1 Introduction

In the field of mathematical-related affect, one main issue is the connection between affective constructs and cognition, as well as the demand of programs and interventions for promoting aspects of affect (Hannula et al., 2019). Technology-mediated learning offers a fertile ground for this purpose (Gómez-Chacón et al., 2016; Bray & Tangney, 2017; Pierce et al., 2007; Reed et al., 2010).

In the geometrical domain, GeoGebra has gained ground among Dynamic Geometry Software for being free and composed of easy-to-handle tools. It enables students to build dynamic representations by trial and error (constructivity); to explore freely and flexibly (navigability); to receive immediate and effective feedback that allows them to become aware of their errors (interactivity); to execute actions with precision (accuracy); and to think in terms of relevant properties in order to build a dynamic representation of a problem (Sánchez, 2001; Santos-Trigo, 2008).

Besides, GeoGebra has a large community of users who provide teaching and technical support and a track record of research on its use in classrooms (Gökçe & Güner, 2022; Wassie & Zergaw, 2018; Zulnaidi et al., 2020). According to these authors, to date, most studies on GeoGebra report its contribution to learning different mathematical topics and problem solving; fewer works focus explicitly on the affective dimension, although the potential of the software to foster students' motivation, autonomy, involvement, etc. is recognized. Nonetheless, the relevance of affective factors and of the affect-cognition relationship in technological learning environments, particularly with GeoGebra, is highlighted in works like those by Gómez-Chacón (2010, 2011) and Gómez-Chacón et al. (2016). Our work follows this line of inquiry, focusing on mathematical attitudes, such as perseverance, autonomy, rigor, flexibility of thought, etc., which are different from attitudes toward mathematics, such as enjoyment or perceived usefulness. Attitudes of the former type, tightly linked to mathematical cognition, are less thoroughly studied than those of the latter type.

The scarcity of studies on mathematical attitudes may be influenced by the difficulty to measure them, since the strong cognitive component of mathematical attitudes implies that their study cannot be carried out relaying only on participants´ declarative statements, but must be complemented with the observation of the students' behaviour while students do mathematics (Castro, 2004; Gómez-Chacón & Marbán, 2019). To date, much literature on mathematics-related affect relies on the participants´ self-reports (Hannula et al., 2019), and there is also abundant literature based on questionnaires about the introduction of digital tools in the classroom (Turner et al., 2010). In both domains, we should be careful when making conclusions by relying only on self-reported data, given the evidences of discrepancy between reported and observed behaviour (Dignath & Veenman, 2021; Pierce et al., 2007).

On the other hand, Hannula et al. (2019) denounce that in much research on mathematical affect, doing mathematics is regarded as a mere context where the mathematical activity is hardly relevant. When that happens, research may miss opportunities to bring to light decisive factors. The same is true about educational research on the use of digital tools for mathematics education that is not based on domain-specific didactical knowledge. General claims may ignore the subtlety of using technology for learning mathematical content, and the fact that its effects depend to an important extent on the specific technological application, the educational setting and the orchestration by the teacher. It is the how that counts (Drijvers, 2018). Hence, the need arises for the collection of observational data, preferably in natural environments (Reed et al., 2010). According to Gómez-Chacón (2011), it is in these classroom environments that mathematical attitudes can and should be fostered, since students need a socialization process to be imbued with certain habits of mind and mathematical attitudes such as those mentioned above. Within the Design Research paradigm, transformative teaching experiments (Confrey & Lachance, 2000) are well suited for this purpose, since they create and study new teaching and learning possibilities in real contexts, in which they seek to understand the situation under investigation by addressing the how question (Cobb et al., 2015; Molina, 2021).

This article presents a teaching experiment aimed at improving perseverance, precision-rigor and autonomy in secondary students by introducing GeoGebra in the classroom. In what follows, we establish the conjecture and objectives of the study and characterize the mathematical attitudes that it considers. Then, we describe the assumptions of the experiment and we use argumentative grammar, proposed by Cobb et al. (2015) for design experiments, to describe what specific forms of behaviour developed among the students, how these forms of behaviour emerged, and how GeoGebra affordances and the learning environment ensured this process.

2 Objectives of the study

There is abundant literature on the characteristic forms of reasoning supported by GeoGebra, which have no equivalent in traditional learning environments (Wassie & Zergaw, 2018). The conjecture of this study is that this software may also foster students’ mathematical attitudes embedded with these forms of reasoning which are usually difficult to develop in learning environments of paper and pencil geometry.

In order to test this conjecture, two sequences of geometrical tasks were designed and implemented at a secondary school: the first one with paper and pencil and the second one with GeoGebra, with the following objectives:

  1. 1

    To improve perseverance, precision-rigor and autonomy in students from working with paper and pencil to working with GeoGebra.

  2. 2

    To describe how these attitudes change due to software-assisted work, and how GeoGebra affordances and the learning environment sustain this transformation.

3 Mathematical attitudes: Perseverance, precision-rigor and autonomy

Attitude as a construct has been a focus of research in mathematics education in the last decades. However, the concept of attitude is problematic in literature and there is no unified framework to delimit this construct (Beltrán-Pellicer & Godino, 2020; Di Martino & Zan, 2010; Hannula et al., 2018). Attitudes are dispositions to act that usually stand on beliefs and values. Nonetheless , while the latter represent convictions, attitudes are expressed in behaviours. Building on the assumption that no affective behaviour is devoid of cognition (Gómez-Chacon et al., 2016), it is important to note the distinction between attitudes towards mathematics and mathematical attitudes (Feregrino et al., 2020; Gómez-Chacón, 2011; Marbán & Fernández-Gago, 2022; National Council of Teachers of Mathematics [NCTM], 1991). Attitudes towards mathematics refer to the valuation of this discipline, the interest in the subject and the desire to learn it. They stress the affective component —expressed as interest, enjoyment, self-confidence, perceived usefulness, etc.— over the cognitive component. By contrast, mathematical attitudes relate to mathematical processes and activities. They refer to certain habits of mind and a tendency to think and to act in positive ways. Students’ mathematical attitudes are manifested in the way they approach tasks and solve problems; therefore, they are considered by authors like Marbán and Fernández-Gago (2022) as purely mathematical attitudes. Examples of them are flexible thinking, perseverance, independence, accuracy and thoroughness, creativity, and so on, which are consubstantial to mathematical work, and stress the cognitive component over the affective one (Nechache & Gómez-Chacón, 2022) .

According to Gómez-Chacón (2011), the study of attitudes towards mathematics has a long tradition, which is shorter for technological environments, and the tendency to evaluate them has been through questionnaires. In the case of GeoGebra, works such as those of García & Romero (2020), Wassie and Zergaw (2018), Yoganci (2018) and Zengìn (2017) show that it helps to increase the confidence, self-esteem, interest and motivation of students. Notwithstanding, favorable attitudes towards mathematics, with or without technology, are not enough to ensure the mathematical attitudes in the students that support genuine mathematical activity in school settlements (Barnes, 2021; Gómez-Chacón & Marbán, 2019; NCTM, 1991).

Even though the influence attitudes have on the teaching–learning process is well-known, in the field of mathematics, attitudes have been less clearly defined than in the field of psychology (Marbán & Fernández-Gago, 2022). According to these authors, research studies on attitudes in mathematics could be divided into positivist and interpretative. In positivist studies, attitudes are frequently defined implicitly, through the instruments employed to measure them. Moreover, those studies who give explicit definitions don´t share a common characterization and there is no agreement about their observable features (Di Martino & Zan, 2010). This state of the art calls for a clear conceptualization of the components and constructs referenced by every author in their works to avoid possible adverse effects due to lack of consensus (Marbán & Fernández-Gago, 2022).

This study belongs to the interpretative type, since it is focused on the understanding of the phenomenon and has a predominantly qualitative nature. Characterizations of the three mathematical attitudes considered are presented. They are based on grounded theory (Corbin & Strauss, 1990), in which conceptual frameworks, together with the evidence collected in the previous cycles of the experiment, guide the identification of the indicators for each of the attitudes (García, 2011). The indicators provided are consistent with the literature and are stated in such a way that they allow the collection of evidence of the students' behaviours in the class group. For this reason, to facilitate the observational recording in the classroom, positive and negative indicators are provided. In this way, the present work meets the demand made by Hannula (2018) for research on affect in mathematics education on group level dynamic processes, which is poorly explored, but deemed necessary, since it will help to create classrooms that are able to affectively engage our students.

3.1 Perseverance (PE)

Problem solving is central to learning mathematics and in numerous international curricula children are expected to persist when faced with difficult problems. For example, in Singapore, mathematics frameworks center on problem solving where students develop an "attitude of perseverance" (Sengupta-Irving & Agarwal, 2017). Perseverance is an attitude that implies remaining constant when carrying out an activity or accomplishing a set target. Bettinger et al. (2018) define a student with high perseverance as someone who stays focused on challenging tasks, works hard, and does not give up. However, it is necessary to distinguish between perseverance and persistence, the latter meaning inflexible perseverance, since striving might be interpreted as keeping going, irrespective of the quality of findings of each try and, therefore, persistent behaviours in mathematical reasoning may limit the application of learning from trials (Barnes, 2021; Jiang et al., 2021). Therefore, in our study, we adopt the definition of perseverance by Thom and Pirie (2002) as the student’s sense in knowing when to continue with, and not to give up too son on a chosen strategy or action, and at the same time, knowing when to abandon a particular strategy or action in the search for a more effective or useful one.

In spite of curricular efforts, in many mathematics classes, students in general show limited persistence when facing non-routine tasks or problems. In our experience, there are those students who, when facing a task that they cannot soon resolve, are prompt to give up. Other students find a solution, but if they realize it is not the right one, either by themselves or because someone else points it out, they refuse trying new approaches and settle for the wrong response.

From this perspective, we propose the following indicators for perseverance:

  • PE1: Tendency to give up easily before finding the solution to problem

  • PE2: Willingness to settle for the wrong answer when failing to find the right one and reluctance to try other approaches

  • PE3: Refusal to give up until a solution to the problem is found which is satisfactory and considered correct, changing approaches when necessary

3.2 Precision-Rigor (PR)

Precision and rigor can be considered as genuine elements of mathematical work (Gómez-Chacón, 2009; Marbán & Fernández-Gago, 2022). These two elements often appear linked when regarded as a mathematical attitude; in fact, to define one of them the other is usually employed (Allende, 2004). In the Common Core Standards for Mathematical Learning (Klirs, 2013), precision goes beyond skill proficiency and accurate calculations. The standards embody the notion of rigor as intellectual precision, which place an emphasis on mathematical reasoning. The focus shifts from skills and computation to consider skills as tools for reasoning and understanding (Hull et al., 2013). As for what rigorous reasoning means, Gómez-Chacón (2009) and Dedò (2012), consider mathematical rigor as a dynamic construct, with different levels that should accommodate the learner´s evolving maturity. In the same line, Liu (2018) echoes the criticism of authors such as Freudenthal (1973) to the logical validity of arguments and proofs as the main characteristic of rigor in educational contexts. She provides a definition of learner’s rigor as actions one takes at different levels to validate one’s mathematical working (e.g., actions on ensuring accuracy) and the criteria one adopts for judging whether certain mathematical responses in a specific context are correct or not.

We consider that a student is not precise, nor rigorous, when he or she performs calculations or geometrical constructions carelessly, thinking that a small error will not be relevant. Also, when the strategy followed by the student is not totally correct and therefore the result obtained is not precise; yet, he or she remains satisfied, despite knowing that the solution could be improved. Conversely, a student shows precision and rigor, when carrying out calculations and geometrical representations, and obtaining a result that does not fit, he or she discards it and repeats the process until satisfied with the outcome. Likewise, when the calculations and representations obtained are not accurate, possibly because they lean on a mistaken reasoning, the student continues to refine his or her reasoning, and works until the intended precision is attained.

Taking the above into account, we propose the following indicators for precision-rigor:

  • PR1: Failure to view errors as important

  • PR2: Willingness to settle for approximate solutions, without considering its righteousness or adequacy

  • PR3: Attention to geometric constructions and arithmetic operations, scant tolerance of errors, and inquiry on the strategies behind them

3.3 Autonomy (AU)

Autonomy is considered by advocates of self-determination theory as one of the three fundamental psychological needs that motivate behaviour (Deci & Ryan, 2012; Russo & Minas, 2020).

Developing autonomy in learners has been emphasized as a goal of mathematics education (Sachdeva, 2019). According to this author, autonomy has both psychological and intellectual aspects. From the psychological tradition, the term autonomy literally refers to regulation by the self (Ryan & Deci, 2006). Referred to the learning process, autonomy is one of the students’ basic psychological needs, which facilitates student self-regulation for learning, academic achievement and well-being (Wilujeng, 2018; Flunger et al., 2022). As important as it is to develop such a psychological relationship with one´s mathematical learning, students must have more than a sense of acting of their own accord to be considered intellectually autonomous in mathematics (Wood, 2016). In fact, discussions related to autonomy in mathematics education literature have mostly depicted it as an intellectual attribute (Cobb & Yackel, 1998; McConney & Perry, 2011; Wood, 2016; Yackel & Cobb, 1996). Along this line, we consider autonomy in our work, as an intellectual activity that implies students´ willingness to draw on their own capabilities when making mathematical decisions and judgements.

Historically, intellectual autonomy was defined in terms of the individual; however, as researchers adopted sociocultural perspectives, they began to frame autonomy in relation to others (Sachdeva, 2019). From the communicational framework, individual students might maintain autonomy while learning from others (Ben-Zvi & Sfard, 2007). Notwithstanding, despite the need to learn from others, learning mathematics can be autonomous when the learner seeks to explore new ideas for his or herself rather than merely resort to others´ ideas.

In our experience in classroom environments, non-autonomous students, when facing a non-routine task refuse to think for themselves, seek for external help to know how to act and depend on constant support to progress in a task. There are more autonomous students who usually work on their own and, when finding a blockage, seek the way to solve it by devising their own strategies, alone or collaborating with their peers. These students only seek the guidance of the teacher after exhausting their own resources. Finally, there are those students who have a sense of their target when facing a task and are eager to make their own decisions in the process to get it.

According to these observations, we propose the following indicators for autonomy:

  • AU1: Reluctance to think for oneself, preferring to resort to external guidance on how to act

  • AU2: Initiative to surmount intellectual obstacles by drawing on one´s own mathematical ideas or strategies

  • AU3: Eagerness to make one’s own decisions for one’s own purposes

4 Design experiment

This study is part of a teaching experiment (Prediger et al., 2015), conducted in a public school in Spain, whose classes were equipped with one computer for every two students. Its aim was to explore the effect of working with GeoGebra on promoting mathematical attitudes, attitudes toward mathematics and mathematical competence in students (Romero et al., 2015; García & Romero, 2020; García et al. 2021). In accord with the cyclical nature of design research, two successive cycles were carried out.

The first cycle was aimed at designing, implementing in two classes, and adjusting the didactical sequence with GeoGebra together with the observation instruments. The second cycle, which is the object of this article, was conducted in two classes of the same school along two months. Each class had 23 students, aged 14–15, who, as in the previous cycle, had no previous experience with mathematical software.Footnote 1 In all classes, the mathematics teacher was the second author of this article and a member of the research team.

According to Suh et al. (2008), a technology-rich environment influences some critical features of the mathematics classroom, such as the social culture of the classroom, the role of the teacher, and the nature of the tasks. In this section, these features are addressed, along with the material and methods of the experiment.

4.1 Tasks, classroom culture and the role of the teacher

For the second cycle of the experiment, two sequences of geometrical tasks were designed by the research team, which included two university researchers and the above-mentioned teacher-researcher. In the first sequence of 13 tasks, PP tasks, students worked on plane geometry with paper and pencil to become familiarized with all the methodological aspects of the design, except for the use of GeoGebra. In the second sequence of 10 tasks, GeoGebra tasks, students worked on plane tessellations with GeoGebra.Footnote 2 Both sequences were graded by increasing difficulty, starting with tasks where it was easy to progress by trial-and-error and ending with more complex ones; all of them demanded reasoning and argumentation.Footnote 3 All the tasks were contextualized, no resolution strategy was previously taught and they intended to challenge to the students. GeoGebra was the only differentiating factor between both sequences. Observation grids were used in these sessions to assess attitudinal levels in students.

Concerning the classroom culture, a social-constructivist paradigm was adopted. During the experiment, students worked in pairs, sharing a computer in the case the GeoGebra sequence. The tasks were intended to promote social interaction between peers, to have them talk about their mathematical explorations, to refine their successive intuitions in the problem-solving process, and to take the lead in their learning. However, the teacher’s role was decisive to orient students’ activity according to the parameters of mathematical knowledge. The teacher-researcher took the role of guide, resource provider, learning motivator, and constant appraiser of the students as they worked on all tasks. At the end of each task, she orchestrated the whole-class discussion so that the students could attain a better conceptualization of the emerging mathematical contents. The functions of teacher and students in technological environments are coherent with this methodology (Barberá et al., 2008; Lavy & Leron, 2004; Olkunn et al., 2005; Sinclair, 2005).

4.2 Materials and methods

The design research was conducted at two levels: a micro level, referred to classrooms and sessions, and a nano level, referred to individuals and tasks (Prediger et al., 2015).

At the micro level, the overall progress of all students was analyzed in terms of their mathematical attitudes before and during the work with GeoGebra (PP tasks and GeoGebra tasks). Data were taken declaratively, from whole-class semi-structured interviews and from spontaneous interventions of the students in a virtual post-box for suggestions (see Examples 2, 5, 7, 8, 9 and 10 in the results section), and observationally, from attitude grids, student´s productions and from the teacher-researcher´s diaries (Examples 1 and 3). The diaries were written by the teacher-researcher after every session and the observations were based on the indicators described in Section 3. A register was kept for each attitude and completed after each session regarding whether no students, some students (up to two thirds) or the majority (over two thirds) showed the above attitude (Table 2 in Section 5.1).

This register was contrasted with the one filled by an external expert at random sessions, and periodically discussed by the research team. The following is an extract of the teacher’s diary of session 12 for GeoGebra tasks:

Despite not being able to visit the Escher mosaic web due to technical problems, as an inspiration to solve task 10, almost the whole class worked on it with fewer difficulties than expected. Although some pairs have not managed to tile properly with the basic tile created, 38 of the 46 students have not abandoned the task until they have found a solution, changing approaches when necessary (PE3). 34 of them have worked with total precision and rigor (PR3) to get a perfect mosaic and those who have not achieved the desired result, have undone and redone their construction until they achieve it. Most of the students (34 of 46) have worked autonomously (AU3), testing different transformations to get the basic tile, asking for my help only in specific cases.

At the nano level, a representative sample of 12 students (6 in each class) was selected, considering a combination between their attitudinal profile and competence level. They were determined by analyzing the attitude grids (double-entry table with attitude indicators and studentsFootnote 4), the teacher-researcher's diaries, and each student's performance during the first sequence of paper and pencil tasks (PP tasks) prior to the GeoGebra experience, as shown in Table 1. The number of students chosen from each profile was proportional to the number of students with that profile in the total sample of students (N = 46).

Table 1 Sample students´ profiles
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In every session, the students wore microphones connected to the computers to record their dialogues. The teacher-researcher completed an attitude grid, which was contrasted with the one completed by the above-mentioned external expert, reaching an interobserver agreement with a Kappa index = 0.87 (see Figs. 2, 3, 5 and 6 in Section 5). Moreover, she registered individual observations for each sample student in her diaries (Examples 6 and 13).

All these data were triangulated and analyzed using the software for qualitative analysis Atlas.ti. (Fig. 1). The process included: (a) transcribing sample students´ audios of the tasks; (b) reviewing the construction protocol of their GeoGebra files; (c) introducing in Atlas.ti each audio fragment with its corresponding GeoGebra protocol fragment, matching the dialogues of the students with the actions they were performing with GeoGebra; (d) coding the fragments of the Atlas.ti files in which each student manifested perseverance, precision-rigor and/or autonomy, using the indicators of each attitude; (e) adding to each fragment the codes which reflected the factor that was influencing to a greater degree the observed attitude (GeoGebra, peer, teacher, or task), provided that this influence was perceptible; (f) contrasting and triangulating the coding made in Atlas.ti for the attitudes of each task, with the observations registered in attitude grids and in the teacher-researcher´s diaries; and (g) analyzing the co-occurrence between codes of attitudes and codes of influence factors (Tables 3, 4, 5, and Figs. 4, 7 and 8 stem from this analysis). The whole process was reviewed and approved by an external expert in the use of Atlas.ti and in analyzing classroom discourse, in terms of the coding performed, the calculations, and the conclusions obtained.

Fig. 1
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Integration of data from different sources and codification with Atlas.ti (extract of task 9 in which students built the airplane-shaped mosaic of the Alhambra starting from a square and deforming it)

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5 Results and discussion

In this section, global results for all students are presented as a frame for the in-depth analysis of the transformation of students´ perseverance, precision-rigor and autonomy due to GeoGebra. The later analysis is carried out with data of the sample students and shows how the software influenced the attitudinal progress.

5.1 Global results

The global results, from the observations in the teacher-researcher´s diaries, based on the attitudinal indicators, are presented in Table 2. They show a clear improvement of mathematical attitudes in the GeoGebra sessions.

Table 2 Mathematical attitudes in paper and pencil and GeoGebra sessions
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Regarding the sample students, Fig. 2 shows the results from individual observation grids, which are coherent with those of the total. Percentages were obtained by analyzing the number of sessions in which sample students manifested each attitude, when necessary or appropriate, to solve the proposed paper and pencil (PP) or GeoGebra tasks:

Fig. 2
figure 2

Percentages of sessions and attitudes demonstrated by students in the sample (To analyze the attitude grids, the indicators recorded for each student in each session were scored: PE1 = PR1 = AU1 = 0; PE2 = PR2 = AU2 = 1 and PE3 = PR3 = AU3 = 2. \(Studen{t}{'}\;attitude\; X\; percentage=\frac{\sum\; score\; studen{t}{'}\;attitude\; X\; of\; each\; session}{Maximun \;score}\cdot 100\) (X = PE, PR, AU). (Maximum score: PP tasks = 26 points (13 sessions), GeoGebra tasks = 24 points (12 sessions))

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In GeoGebra tasks, the great majority of the students in almost all sessions showed the three attitudes, while in PP tasks, in most sessions none or some students showed them. Example 1, corresponding to for the penultimate PP task, illustrates this situation.

figure a

By introducing GeoGebra in the classroom, this situation improved remarkably, as we discuss in the next sections.

5.2 Perseverance (PE)

During PP tasks, when students tried to solve non-routine problems without succeeding at first trials, many of them usually gave up or settled for careless responses. With GeoGebra, most students persevered until they found a satisfactory answer. Students´ interventions in the whole-class interviews and the teacher-researcher observation in her diary give us a glimpse of this change (Examples 2 and 3).

As with the total of students (Table 2), perseverance rose up significantly for the sample ones. Figure 3 shows the disparity among the values for the sample students during PP tasks, while in GeoGebra sessions they all score very high. By means of the co-occurrence analysis with Atlas.ti, the improvement of Perseverance was associated mostly with GeoGebra (Table 3).

Fig. 3
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Percentages of sessions where sample students manifested Perseverance during PP and GeoGebra tasks

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Table 3 Percentage of influence of the factors that contributed to the sample students showed PE3
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The constructivity of GeoGebra allowed the students to generate a large number of examples over which to reason and to act by trial–error when not having a definite strategy, which kept most of them working on the tasks until completing them. They could venture to test different conjectures, undoing and redoing without wasting much time and effort. Navigability also enabled the students to explore different possibilities. Example 4 (Fig. 4) shows how these affordances allowed sample student S3 to keep working on one of the tasks. Prior to GeoGebra, this student usually abandoned the problems without obtaining any solution, in many cases without even trying to solve them (PE1); therefore, he was an underachiever. However, with GeoGebra he showed a completely different behaviour, not leaving one single task unresolved, testing different approaches when necessary (PE3).

Fig. 4
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Example 4 of the influence of GeoGebra on Perseverance in sample student S3 during task 7, where students were asked to build a semi-regular mosaic with squares and triangles

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The example 4 (and the one in Fig. 1) illustrates the way in which students showed flexibility in their approaches, rather than fruitless persistence (Barnes, 2019; Thom & Pirie, 2002). On the other hand, their enjoyment with computer work did not distract them from their line reasoning; despite difficulties and delays, they remained focused on the task, unlike the case reported by Barnes (2021).

5.3 Precision-Rigor (PR)

Regarding this attitude, in PP sessions, when it was necessary to do new calculations and representations, various students were rather unmotivated and they settled for approximate answers, sometimes because they did not know the root of their inaccuracy and others because they did not consider it was worth the effort to amend it (Example 2). More important, many students performed calculations and made representations without reflecting on their coherence and soundness in the context of the non-routine tasks (Example 1).

The improvement of precision-rigor in sample students and the dispersion of the student´s values is shown in Fig. 5. The co-occurrences analysis with Atlas.ti, for the sample students also associates the improvement of this attitude mostly with GeoGebra (Table 4).

Fig. 5
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Percentages of sessions where sample students manifested Precision-Rigor during PP and GeoGebra tasks

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Table 4 Percentage of influence of the factors that contributed to the sample students showed PR3
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GeoGebra afforded the students to carry out accurately a variety of graphic representations, whose visualization informed them about the degree of precision of their construction methods (interactivity). When students became aware of inaccuracies, it was easy to undo the constructions and build new ones (constructivity), and they were eager to do so (see Fig. 4). Using GeoGebra, they became involved in the tasks and keener on doing them as right as possible (Examples 5 and 6).

Besides, in GeoGebra, representations are consequence of decisions taken and strategies implemented for their construction and can be manipulated maintaining the geometric properties under which they were built. The fact that students could repeat the same process varying the conditions or trying different strategies made them become not only more accurate, but more rigorous. At first, they acted by trial and error but, progressively, they inquired about the pertinence of the strategies behind their approaches and their suitability. Figures 1 and 4, above, illustrate this fact for sample students S8 and S3, respectively. As for their previous profiles, during PP sequence, S8 was always accurate but only occasionally rigorous and S3 did not show any accuracy nor rigor. Both of them showed PR3 in all GeoGebra tasks.

In our experiment, as GeoGebra enabled the students to behave more accurately and rigorously, some of them began to appreciate the value of being precise as a tool for understanding (Hull et al., 2013). One of the students says it this way in the whole-class interview (Example 7):

5.4 Autonomy (AU)

When non-routine problems were posed to students in PP tasks, many got blocked after reading the outline and requested for help immediately, as illustrated in Example 1. With the introduction of GeoGebra, autonomy was registered in almost all sessions for nearly all the students (Table 2). From the beginning, they tried to solve each task with their respective peer, testing different strategies, and they only asked for the teacher’s help after various failed attempts. Students’ comments illustrate this (Examples 8, 9 and 11).

In the student´s words of Example 9, we can appreciate how she begins to feel part of a learning community where students are able to contribute and to benefit from each other´s understanding (Barberá et al., 2008; Preiner, 2008). Despite the need to learn from others, the student´s sense of autonomy is reflected in the drawing of her own conclusions from a variety of points of view (AU2) (Ben-Zvi & Sfard, 2007). This, in turn, contributed to increase her self-confidence and motivation. The increase in motivation is also illustrated in Example 10. The fact that GeoGebra creates impersonal situations where students can make mistakes in private (Preiner, 2008) impacts on their involvement and autonomy.

All sample students, except for S7 and S11, reached high levels of autonomy (Fig. 6). Again, GeoGebra was the most important factor influencing this attitude (Table 5).

Fig. 6
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Percentages of sessions where sample students manifested Autonomy during PP and GeoGebra tasks

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Table 5 Percentage of influence of the factors that contributed to the sample students showed AU3
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The initial low level of some students’ autonomy obeys two different profiles: some were barely autonomous due to a lack of motivation, which led them to not working on the tasks or, if they did, expecting the teacher or other classmates to tell them the way to go. Other students, despite being interested in the tasks, had difficulties defining their own strategy and requested for external help continuously. In the students matching the first pattern, an increase of motivation towards GeoGebra triggered a more autonomous behaviour. Students fitting the second pattern (cognitive deficiencies), except S11 who suffered more severe limitations, also experienced a considerable improvement. GeoGebra enabled the students to become responsible for their own learning, affording them to remain active throughout the problem-solving process (constructivity), providing constant feedback (interactivity), and allowing them to explore different possibilities (navigability), as we illustrate with sample students S7 and S8.

Prior to GeoGebra, S7 always preferred not to think for herself. She asked her classmates or the teacher what to do and often copied the answers from others without understanding them (AU1). In her case, constant use of GeoGebra helped her to improve progressively; even if she did not finally reach high autonomy levels, she became more self-sufficient, asking for help only after trying several ways and being unable to continue (Example 11 in Fig. 7).

Fig. 7
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Example 11 of the influence of GeoGebra on Autonomy in sample student S7 during task 5, where students were asked to obtain the three regular mosaics that tessellate the plane, discussing their existence and uniqueness

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The case of student S8 was different. He had shown initiative to surmount obstacles during PP tasks, although he needed support in complex ones (AU2). With GeoGebra, he grew in intellectual autonomy (Wood, 2016), developing a sense of his own inquiries. Navigability afforded him the freedom to explore different strategies and tools, resorting to the teacher only to explain his own initiatives to her (Example 12 in Fig. 8), or even ignoring her when pursuing his interest (Example 13).

Fig. 8
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Example 12 of the influence of GeoGebra on Autonomy in sample student S8 during task 4, where students were asked to to build regular polygons in order to make the tiling in task 5 (S8 had previously built a regular pentagon with 72º rotations)

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6 Conclusion

This study addresses a double demand in the affect agenda in mathematics education: the need of research on transformative interventions that promote affect in mathematics learning, and the need to link affect and cognition, transcending the lack of mathematical specificity denounced by Hannula et al. (2019) in much research on mathematical affect, where doing mathematics is simply a context, but mathematics has little explicit relevance.

Our initial conjecture was that mathematical attitudes, such as perseverance, precision-rigor and autonomy can be fostered in students by the use of GeoGebra. Quantitative results for the students of the two secondary classrooms under study informed that over two thirds of the population manifested adequate levels in these attitudes for almost all sessions working with the software, something that had not occurred when doing geometry with paper and pencil. Qualitative analysis, performed on data from a representative sample of students, showed how the introduction of GeoGebra brought about important changes regarding these attitudes.

When doing mathematics, and particularly geometry, students felt more or less confident learning skills and procedures, but when non-routine problems were posed, most of them got blocked after the first attempts, resorted to external help repeatedly, and were willing to settle for inaccurate or even incoherent responses, showing no genuine involvement in open tasks. Working with GeoGebra, new forms of behaviour emerged.

From the first tasks, the constructivity of the software kept most students busy, generating a variety of figures over which to reason. They enjoyed the process, but this did not distract them from their focus on completing the tasks. Since these tasks were properly graduated, many students could succeed from the first sessions. The constructivity and navigability of GeoGebra allowed them to persevere with fluency, changing approaches and strategies when necessary and showing flexibility of thought for solving the proposed problems.

The interactivity of GeoGebra also afforded the students to become aware of the inaccuracies of their constructions and constructivity allowed them to easily redo them. Most of them naturally developed an inclination to make geometric constructions in a precise way. Moreover, since GeoGebra representations are a consequence of decisions taken and strategies implemented for their construction, various students could realize the importance of precision to carry out adequate reasoning for solving the problems. This feature, exploited by the teacher´s questions and guidance, increased the level of precision-rigor in the classrooms, by considering skills as tools for reasoning and understanding.

Besides, GeoGebra afforded the students a collaborative space where they could share perceptions and ideas. They were willing to collaborate with their peers, take advantage of other´s points of views and suggestions and help others. This provided the whole classroom a great deal of autonomy from the teacher, which she employed to guide individual processes, by offering questions and suggestions that could direct the inquiries of the students in the right direction. Sometimes it was observed how students pursued their own insights, exploring them till they were satisfied. The constructivity, navigability and interactivity of the software afforded them this degree of autonomy.

These results reflect a positive turn in the affective-volitional relation between students and mathematics that hinges around classroom interactions deeply influenced by software-assisted work (Roth & Walshaw, 2019). In our case, the introduction of GeoGebra in the classroom, with properly graded tasks and adequate classroom management, enabled the teacher-researcher to capitalize on the students´ motivation towards the use of computers to promote mathematical attitudes, such as perseverance, precision-rigor and autonomy, necessary to sustain mathematical thought processes of increasing complexity (Gómez-Chacón et al., 2016; García et al., 2021).