s is infinite, and the points can ‘move’ infinitely on the trajectories. In line 44, Rio struggles to discard WP2, even though it conflicts with WV6 and WP6. However, ultimately, Rio reinterprets the constants in the ordered pairs as the trajectories’ distance from the axis, and WP2 evolves into WP8.

Step 3: Warrants as Windows on the Components of Students’ Schemes

I now turn to the evolution of warrants in the IJ process, by considering the components of the scheme (Vergnaud, 1997, 1998) as the students’ progress in their IJ process. In the excerpts, Rio is both the active user of the tools and the most articulate. Consequently, most inferred warrants can be connected to his schemes alone. In sub-process 1a, Lev does challenge Rio’s justification about the limit of four, which shows us a little about Lev’s warrant at that specific point in the process. We cannot know the extent to which Lev assimilates his warrants according to Rio’s justification; we can only observe that Lev does not object any further. Thus, in the following analysis, I will only consider Rio.

Remember that schemes are goal-oriented concerning the task at hand (Vergnaud, 1997)—in this case, the goal is putting forward a prediction and justifying that prediction by changing the epistemic value. Such activity involves both rules-of-action and theorems-in-action about relevant concepts-in-action: variables and ordered pairs as points in the co-ordinate system. Thus, it is possible to elaborate on warrants as rules-of-action generating techniques relying on theorems-in-action about concepts.

Rules-of-Action

Recall that rules-of-action are implicit propositions concerning the appropriateness of actions for a particular situation (Vergnaud, 1997, 1998). Consequently, rules-of-action can be appropriate and efficient, or irrelevant or inefficient. Some inferred warrants (see Appendix 1) can be considered rules-of-action as they are mobilised into different techniques. The warrants WV1: ‘A variable should take a random value’ and WP1: ‘Ordered pairs of numbers correspond to the x-co-ordinate and y-co-ordinate of a point in the coordinate plane’ are mobilised as techniques for plotting points for randomly picked values of the variables, initially, by counting the distance from the axes to place points A and B in the co-ordinate plane and, then, as imagined points in the co-ordinate plane or by moving the points to new positions in the co-ordinate plane corresponding to other random values of the variable. However, these plotting techniques are deemed ineffective in sub-process 1b. Instead, the rules-of-action WV4: ‘The variable should take an interval of values’ and WP3: ‘An ordered pair with a variable corresponds to a set of points in the co-ordinate plane’ are mobilised as techniques for sketching trajectories. At first, this is done by tracing and moving a point, but, as the tracing is difficult to control, this is too ineffective. The rules-of-action are, however, still relevant and they are mobilised as drawing trajectories with the pen tool instead.

Inferences Drawn Between Theorems-in-Action About Concepts-in-Action

Now, let us turn to the theorems-in-action (Vergnaud, 1997, 1998). Recall that theorems-in-action are held to be true propositions about concepts-in-action. Clearly, concerning mathematical theory, theorems-in-action can be false, partly true or true. The warrants not already identified as rules-in-action are theorems-in-action (see Appendix 1). In the students’ IJ process, most theorems-in-action are false, or only partly true propositions, and are disregarded during the IJ process, starting from WV3 until the students reach the true proportions of WV8 and WP7 + 8 in sub-process 2. However, in sub-process 1a, WV2: ‘A variable represents the same value wherever it appears within the same problem’ is true and undergoes no evolution. Rio mobilises this theorem-in-action to interpret the co-variance of the points as a pattern of perpendicular trajectories, which remains consistent throughout the IJ process.

To reach a justified prediction, Rio uses inference possibilities to infer properties about concepts-in-action. In the following analysis, I attempt to understand better Rio’s IJ process by suggesting what inferences he has made. I do so by considering the order of appearance (see Appendix 1) of the warrants that are theorems-in-action in the IJ process as the line of thought. This approach has a weakness, in that the order of gestures and speech is not necessarily the order of thought. However, observing these actions is our only possible observation to understand how the students’ schemes evolve in the process. I then chart the adopted possibilities of inference (Vergnaud, 1997, 1998), since a proposition about one concept is inferred into a proposition about another in support of either claim 1 or 2.

  1. A)

    Inference chain warranting claim 1

    • As WV3: When the variable changes, it increases in value.

    • WP2: The shared position between points is a ‘starting point’.

    • (And could be that WP4: A set of points is limited by the values we pick for the variable?)

    • No, so WV5: The variable can increase infinitely but is limited by the ‘starting point’.

    • WP5: An ordered pair with a variable corresponds to a set of points in the co-ordinate plane and is limited by the ‘starting point’.

  2. B)

    Inference chain warranting claim 1

    • As WV6: The variable can increase infinitely and decrease to the limit.

    • And WP2: The shared position between points is a ‘starting point’ defined by the common constant in the ordered pairs.

    • It must be that WP5: An ordered pair with a variable corresponds to a set of points in the coordinate plane and is limited by the ‘starting point’.

  3. C)

    Inference chain warranting claim 2

    • But WP6: An ordered pair with a variable corresponds to a set of points in the co-ordinate plane, which can include the ‘starting point’.

    • So, WV7: The variable can increase infinitely and decrease to zero beyond the limit.

  4. D)

    Inference chain warranting claim 1

    • •But WP2: The shared position between points is a ‘starting point’ corresponding to the constant in the ordered pair.

    • •And WP5: An ordered pair with a variable corresponds to a set of points in the co-ordinate plane and is limited by the ‘starting point’.

    • •So, WV6: The variable can increase infinitely and decrease to the limit.

  5. E)

    Inference chain warranting claim 2

    • As WV8: The variable can infinitely increase and decrease.

    • So, WP7: An ordered pair with a variable corresponds to an infinite set of points on a trajectory in the co-ordinate plane.

The students’ IJ process starts with the false theorem-in-action WV3, creating two issues that the students must resolve to reach claim 2: the direction of the movement and the limits/starting point. These two issues are two sides of the same coin. If we consider WV3 and WP2, the inference could be along the lines of ‘because the variable (only) increases in value, the variable must have a starting point’. As there is little other information provided by the task, the shared position of the points in (1, 1) (the intersection of the trajectories of points A and B) is interpreted as this starting point. It is also possible that the inference moves from identifying (1, 1) as the starting point to inferring the direction of the movement as only positive.

In both cases, how can we understand these false theorems-in-action? They may relate to which properties of the variable are relevant in this situation. The students will have encountered situations in which the variable is a placeholder for a value (e.g. in the context of formulas). In such situations, one must select relevant numeric information from a context to replace the variable, which is a rule-of-action. By mobilising such a rule-of-action, Rio attempts to select numeric information from the ordered pairs of points A and B, which are constants of one in this case. From that perspective, the theorems-in-action WV3 + 5 and WP2 + 4 + 5 combine the properties of the variable as a placeholder and the properties of the variable as a general number. Such a warrant has not been inferred in the IJ analysis and is speculative.

In inference chain B, WV6 indicates a turning point. Rio recalls that the variable can increase and decrease, which is inconsistent with inference chain A. It seems that Rio struggles to accommodate his scheme through inference C–E, moving back and forth between justifying claim 1 and claim 2. In inference chain C, he infers properties from WV6 to properties of the points in WP6, but still refers to WP2 and WP5. In inference chain E, Rio realises the full extent of the variable as a generalised number in WV8, rejects any limits on the variable and reinterprets the constants in the ordered co-ordinates of points A and B.

Discussion

I first address the first research question by discussing the interplay between tool use and the justification process. I then address the second question via a discussion of the applied framework. Then, I consider the prediction task as the prediction of dynamic behaviour and, finally, I comment on the limitations of the study. The notion that instrumental genesis is goal-oriented is a core assumption of the IAME. The findings show how the goal of justification results in a particular process of instrumental genesis. By extending the IAME with an analytical tool for IJ, with the goal of changing the epistemic value of claims (Duval, 2007) from the perspective of the students (Duval, 2007; Stylianides & Stylianides, 2022), we can see how the instrumental genesis unfolds through the production and interpretation of data. The change in epistemic value occurs through an interplay of producing data and interpretation through inference between the operational invariants. The inference allows the production of additional supportive or contradictory data. This cycle continues until the epistemic value is changed.

As argued by Shvarts et al. (2021) and many others, the educational value of using tools relies on the epistemic processes that allow for students’ conceptual development. The conceptual element of IJ is considered through the warrants inferred and the analysis of operational invariants. Through such analysis, the case provides an example of the epistemic use of tools as the students’ progress in the complexity of techniques used to produce data and in their conceptual understanding of the interpretation of data. In addition, the analysis of operational invariants shows that progression in conception emerges through inferential possibilities. What drives the development of instrumental genesis from one artefact to the next?

We know from Vergnaud (1997, 1998) that rules-of-action concern the appropriateness of actions for a task and can be efficient or inefficient. In this case, this is evident in the first progression, from placing points to tracing the trajectory of the points. However, inefficiency is relative to the students. The plotting of points can, for other students, be considered efficient. Moreover, the progression from tracing to drawing is not connected to a change in the rules-of-action. Rather, it is the artefact and technique that is inefficient. This drives Rio to try a different technique that more efficiently produces data and is coherent with the rule-of-action. From this, we can argue that the inefficiency of both rules-of-action and the constraints of an artefact can drive the development of instrumental genesis.

What is particular to an IJ process of predictions is that inefficiency is related to the production of data to represent or contradict a prediction and the goal of changing the qualifier. Changes in technique, for example, from discrete to continuous dynamic movements, and artefact produce new types of data. The fact that inefficient rules-of-action and techniques advance instrumental genesis and the IJ process supports the idea of a scheme/technique duality, as proposed by Drijvers et al. (2013). In addition to compliance with IAME, this also shows that the constraints and possibilities of the artefacts influence the process of instrumental genesis and, consequently, IJ.

The IJ analysis tool makes a significant contribution to our understanding of the epistemic use of artefacts, representing a step forward in comprehending students’ justification processes. Notably, it brings into focus the students’ utilisation of tools with an orientation toward the production of data. Furthermore, the model underscores the importance of discerning how students interpret the generated data. In essence, the tool emphasises the student’s perspective on instrument use by inferring warrants. These warrants play a pivotal role in interpreting data in alignment with a claim. This nuanced approach enables us to deliberate on the co-evolution of students’ tool use, considering both the evolution of techniques and the evolution of justifications. In doing so, the IJ analysis tool provides a valuable framework for delving into the intricate dynamics of students’ interactions with tools in justification processes.

The IJ tool links student’s reasoning competency to the use of artefacts. In the concrete case, we see that students broaden their radius-of-action by engaging in IJ processes, which also reflects the technical dimension of students’ competency, as they progress in terms of the complexity of techniques. The analysis of scheme elements shows that such a progression goes hand-in-hand with the conceptual development that emerges from the inferences drawn between operational invariants. This analytical step enables us to contemplate how the evolution of concepts is intricately linked to the exercise of reasoning competency. In this way, the additional analysis of scheme elements is a valuable tool for illuminating the nuanced interplay between students’ reasoning competency and conceptual development when using tools.

For the students, the prediction task has a familiar theoretical component of ordered pairs and points in the co-ordinate system, with which they have several years of procedural experience. It also has a less familiar component, because the students have no experience with operationalising variables in this context. This balance between the familiar and unfamiliar allows students to use the co-ordinate system to identify patterns in symbolic terms, as this can be from a procedural conception. In the introduction, it is hypothesised that predicting the dynamic behaviour of objects in a DGAE will allow students to reason about algebraic properties based on their mathematical and conceptual knowledge, while capitalising on virtual objects’ realness via their dynamic properties. In this case, the students (or at least Rio) recognise the constant as the invariant pattern of perpendicular movement and the variable as the infinite movement of the points.

As Noss et al. (2012) maintain, such an inference would not be possible in a paper-and-pencil environment, as both the constant and the variable would be represented statically. Predicting movement, rather than asking for a translation of the variable points, allows students to capitalise on the dynamic behaviour and provides a context in which they can develop their theories-in-action about terms, and move on to the following task about variable points.

Nevertheless, one concern is issues of the translation of representation, such as the one-to-one mapping of terms, which Duval (2006) problematised in relation to dynamic environment. In this case, how do the students perceive A and B in the final prediction? Do students perceive A and B as particular points that move, or have they objectified A and B as structural patterned movement or, possibly, a hybrid of the particular and generalised? Such questions could be addressed by observing the students’ progression in the prediction of other variable points.

Another issue we witnessed in the case was how phenomenological impressions (Baccaglini-Frank, 2019) can be a stumbling block to students’ reasoning processes. This relates to the issue of a starting point. The students’ experience of the physical world is that things that move begin moving while constrained by time and space. Consequently, the students misinterpreted the starting point from the common constant in symbolic terms in the variable point. However, through inference, the students overcome this misinterpretation, reaching an interpretation of the data and a prediction based on mathematically true theorems-in-action. This advances the hypothesis that prediction tasks can address students’ tendencies toward phenomenological justification.

Some limitations of the results should be clarified. As the results are only based on one case, they are suggestive regarding the progression the students portray. Analysis across a wider set of cases will allow researchers to reveal the relationships between progression in tool use, issues that arises concerning the prediction of dynamic behaviour and student conceptual development in justification processes. Similarly, considering students’ instrumental genesis during similar prediction tasks could reveal how invariant behaviour affects the prediction of dynamic behaviour of variable points.

Conclusion

This study offers an analytical tool that can increase our understanding of students’ use of tools, in interplay with their reasoning competency, from a student-centred perspective within the IAME. Via instrumented justification, we can consider tool use as a particular use in justification processes, and as structuring such processes because of the production of data through techniques and the interpretation of data, as evidence, through warrants. In addition, an analysis of warrants as the generative and epistemic components of schemes (Vergnaud, 1998) provides insights into the progression of students’ conceptual understanding as a result of inferences drawn between theorems-in-action about concepts-in-action, which co-evolve with the progression of techniques. In addition, the progression of instrumental genesis is driven by students’ experience of the inefficiency of both rules-of-action and the constraints of the artefact pertaining to the goal of changing the epistemic status of a claim.

Inefficiency drives students to progress to other techniques and artefacts, ultimately advancing their instrumental genesis. This result aligns with the scheme/technique duality, which is a component of instrumental genesis as proposed by Drijvers et al. (2013). Such a result can inspire task design that intentionally provides students with ineffective tools, with a view toward progression to more advanced tools and the conceptual development that comes with this. Altogether, the proposed framework links students’ progression in the radius of action in their reasoning competency and the use of tools to inferences drawn between theorems-in-action. Furthermore, the prediction task provides context for students engaged in IJ.

The prediction task is particularly valuable, as the prediction of dynamic behaviour reveals properties of the variable, as a concept, in this case infinity. In addition, the students must interpret both constant and variant terms, explicating their structural properties in the very simple representational form of points moving in a co-ordinate system. Such tasks may be useful in developing a structural conception of the variable. Within a more general perspective, the prediction of dynamic behaviour that requires the translation of representations can challenge students’ phenomenological impressions of dynamic behaviour and help them move toward a theoretically grounded justification.