Introduction

The emergence and development of information and communication technologies (ICTs) transform the way students learn, driving education reform and change globally (Wang et al., 2024). The integration of ICTs can enhance students’ learning processes (Peng et al., 2023) by supporting students’ active engagement, participation in groups, frequent interaction and feedback, and connections to real-world contexts (Clark-Wilson et al., 2020). Several studies have shown that when students learn mathematics with the help of technology, they can turn technology as an artifact (a given object) into technology as an instrument (a psychological construct: an instrument = an artifact + schemes and techniques) serving mathematics learning (Trouche, 2004; Drijvers et al., 2009; Pittalis and Drijvers, 2023). These processes are so-called “instrumental geneses” (Rabardel, 2002, p. 77), which are helpful in making interactions between learners and artifacts more meaningful and finally resulting in shaping the artifacts’ use and the learners’ thinking (Bozkurt and Uygan, 2020). In other words, when learners interact with the artifacts, they are likely to emerge with some new ideas for solving a problem or gain a deeper understanding of a concept, which in turn guides the learners’ manner of manipulating the artifacts. For example, Yao (2020) found that three preservice mathematics teachers gained a new understanding of geometry through experiencing instrumental geneses with dynamic geometry software, and these newly developed understandings further guided teachers’ subsequent interactions with the software. Given the importance of instrumental genesis, Artigue (2002) explored its complexity within a computer algebra systems environment and clearly clarified the pragmatic and epistemic value of techniques. Since schemes are invisible whereas techniques are visible (Drijvers et al., 2009), this article focuses on techniques only and adopts the definition of “technique” from Artigue (2002), regarding it as “a manner of solving a task” (p. 248). According to Guin and Trouche (2002), human ordinary activities can be described as techniques. As an example, if someone organizes action in order to realize a given task, a technique may appear as a combination of several gestures (Guin and Trouche, 2002).

During the past two decades, an increasing number of studies have examined the processes of instrumental geneses, particularly in the environment of ICTs, to analyze the important roles played by them (van Dijke-Droogers et al., 2021; Shvarts et al., 2021). Although the genesis of an instrument is beneficial to students’ learning, it is not natural or spontaneous; instead, it needs to be steered, mastered, or guided by teachers (Bozkurt and Uygan, 2020). Just as Trouche (2004) stated, “one of the key elements for a successful integration of artifacts into a learning environment is the institutional and social assistance to this individual command process” (p. 304). These words suggested the necessity of teachers’ integration of technology into teaching. To use a more technical term, it is “instrumental orchestration”, which was first introduced by Guin and Trouche (2002) and was identified as a plan of action that participates in a didactic exploitation system organized by an institution with the goal of guiding students’ instrumented action. In other words, it is defined as “the teacher’s intentional and systematic organization and use of the various artifacts available in a—in this case, computerized—learning environment in a given mathematical task situation, in order to guide students’ instrumental genesis” (Drijvers et al., 2010, pp. 214–215). In the beginning, this term was defined by four components: (1) a set of individuals; (2) a set of objectives (related to the achievement of a type of task or the arrangement of a work-environment); (3) a didactic configuration (a general structure of the plan of action); and (4) a set of exploitations of this configuration (Guin and Trouche, 2002). Soon after, this term was used by Trouche (2004) again to stress the neglected necessity of the external steering of students’ instrumental geneses, and he took didactical configurations and exploitation modes as two main components of it. A didactical configuration refers to an arrangement of the teaching setting and the artifacts involved in it (Drijvers et al., 2010); that is, the teacher needs to anticipate what artifacts will be used by the teacher or student in teaching a specific lesson, think about how to lay out these artifacts, and consider the way of arranging the artifacts, such as by the whole class, by individual students, or by group. An exploitation mode refers to “the way the teacher decides to exploit a didactical configuration for the benefit of his or her didactical intentions” (Drijvers et al., 2010, p. 215), including thinking about the possible roles of the artifacts to be played, considering the schemes and techniques to be developed and established by the students, and deciding the way a task is introduced and worked through. For example, based on the didactical configuration, a teacher needs to think about the role played by the artifacts he/she chooses; and consider how he/she should operate the artifact in order to help students develop a type of technique that is useful in the process of their mathematics learning with the same artifact, such as how to draw a function graph in dynamic mathematics software and scale it to find the asymptote; besides, the teacher needs to design the way of managing the mathematics tasks in advance.

These key points above constitute the original definition of instrumental orchestration. Several years later, Drijvers et al. (2010) pointed out that an instrumental orchestration was partially prepared in advance and partially created on the spot while teaching, so didactical performance was added to instrumental orchestration, which refers to “the ad hoc decisions taken while teaching on how to actually perform in the chosen didactic configuration and exploitation mode” (p. 215). For example, teachers pose questions temporarily, deal with unexpected aspects of mathematics tasks, or achieve other emerging goals. Since then, a consensus that instrumental orchestration is defined by didactical configurations, exploitation modes, and didactical performance has been reached.

Instrumental orchestrations have appeared in several mathematics education studies with different ICTs, such as the dynamic geometry software (Erfjord, 2011; Ndlovu et al., 2013; Savuran and Akkoç, 2023), PhET interactive simulations (Findley et al., 2019), programming (Buteau et al., 2022), and designed digital games (Trgalova and Rousson, 2017). These studies implied that teachers’ instrumental orchestrations are critical for students’ learning in an ICT-supported environment. Moreover, instrumental orchestrations have been considered as a possible way to overcome the didactic stagnation that has accompanied technological evolution (Trouche, 2003).

Since teachers’ instrumental orchestrations are so important, possible ways should be found to improve them. Trouche (2003) proposed that teacher training could be used as a prerequisite for teachers to introduce instrumental orchestrations in technology-integrated classrooms. After teacher training, preservice and in-service teachers will be able to manage and organize artifacts at the mathematical, technological, and psychological levels. Drijvers et al. (2020) also claimed that the concept of instrumental orchestration may be a very appropriate starting point for teacher professional development. Although some other theories relating to technology, like Technological Pedagogical Content Knowledge (TPACK), have been used by studies about teacher professional development institutes for analyzing teachers’ use of technology (Voogt and McKenney, 2017), the focus is rarely on the teacher’s behavior but on the teacher’s knowledge, which is not suitable enough for characterizing a teacher’s process of using technology, resulting in a lack of clarity in describing the teacher’s behavior. However, the instrumental orchestration theory is born to depict the teacher’s intentional and systematic behavior of organizing and using the artifact, as defined, so it has a natural advantage to be adopted in a professional development program for describing the teacher’s integration of technology into teaching, as it is definitely an intentional and systematic behavior. Previous studies have highlighted the significance of instrumental orchestrations and been aware of this concept’s availability in teacher professional development, but the fact is that limited studies have conducted professional development programs focusing on inviting changes in teachers’ instrumental orchestrations, let alone the exploration of the possible ways of improving teachers’ instrumental orchestrations. Thus, based on the definition of instrumental orchestration, this study designed and implemented a professional development course aimed at exploring whether the designed course influenced preservice mathematics teachers’ integration of technology into teaching, particularly the aspect of instrumental orchestrations. The research questions are as follows:

(RQ1) After participating in a professional development course, what are the changes in the types, frequencies, and details of instrumental orchestrations preservice mathematics teachers used?

(RQ2) What elements in the professional development course invite changes in preservice mathematics teachers’ instrumental orchestrations?

Theoretical framework

Using the concept of instrumental orchestration, nine types of instrumental orchestrations for whole-class teaching were identified before, namely, Technical-demo, Explain-the-screen, Discuss-the-screen, Link-screen-board, Spot-and-show, Sherpa-at-work, Guide-and-explain, Board-instruction, and Work-and-walk-by (Drijvers et al., 2010, 2013). For each orchestration type, the first two components of an instrumental orchestration, i.e., didactical configuration and exploitation mode, were described briefly by researchers (Drijvers et al., 2010, 2013). The didactical performance was not included because it has an ad hoc nature. The descriptions of these nine orchestrations are shown in Table 1. In addition, the Work-and-walk-by orchestration was more finely divided, like Technical-demo, Guide-and-explain, Link-screen-paper, Discuss-the-screen, and Technical-support.

Table 1 Descriptions of nine whole-class orchestrations (adapted from Drijvers et al., 2013).
Full size table

As mentioned in the introduction, instrumental orchestrations were often used in the ICT-supported environment, especially in the environment containing the dynamic mathematics software GeoGebra (GGB) (Bozkurt and Ruthven, 2018; Savuran and Akkoç, 2023; Ratnayake et al., 2023). As a subject-specific digital tool for teaching and learning, GGB has the power to help teachers create meaningful learning environments that allow students to flexibly explore geometry objects, mathematical concepts, and relationships between different mathematical phenomena so as to develop their schemes (Martinovic and Karadag, 2012; Venturini and Sinclair, 2017). In this regard, teachers’ orchestrations of GGB should be of great concern. According to the definitions of didactical configuration and exploitation mode, some observable elements can be extracted. Specifically, a didactical configuration can be described by the overall layout of GGB, the involving GGB tools (such as move, point, line, perpendicular line, polygon, slider, etc.), and their arrangement pattern (a whole-class arrangement, a group arrangement, or an individual arrangement). Here, GGB tools refer to all the tools available in the GGB software. They enable the user to produce new objects, get information, or perform actions, and can be activated by clicking on the corresponding buttons on the toolbar. In the GGB software, each toolbar is divided into toolboxes, containing one or more related tools. There are four kinds of toolbars: Graphics View Toolbar, 3D Graphics View Toolbar, CAS View Toolbar, and Spreadsheet View Toolbar (see https://wiki.geogebra.org/en/Tools, for more details). An exploitation mode can be described by the roles of GGB tools, the techniques teachers used, and the management of mathematics tasks. Here, the “scheme” mentioned in the definition is removed because it is not visible (Drijvers et al., 2013). As for “technique”, it refers to teachers’ manners of using GGB to solve mathematics tasks in a lesson, and the same or similar techniques can also be developed or established by students. In other words, if students have access to the GGB file produced by their teacher, they can try to make some similar interactions with GGB, leading to an exploration of a given mathematics task. Drawing on previous studies that took instrumental genesis as the main framework (Ruiz-López, 2017; Yao, 2020), techniques in a GGB environment could be regarded as the combination of dragging, measuring, constructing, and other techniques demonstrated in teaching. Different types of them have been identified in the literature (Arzarello et al., 2002; Olivero and Robutti, 2007; Laborde, 2005), as shown in Table 2. Regarding the role of GGB, there have been researchers characterizing several roles of ICTs in mathematics education. Based on the Roschelle et al. (2017) classification of the roles of ICTs, drawing on some other findings (Aldon, 2010; Daher, 2010; Li et al., 2010), Fan et al. (2022) summarized the roles of ICTs in general learning, which include developing conceptual understanding, practicing problem-solving skills, promoting interest in mathematics, strengthening inquiry-based learning, and enhancing communication and collaborative learning. In consideration of the affordances of GGB, these roles are adopted. Additionally, the role of GGB in doing mathematics is retailed because it can help people offload labor that needs to be done by hand, like calculating (Roschelle et al., 2017). Therefore, an analytical framework of instrumental orchestrations within a GGB environment was constructed, as presented in Table 3. As for didactical performance, it has the nature of being unpredictable, which makes it unsuitable for determining some observable elements in advance.

Table 2 Different types of dragging, measuring, and constructing.
Full size table
Table 3 The analytical framework of instrumental orchestrations within a GGB environment.
Full size table

Methods

Participants

The participants of the study were 26 preservice mathematics teachers studying at a teacher training university located in the middle of China, ranging in age from 20 to 22. They were completing a 4-year bachelor’s program that provides courses about mathematics, general psychology and education, and mathematics education. In their pursuit of a bachelor’s degree, they all need to go to secondary schools to complete an internship, which is required to last for at least two months. Most of them will become mathematics teachers as soon as they graduate, and some of them will become teachers after continuing to complete a 2-year master’s program. In fact, not all of the participants were active, but most of them were willing to do some activities that would help them in their future careers as mathematics teachers. What’s more, when this study recruited participants, only 26 preservice mathematics teachers volunteered to participate and signed an informed consent. During their participation in this study, they did not participate in other training. This study was conducted to explore whether a designed professional development course could change preservice mathematics teachers’ integration of technology into teaching. The focus was on changes, not on their learning about integrating technology. Therefore, the target population of this study was preservice mathematics teachers who were able to integrate technology into their teaching before participating in this study. The sample of 26 participants was related to the target population because they had all enrolled in a course relating to how to use technology in mathematics teaching and passed the final exam before this study. That is to say, they were familiar with the software GGB and had the ability to use its basic tools, simple instructions, and scripts. Additionally, they had never heard of the notions of instrumental geneses and instrumental orchestrations before.

The implementation of a professional development course

When participants took part in the professional development course implemented during summer vacation, they were distributed in 18 different cities. Therefore, the professional development course was carried out through an online platform. Learning from the opinions of Joyce and Showers (1980), theory, demonstration, practice, feedback, and guidance were integrated into the course design to improve its effectiveness. The specific arrangements are shown in Table 4. There were theories and cases of instrumental genesis, the demonstration of software GGB, two rounds of simulated teaching practice, feedback on discussion and teaching practice, and the guidance of mathematics instructional design and GGB operations. The reasons why the professional development course was arranged like this are as follows: First, two rounds of simulated teaching were arranged on the first and last days of the professional development course, in order to examine preservice teachers’ changes in instrumental orchestrations. In the two rounds of simulated teaching practice, a peer-teaching approach was used, because no real students participated in this study. Second, the guidance of mathematics instructional design was provided to ensure all participants knew what basic elements should be included in an instructional design. Third, the extent of mastering GGB was considered because a few participants may forget some operations of GGB although they have learned, so the main function of this theme was to review. Fourth, based on the view of earlier researchers that a key cause of the difference between researchers’ optimistic opinions of technology’s affordances and the real obstacles in integrating technology was the unawareness of instrumental geneses (Artigue, 1996; Rabardel, 2002), the suppose that teachers’ instrumental orchestrations may be improved when they know more about instrumental geneses was proposed. So, the instrumental approach was introduced to preservice teachers, and some cases about instrumental geneses in literature were provided. Fifth, the possible benefits of community were considered, so discussing and sharing ideas within the group and class was designed.

Table 4 The specific arrangements of a professional development course.
Full size table

A month before the start of the professional development course, five secondary mathematics teaching topics suitable for technology integration were provided. There were both junior and senior high school contents, including algebra (inequality properties and power functions) and geometry (isosceles triangles, rotation of graphs, and ellipses). The participants were required to choose one of the provided topics and prepare a 45-min mathematics instructional design that GGB must be integrated into. Given that there were no real secondary school students present, the simulated teaching was expected to last for about 25 min, less than in the real class. Additionally, the participants were asked to highlight the use of GGB in teaching new knowledge as much as possible and to spend less time on exercises that don’t need to use GGB. According to the selection of teaching topics, 26 participants were divided into five groups, and team leaders were determined by themselves, with 5 or 6 members in each group. All participants were required to submit the relevant documents and materials of the first round of teaching design three days before the start of the professional development course. On the first day of the professional development course, they all went through the first round of simulated teaching practice. In the next few days, they needed to reflect on what they have learned every day, submit the day-by-day self-reflections, and use their spare time to revise the initial documents and materials submitted. On the last day of the professional development course, all participants submitted the second teaching design and carried out the second round of simulated teaching practice. After a week, a summary reflection report was asked to be submitted by each participant. The research process diagram can be seen in Supplementary Fig. 1. It did not matter that there were five teaching topics that had different demands on the use of technology because the aim of this study is not to compare the impact of the designed professional development course on the changes in teaching different topics, but to explore whether the designed professional development course can change preservice mathematics teachers’ instrumental orchestrations, by comparing two rounds of teaching design and implementation of the same teacher and finding out the difference. That is, comparisons were made at the individual level, not the collective level.

Data collection and analysis

Data were collected by computer screen recordings, questionnaire surveys, written interviews, and document collection. When designing the semi-structural questionnaire and interview outline, the “orchestration chart” proposed by Drijvers et al. (2014) was adopted. Since there were 26 participants, which suggested the difficulty in determining who would provide valid information in advance, this study adopted the form of written interviews. All the interview data was browsed immediately. When there was an ambiguous answer, the respondent would be found and asked to answer questions again until he/she gave a clear answer. In Table 5, the collection methods, forms, and main contents of the data are shown. Because none of the participants dropped out of the professional development course, the number of every kind of data we collected from participants is 26, including teaching design documents and corresponding slides, GGB files, questionnaires, interview documents, and videos of simulated teaching in each round, as well as daily reflection reports during the professional development course and the final summary reports. About the videos of simulated teaching, most of the videos lasted between 20 and 30 min, but there were still a few videos of shorter duration, with the shortest being about 13 min. The first author was responsible for collecting the data from participants, and the corresponding author was responsible for recording the professional development course.

Table 5 Collection methods, forms, and main contents of the data.
Full size table

When analyzing the data, the starting point was to create a folder for each participant and then organize his/her materials according to the time order. The second step was to transcribe the videos of two simulated lessons and import the transcripts into the qualitative data analysis software NVivo. Next was the most critical step: to code the types of instrumental orchestrations each participant used. The rationale behind the coding was the descriptions of nine whole-class orchestrations, as detailed in Table 1, which was summarized from the whole-class orchestrations overview and descriptions (Drijvers et al., 2013). In the coding process, each sentence in the transcripts was checked according to Table 1. If orchestrations were observed multiple times in a sentence, then this sentence would be coded multiple times. At the same time, the teaching design documents, questionnaires, and interview documents were also checked as multiple supporting pieces of evidence, in order to conduct triangulation and provide possible extra information. The interval between the two codings was more than 3 months. After coding, in order to measure the degree of agreement between the two codings, NVivo was used to run a coding comparison query by calculating the Kappa coefficient individually for each combination of node (code) and source (content of transcripts). The Kappa coefficient between 0.4 and 0.75 indicates fair to good agreement, and the Kappa coefficient over 0.75 indicates excellent agreement. In this study, the Kappa coefficient for each node was >0.72, which meant the coding consistency was relatively good. For the difference in the coding, the researchers discussed it until a consensus was reached. Based on the coding results, the types and frequencies of the participant’s instrumental orchestrations in each round of teaching can be obtained, so as to analyze the changes quantitatively. And then, according to Table 3, the detailed changes in instrumental orchestrations were analyzed qualitatively. Considering that there may be various instrumental orchestrations in solving one mathematics task and that it is very likely that the didactical configurations of these different instrumental orchestrations may be the same, in order to avoid redundancy, the types of instrumental orchestrations were not distinguished when analyzing the didactical configurations. At last, by tracing various documents, the elements inviting changes in preservice mathematics teachers’ instrumental orchestrations were inferred.

Results

In this section, three participants (2 females and 1 male) are presented as the main cases to describe the changes in preservice mathematics teachers’ instrumental orchestrations. It was a purposeful selection because more information about changes was expected, and different genders were also hoping to be included. It happened that two of the three selected cases were group leaders, but in fact, we did not purposefully select group leaders. The criterion for case selection was to identify preservice mathematics teachers who could provide as much information as possible. There are three reasons why the three cases were ultimately selected: First, they all performed actively in the designed course, completed all tasks with high quality, and provided relatively sufficient data. Second, their changes in the types and frequencies of instrumental orchestrations were similar to those of most participants, suggesting that the three cases were representative to some extent. Third, their changes in didactical configurations and exploitation modes were different, and they also differed in what course elements they found useful because of their personal circumstances, which reflected the heterogeneity of the three cases. Here, a posteriori explanation is chosen to explain the use of 3 out of 26 participants. That is because, before reviewing the collected data from participants, it was not known which preservice mathematics teachers’ instrumental orchestrations would change and the extent of those changes; in other words, the participants who could provide more information were unknown; additionally, the heterogeneity of the selected cases would not be known until examining their recordings and reports.

Here are the descriptions of the selected cases. T1, a 21-year-old girl, chose the topic of ellipses. She was the leader of her group, spoke actively in the designed course, and used technology for a long time in the two simulated teaching practices. In addition, her self-reflections were very detailed. Before participating in the professional development course, she had won several prizes at the school and university levels in teaching competitions. T2, a 22-year-old boy, was also the leader of his team, whose teaching topic was isosceles triangles. He also spoke actively, and his words were closely related to what he had learned that day. His reflections were concise, but all the essential points were present. After participating in the professional development course, he thought it was necessary to integrate technology into his teaching topic, which was in sharp contrast to his previous view that his teaching did not need the use of technology. T3 is a 21-year-old girl who chose the topic of power functions. Her performance was relatively ordinary, and she almost did not speak in the professional development course, but she also provided detailed self-reflection.

Types and frequencies of instrumental orchestrations

According to the meaning of different kinds of orchestrations in Table 1, the orchestration types observed in the three preservice mathematics teachers’ teaching were identified, and the counting frequencies were obtained correspondingly. Table 6 shows the initial and final types and frequencies of instrumental orchestrations for each of the three preservice mathematics teachers. There were only five orchestrations observed. Some real coding examples of each observed orchestration are as follows:

Table 6 Orchestration types and frequencies distribution by teachers.
Full size table

(1) Technical-demo: “Now let’s start the motion of this moving point M, which represents the motion of our pen.” (Count once.)

(2) Guide-and-explain: “Class, please observe if they have any common points? Well, you have sharp eyes, you tell me that point (1,1) is the common point, this is what we get from the graph, then can you tell me why the point (1,1) is the common point from an algebraic perspective? Right, it is obvious because…” (Count twice, as the two different questions meant the teacher guided and explained twice.)

(3) Explain-the-screen: “Look, as point A changes, the shape of the isosceles triangle changes, so does its angle. But the degrees of angle C and angle B are always the same; see, now the degree is 48.1, (let point A move further back), now the angle is 27.39, they (refers to the degrees of angle C and angle B) are the same.” (Count once, as these words were all used to explain the same phenomenon.)

(4) Discuss-the-screen: “Class, please discuss what is the effect of the change of α on the graph. Ok, let’s start the discussion.” (Count once.)

(5) Work-and-walk-by: “Ok, I just went around and saw that everybody had finished.” (Count once.)

As shown in Table 6, there was little change in the types of instrumental orchestrations, but the total frequencies increased significantly. Three kinds of orchestrations could be observed in the simulated teaching of T1, T2, and T3, namely, Technical-demo, Guide-and-explain, and Explain-the-screen. In T3’s simulated teaching, Discuss-the-screen and Work-and-walk-by occurred, although their frequencies were very low. Besides, the changes in T3’s frequency of using them were also not obvious. What stands out in this table is the quantity of Technical-demo orchestration used in the two rounds of simulated teaching. It is clear that Technical-demo was the most frequently used orchestration type for every preservice teacher, and its increase was most noticeable. In particular, the number of Technical-demo orchestration used by T3 augmented sharply after participating in the professional development course, up to 67. As for Guide-and-explain orchestration and Explain-the-screen orchestration, the frequency of using them went up to some extent. The other orchestration types, like Link-screen-board, Spot-and-show, Sherpa-at-work, and Board-instruction, were not observable in all three preservice mathematics teachers. That was because the two simulated teaching practices were implemented on an online platform that had no board, and there were no real secondary school students. Essentially, a peer-teaching approach was used there instead of classroom-based intervention. These may also be the reasons why the Discuss-the-screen and Work-and-walk-by orchestrations were hardly used.

Changes in details of instrumental orchestrations

Changes in didactical configurations

After participating in the professional development course, T1’s GGB files changed greatly. In the second simulated teaching practice, T1 designed one more GGB file for demonstrating the process of a plane truncating a cone (see Supplementary Fig. 2). In the class discussion, T1 mentioned more than once that the teaching topic of ellipses was located in the first section of the chapter “The equations of conic sections”, and it was necessary to show the process of a plane truncating a cone. By doing so, she believed “the origin of conic sections can be shown, which laid the foundation for the following learning in the chapter”. It suggested that T1 had a different idea of teaching design and implementation from the first time, and she began to pay attention to the chapter introduction of the textbook. In terms of GGB’s layout, as shown in Supplementary Fig. 2, she made the 3D graphics and the view of the plane present at the same time, so that the section locus became more intuitive. She chose to use three kinds of tools: the slider, button and “move” tool. The arrangement of these tools is a whole-class arrangement.

For the same mathematics task, the GGB files submitted by T1 before and after participating in the professional development course were also quite different, as shown in Supplementary Fig. 3. It is obvious that more sliders were used in the final GGB file, which meant that more variables were added. In the initial GGB file, MF1 plus MF2 was a fixed value, which meant the rope length was fixed. The only variable was the distance between the two thumbtacks (F1 and F2), which was controlled by the slider “c”. The two thumbtacks could only move on the hidden X-axis, ignoring the ellipse where two focuses were elsewhere in the plane. By contrast, in the final GGB file, both thumbtacks could be moved randomly, and the length of the rope could also be changed. In addition, the length of the rope and the distance between two thumbtacks were expressed in the form of formula text, making the size of the two data clear at a glance. T1 also adjusted the position and the text color of the conclusion so that they were more eye-catching. The GGB tools involved in the two teachings were almost the same, namely, the slider, button, check box, and “move” tool. They were all arranged in a way that faced the whole class.

T2’s initial and final GGB files can be seen in Supplementary Fig. 3. In the initial GGB file, points A–C could be dragged. Point A controlled the length of the height AD of the isosceles triangle ABC, point B controlled the length of the waist and base, and point C controlled triangle ABC to fold along the dotted line AD. In the final GGB file, T2 added more interactive controls, showing the process of folding and cutting a rectangular piece of paper, which is an exploratory activity in the textbook. Point E could be moved on the line segment D1A1, driving the movement of the “crease” ED, which allowed students to change the position of the crease. The slider “paper” could change the size of the rectangular paper. Both the slider “cut” and button “cut” can show the process by which a rectangular piece of paper was folded and then cut to create an isosceles triangle. The slider “unfold” controlled the folding and unfolding of the newly created isosceles triangles. The button “recover” could make the rectangular paper instantly return to its unfolding state. Apparently, the GGB tool involved in the first teaching design and implementation was the “move” tool, while in the second teaching, apart from the “move” tool, the involving GGB tools include three sliders, two buttons, and an “angle” tool. Their arrangements were whole-class arrangements.

T3’s initial and final GGB files are presented in Supplementary Fig. 3. In the two GGB files, the check box “plot points” and the button “start” were used to demonstrate the process of obtaining a graph of the quadratic function \({y}={x}^{2}\), and the button “recover” was used to highlight some critical points in the quadratic function plot while hiding others. The difference was, in the final GGB file, the buttons “start” and “recover” were controlled by the check box “plot points”, making the GGB interface more concise. In the initial GGB layout, five specific power functions were plotted in five different colors, each of which was controlled by a check box. Whereas, in the final GGB layout, the presentation of those five power functions was controlled by only one check box named “5 concrete power functions”. In the final GGB file, the inquiry into power functions was no longer limited to five specific cases but focused on more general power functions. Compared with the first GGB file, there were more exploratory mathematics tasks in the final GGB file, such as exploring the graph and properties of \({y}={x}^{\alpha }\) when α is constant, including α is an integer and a fraction. The power exponent α could be changed by inputting different numbers in the input box or dragging the slider “α”. Two sliders, “m” and “k”, corresponded to the check box “\({{y}}^{{\prime} }={x}^{\frac{m}{k}}\)” and were used to explore the power functions when the power exponent is a fraction. Besides, three tables were used to summarize the properties of power functions, and they were controlled by check boxes. The types of GGB tools involved have increased. Check boxes, buttons, and a new self-created tool, “draw a Cartesian coordinate system,” were contained in two rounds of orchestrations, but the input box and sliders were added to use in the final orchestrations. The way of arranging all these GGB tools was changed from whole-class arrangement to a combination of whole-class arrangement and group arrangement.

Changes in exploitation modes

In the first teaching, T1 adjusted the distance between F1 and F2 and guided the students to explore the question: When the distance between F1 and F2 is greater than or equal to or less than the fixed value of MF1 plus MF2 (i.e., when the distance between the two thumbtacks is greater than or equal to or less than the length of the rope), what does the trajectory of point M look like? In this process, T1 used the Technical-demo orchestration several times to lead students’ inquiry about the relationship between \(\left|{F}_{1}{F}_{2}\right|\) and \(\left|{M}{F}_{1}+{M}{F}_{2}\right|\), which meant GGB was used for strengthening students’ inquiry-based learning. After this, students would understand the constraints in the definition of an ellipse, which meant GGB was useful in developing students' conceptual understanding. In the second teaching, T1 used more orchestrations, but the roles played by the GGB remained unchanged. However, in the second teaching, T1 added a task about a plane truncating a cone in the section on “introducing new knowledge”. It explained why ellipses are called conic curves, developing students’ conceptual understanding. Additionally, it promoted students’ interest in mathematics through animation effects. As for techniques, T1’s initial and final constructions were robust, and she used guided measuring to make the distance between F1 and F2 and the value of MF1 plus MF2 clearly visible in two rounds of teaching. The most frequent technique used by T1 was dragging. In the initial orchestrations, T1 used wandering dragging and guided dragging. She randomly dragged the slider “c” to show the relationship between the value of the slider and the distance between two points F1 and F2. Besides, she dragged the slider “c” to make it reach certain values to explore the definition of an ellipse. In the final orchestrations, T1 randomly dragged slider “m” to control the value of \({M^{\prime} F}_{1}^{\prime}\) plus \({M^{\prime} F}_{2}^{\prime}\) (i.e., the length of the rope). Apart from that, T1 dragged the point \({F}_{1}^{\prime}\) and \({F}_{2}^{\prime}\) randomly to reflect the two focuses of an ellipse, which can be anywhere in the plane. These all demonstrated the use of wandering dragging. In addition, she also used guided dragging as in the first teaching. The added task “a plane truncating a cone” contributed to other techniques used by T1, like rotating the 3D graphics, zooming in/out of the view of the plane, and so on. The management of mathematics tasks was reflected in the Guide-and-explain orchestration and Explain-the-screen orchestration. In two rounds of teaching, T1 posed questions to guide students on what they should pay attention to, but her questions in the second teaching were more directed and explicit, which showed the proficiency level of her subject matter knowledge and pedagogical content knowledge had improved. For example, when she led the students to explore the conclusion that “when the length of the rope is equal to the distance between the two thumbtacks, the trace drawn is a line segment” in the first teaching, she just posed the question, “What will happen if the distance between the two thumbtacks is increased?” This question is characterized as content-unspecific because two conditions will occur if the distance between the two thumbtacks continues to increase. As a comparison, in the second simulated teaching, she first summed up the conclusion that “when drawing an ellipse, the length of the rope needs to be longer than the distance between the two thumbtacks”, and then guided students to guess, “What will happen if the length of the rope is equal to the distance between the two thumbtacks?” This question covered only one situation, which showed more content-specific strategies encompassed by T1’s second teaching. Accordingly, T1’s explanations were more targeted.

In T2’s two rounds of teaching, he used GGB tools for guiding students to explore a task: Fold an isosceles triangle in half along its center line and then find the overlapping line segments and angles. This explorative task implied that GGB played a role in strengthening inquiry-based learning. According to the above inquiry-based learning, T2 guided students to pose the proposition that the two base angles of an isosceles triangle are equal before giving a mathematical proof of that, which meant developing a conceptual understanding of the isosceles triangle. Although T2 presented more Technical-demo orchestrations and used more techniques in the second teaching, the key mathematics task was the same as the first teaching, making the GGB roles played remain unchanged. As for the techniques T2 used, the constructions were robust, but the techniques of dragging and measuring have changed to some extent. In the first teaching, T2 used only dragging techniques. He dragged point C to fold and unfold the isosceles triangle back and forth, which represented the dummy locus dragging because the motion trajectory of point C is an arc with the vertex of the isosceles triangle as the center and the waist length as the radius. T2 also used guided dragging. He dragged point A to obtain another isosceles triangle whose shape was different from the original one and then asked students to repeat the previous exploration. In the second teaching, T2 added dragging two sliders named “cut” and “unfold” to demonstrate the process of folding a rectangular piece of paper and then cutting it to create an isosceles triangle, as seen in Supplementary Fig. 4 (Supplementary Fig. 4 was taken from the teaching video, so the original Chinese language was not transformed into English in order to ensure authenticity), which represented the use of guided dragging. Beyond that, T2 made the same operation as the first teaching, so he also used dummy locus dragging and guided dragging, and the only difference was that the object being dragged was changed from the point “C” to the slider “unfold”. As T2 led students to conjecture the property that the two base angles of an isosceles triangle are equal, he added using some techniques of measuring (see Supplementary Fig. 4), which could be seen as the combination of perceptual measuring and validation measuring. For one thing, T2 measured the exact degrees of the two base angles of some isosceles triangles, allowing students to go through the process from a qualitative perception to a quantitative one. For another thing, the process of measuring validated the property that was conjectured before. In terms of the management of mathematics tasks, there were also many differences between the two rounds of teaching, reflected in the use of Technical-demo, Guide-and-explain and Explain-the-screen orchestrations. The adding use of measuring techniques reflected one of the changes in the management of mathematics tasks. Furthermore, for the same mathematics task, more interaction and explanation existed in the second teaching. Table 7 shows two excerpts of his working through the same mathematics task in two rounds of teaching. The mathematics task is “In the process of folding and unfolding the isosceles triangle, which line segments are equal?” As it can be seen, T2 recounted some factual knowledge with a lack of interaction with students and detailed explanation in the first teaching, whereas his second teaching presented more questions and explanations in order to reduce students’ confusion. These changes indicated a higher proficiency level of pedagogical content knowledge in T2’s second teaching.

Table 7 Excerpts of T2’s working through the same mathematics task in two rounds of teaching.
Full size table

In T3’s first teaching, she used GGB tools to demonstrate techniques for carrying out two mathematics tasks. One is to review the steps of plotting the quadratic function \({y}={x}^{2}\). The operation of using GGB to plot an infinite number of points offloaded teachers’ labor, which meant doing mathematics. The other is to explore the graphs and properties of five power functions, namely, \({f}_{1}\left(x\right)=x\), \({f}_{2}\left(x\right)={x}^{2}\), \({f}_{3}\left(x\right)={x}^{3}\), \({f}_{4}\left(x\right)={x}^{\frac{1}{2}}\), \({f}_{5}\left(x\right)={x}^{-1}\). This exploration represented GGB’s role in strengthening inquiry-based learning and developing students’ conceptual understanding of what power function is. When exploring the effects of the power exponent, she used the Guide-and-explain orchestration and Discuss-the-screen orchestration. In T3’s second teaching, she used more GGB tools, increasing the frequencies of the Technical-demo orchestration, not only for completing the two mathematics tasks above, but also for expanding the exploration about power functions. She used a lot of Technical-demo orchestrations to explore how the power exponent affects the graphs and properties of power functions, and made conclusions about the domain, range, parity, and monotonicity of power functions in combination with using other orchestrations. It meant that GGB tools played a more important role in strengthening inquiry-based learning, and the role of doing mathematics and developing conceptual understanding was still retained in the meantime. It was worth pointing out that in the second teaching design, T3 divided the whole class into groups, and the members of each group worked together to use GGB to explore, which embodied the role of enhancing communication and collaborative learning played by GGB. The most used technique in T3’s two rounds of teaching was to tick or not tick the check boxes, followed by moving the graphics view so that students could better focus on the power function being explored, and the techniques of clicking buttons were used sparingly. In T3’s second teaching, she increased the use of input boxes and sliders, which contributed to the increase in using the Technical-demo orchestration. The techniques of dragging sliders were classified into two types: wandering dragging and guided dragging. The use of input boxes and guided dragging was to explore the properties of the power function \({y}={x}^{\alpha }\) when the power exponent α is a specific value, and wandering dragging was used to explore the properties of a certain class of power functions when the power exponent satisfies certain conditions. There was no measuring or constructing technique in T3’s two rounds of teaching. When exploring five specific power functions in two rounds of teaching, T3 both guided students to draw the graphs of them on paper, and used the Work-and-walk-by orchestration and then took turns exploring their properties on the screen, but the techniques used went from ticking check boxes to inputting a number in the input box. The changes in the management of mathematics tasks were reflected in T3’s Guide-and-explain orchestration, Explain-the-screen orchestration, and Work-and-walk-by orchestration. For instance, in exploring the mathematics task about “the influence of the power exponent on the graph and properties of power functions”, T3 guided the whole class to discuss and draw general rules from the learning of five specific power functions in the first teaching. However, in the second teaching, she divided this task into two small tasks, i.e., exploring the graph and properties of power functions when the power exponent is (1) an integer and (2) a fraction. T3 dragged the sliders so as to show the whole class enough graphs of power functions when the power exponent was an integer, and then guided them to summarize the corresponding conclusions. When exploring the case of the power exponent being a fraction, she asked students to work in small groups using GGB for self-directed exploration. In this process, she walked around the classroom. These changes revealed that T3 interacted more with students and was more student-centered when managing mathematics tasks. T3’s act of subdividing a mathematics task was associated with her reflections after participating in the designed professional development course, resulting in a more content-specific strategy to structure the lesson. In addition, T3 used three tables to help explain and conclude the mathematical content, which was the main reason for the increase in the number of using the Explain-the-screen orchestration. This behavior also indicated an improvement in T3’s topic-specific pedagogy; that is, the properties of a class of power functions with the same characteristics are more suitable for summarization in a table.

Changes in didactical performance

For the three preservice mathematics teachers, there were subtle differences between their teaching design and practice in each round. When they implemented simulated teaching, something unexpected happened. For example, T1 forgot to uncheck the check box and directly presented the answers to students; T2 was preset to display the length of each line segment in the isosceles triangle in the GGB file, but it was not achieved in the actual simulated teaching; T3 wanted to ask students a question, but she accidentally gave the conclusion directly. However, those unexpected events in the two simulated teachings were not the same, so it was difficult to claim there were obvious changes in didactical performance. In addition, there were no real students in two rounds of simulated teaching, and the other preservice mathematics teachers participating in the professional development course acted as students, so the three preservice teachers followed the preset teaching design relatively smoothly, which made it more difficult in catching the changes in didactical performance. Fortunately, according to preservice mathematics teachers’ GGB files and teaching design documents, it was evident that they used more GGB functions in the second simulated teaching and increased more interactions with students, as reflected in other documents similarly. For example, T1 thought that “the disadvantage of my first simulated teaching was that there was no interaction between students and GGB, lacking the process of allowing students to operate GGB… I will try to set buttons for students to be involved in the use of GGB in the second teaching.” T2 argued that “the important change in my second teaching is using GGB more frequently and more appropriately.” T3 planned to “let students drag sliders themselves in the second teaching.” All these suggested they may have more improvisational decisions and behaviors in the second simulated teaching than in the first one, if there were real students here. That said, after participating in the professional development course, preservice teachers’ didactical performance was highly likely to change.

Elements in the professional development course inviting changes in orchestrations

For each of the three preservice mathematics teachers, the changes in didactical configurations and exploitation modes are shown in Table 8. Based on these changes, the documents they produced were examined carefully to seek out which course themes and elements had an impact on the above changes.

Table 8 Three preservice mathematics teachers’ changing aspects in instrumental orchestrations.
Full size table

The most striking thing about T1’s changes was the addition of a GGB file describing the process of a plane truncating a cone, which directly led to an increase in the frequencies of instrumental orchestrations, especially the Technical-demo orchestration. By checking out the documents T1 produced, it was figured out that the course themes of problem discussion and feedback and group independent discussion on the fourth day helped T1 change the initial instrumental orchestrations. She wrote, “After having the course problem discussion and feedback, I think I should show the content of this chapter from the macro level, that is, demonstrate the process of a plane truncating a cone… I will modify my GGB files.” “After having the course group independent discussion, I think I should present the 3D graphics and the view of the plane in the meantime so as to give students a visual impact and intuitively connect with the knowledge to be learned next… In the next simulated teaching, I will use a slider to control the length of the rope.” Combined with T1’s second simulated teaching and the corresponding GGB files, it was found that she really put the above ideas into action. That’s why T1 changed her instrumental orchestrations, especially in the facet of GGB layout, techniques used, and management of mathematics tasks. Besides, introduction and cases of the instrumental approach on the third day helped a lot with those changes. Just as T1 wrote, “After having the course introduction and cases of the instrumental approach, I will modify my next teaching design and simulated teaching.” Even though she didn’t describe in detail how to modify it, she expressed after finishing the second teaching design, “Compared with the first teaching design, an important change in the second teaching design was to use instrumentation theory to guide my teaching”, which demonstrated the impact of her emphasis on students’ instrumental geneses on her changes in teaching design and implementation, resulting in changes in instrumental orchestrations. Additionally, she wrote in the end-of-course reflection, “Teachers need to set up appropriate questions and scaffolds in order to steer students’ instrumental geneses.” That’s actually how she did in the second simulated teaching, which also showed the connection between her emphasis on students’ instrumental geneses and her changes in instrumental orchestrations.

According to T2’s documents, he was inspired by the first round of simulated teaching displayed by a member of his group. This member used the “angle” tool in GGB to measure the two base angles of some isosceles triangles, which enlightened him and promoted him to use measuring techniques in his second teaching, resulting in his changes in instrumental orchestrations. He wrote more than once in his reflections: “A member of my group inspired me to use GGB more… Using GGB to measure the exact degree of angels will be more convincing than students’ perceptual guesses.” The course themes of problem discussion and feedback and group independent discussion were also helpful in changing T2’s instrumental orchestrations. In the class discussion, he expressed his thought that “In the teaching topic ‘isosceles triangles’, there can be no use of GGB or other ICTs.” He said, “GGB in this teaching topic did not play a big role. I feel that my teaching can be carried out normally without GGB, and it is very far-fetched to use it.” After he finished speaking, many participants in the professional development course offered their opinions on this idea. As a result, T2 was aware of the need to use GGB in his teaching topic. He wrote, “After having the course themes of problem discussion and feedback and group independent discussion, I know GGB can be used to demonstrate the process of folding a rectangular piece of paper and then cutting it to create an isosceles triangle, leading students to get a more general conclusion. I will design the GGB file from scratch.” These ideas were put into action in his second simulated teaching, resulting in changes in instrumental orchestrations. These changes were also based on the knowledge he learned in introduction and cases of the instrumental approach. Just as he wrote in his reflection report that day, “Instrumental approach prompts me to think about whether I should use GGB and how to use it, and to avoid making decisions based solely on feelings… When making teaching design, I will think about which teaching sessions are suitable for using GGB tools and consider the roles of these tools.”

In regard to T3’s changes in instrumental orchestrations, the course theme of the first round of simulated teaching played a big role. In the interview, after T3 finished the first teaching, she wrote, “I was greatly inspired by a member of my group. GGB can not only be used to show function graphs, but also show summary conclusions, and it can also dynamically show the influence of the change of the power exponent on the trend of the power function graphs through some sliders.” “I will modify my GGB files. I will combine with more about dynamic demonstrations in GGB to explain the general properties of power functions.” Indeed, T3 did what she wrote above in the second teaching, which caused the changes in instrumental orchestrations. The course introduction and cases of the instrumental approach were also helpful to T3’s changes in instrumental orchestrations. In the end-of-course reflection, she stated clearly, “I will use the instrumental approach to improve my next teaching design and implementation… I want to design sliders that are used for exploring the influence of constant α (refers to the power exponent) on the graphs of power functions, letting students drag sliders themselves to help them go through the process of instrumental geneses, and finally promoting a better understanding of the properties of power functions.” These words were consistent with T3’s changes in the arrangement of GGB tools and the roles of GGB tools played. That is, she arranged for students to use GGB in groups during the second teaching.

In summary, the course themes of the first round of simulated teaching, introduction and cases of the instrumental approach, problem discussion and feedback, and group independent discussion had a direct impact on inviting changes in orchestrations. The course Introduction and cases of the instrumental approach were useful to three preservice mathematics teachers, which represented the theoretical support for technology integration. The course theme of the first round of simulated teaching was believed to be helpful by T2 and T3, which represented the “Same Content Different Designs” activity (or Tong Ke Yi Gou, in Chinese). It was considered useless by T1, probably because T1’s teaching was at a relatively high level before participating in the development course, and other members of her group were not as good as her at teaching. In fact, many participants gave positive comments about the “Same Content Different Designs” activity, which was an important determinant of changes in group performance, as reflected in their reports and within group discussions. In the second simulated teaching, the designs of the chosen teaching topic were more similar among the members of a group, including the didactical configurations and exploitation modes, because of the influence of each other. Taking T1’s group as an example, during the group discussion, everyone offered their own opinions on what contents should be taught by integrating GGB, and finally, they reached an agreement, resulting in similar teaching activities integrating GGB. Here are some words from different members of T1’s group that indicate the agreement between group members. “It is not necessary to use GGB just for the sake of using technology, but to consider whether to use GGB based on various aspects such as teaching objectives, key contents of teaching topics, and students’ cognitive ability.” “We should take into account the knowledge base and cognitive level of students when integrating GGB.” “I found that when making GGB files or explaining what happens on GGB, we teachers should stand in the shoes of students and think about how to inspire students by interacting with GGB.” Problem discussion and feedback and group independent discussion were considered useful by T1 and T2. These course themes meant communicating and discussing with others in the professional development course, like in a community. T3’s reflection reports did not show these themes were useful, probably because she was not the team leader, and she just listened to others without expressing her own views. These two themes, along with the theme of the first round of simulated teaching, acted as community support. As for other course themes, they helped preservice mathematics teachers gain more knowledge but had no obvious effect on changing the orchestrations. The reason proficiency in the software GGB did not work was probably because the three preservice teachers had already known the fundamental operations of GGB before participating in the professional development course, and this theme served only as a review. However, if participants were new to GGB, they would need knowledge about how to use it so as to better integrate GGB into teaching. Thus, proficiency in the software GGB, as technical support, was still considered useful because it was the basis for using orchestrations within a GGB environment.

Discussion

Several reports have shown the importance and necessity of integrating technology into learning at different school levels (Song and Wen, 2018; Hew and Brush, 2007; Turugare and Rudhumbu, 2020). Technology can be transformed into an instrument that influences and shapes students’ thinking (Artigue, 2002; Ritella and Hakkarainen, 2012). The notion “instrumental genesis” has been used many times to describe the process by which technology and students’ thinking influence each other (van Dijke-Droogers et al., 2021), but the process needs to be guided by teachers (Trouche, 2004; Bozkurt and Uygan, 2020). Instrumental orchestration, defined as teachers’ intentional use and systematic organization of artifacts, especially ICTs (Drijvers et al., 2010), is a very appropriate concept to describe teachers’ integration of technology into teaching. As mentioned before, some researchers have realized the value of the instrumental approach and designed courses involving instrumental geneses and instrumental orchestrations in teacher professional development (Drijvers, Tacoma, et al., 2013; Filho and Gitirana, 2022; Ndlovu et al., 2013). However, few researchers analyzed teachers’ changes in integrating technology into teaching after participating in these courses, especially through the lens of instrumental orchestration. Thus, this study set out with the aim of using the notion of instrumental orchestration to elaborate on preservice mathematics teachers’ changes in integrating technology into teaching after they participated in a professional development course and finding out some elements embedded in the course inviting those changes. In this study, technology refers to the software GGB, which was widely used by many studies regarding education and information technologies in recent years (Zengin, 2022; Azimi et al., 2023).

This study reported that after participating in a professional development course, each of the three preservice mathematics teachers’ frequencies of instrumental orchestrations increased significantly. The largest increase was in the use of Technical-demo orchestration, which is similar to the result reported in the study of Drijvers et al. (2010) that Technical-demo was the most frequently used orchestration type. The types of instrumental orchestrations did not change, and many kinds of orchestrations could not be observed in this study. These are disappointing results, which may be somewhat limited by the three cases presented only and the lack of board and real students. And these may also be the reasons why changes in didactical performance were difficult to identify. In the aspect of didactical configurations and exploitation modes, three preservice mathematics teachers created a more informative GGB layout, used more techniques, and had more interactions with students in the second teaching. Just as the study of Bozkurt and Koyunkaya (2022) noted, they considered more about teaching objectives, technological actions, prompts, and potential students’ responses. In this regard, we think, as Trouche (2020) claimed, the design of instrumental orchestrations needs to choose appropriate mathematical problems, consider the affordances of artifacts, and anticipate the possible instrumental geneses of students.

It is indicated that the professional development course had an impact on changing preservice mathematics teachers’ instrumental orchestrations, which reveals the importance of conducting serious professional development lessons for teachers. Zengin (2023) examined the effect of a professional development course based on ICTs on mathematics teachers’ designing tasks, and results showed that it played a significant role in improving the competency of designing high-quality technology-enhanced tasks. Another important finding in this study was to figure out the course themes in the professional development course that were useful for inviting changes in instrumental orchestrations. According to the results, the themes of introduction of the instrumental approach and cases of instrumental geneses were useful as theoretical support, which corroborates the views of earlier researchers that the unawareness of processes of instrumental geneses was an important source of the discrepancies between the innovators’ positive perception of technology’ affordances and the real difficulties in integrating technology into teaching (Artigue, 1996; Rabardel, 2002). Also, it is consistent with the opinion that anticipating students’ instrumental geneses was necessary for designing instrumental orchestrations (Trouche, 2020), which required teachers to understand the meaning of instrumental geneses. The theme of the first round of simulated teaching represented the “Same Content Different Designs” activity, which was also very helpful. It was often used in China, which provided an opportunity for a teacher to see other teachers’ designs for the same lesson and feel their own strengths and weaknesses (Yuan and Li, 2012, 2015), as it did in this study. The themes of problem discussion and feedback and group independent discussion also contributed to inviting changes in orchestrations, and they, together with the first round of simulated teaching, represented community support, which has already been proven useful in teacher professional development (Lachance and Confrey, 2003; Matranga and Silverman, 2022). In addition, the theme of proficiency in the software GGB was also considered indispensable technical support, relating to technical knowledge (TK), a component of TPACK, which suggested the promising combination of the theory of instrumental orchestration and the TPACK model, as Drijvers, Tacoma, et al. (2013) stated in a study.

Additionally, although teachers’ knowledge was not the focus of this study, the results still revealed teachers’ pedagogical content knowledge (PCK) and subject matter knowledge were important to the design of learning environments and the adoption of teaching strategies (e.g., using appropriate didactical configurations and exploitation modes) for their students, which is similar to the conclusion that teachers’ analysis of a specific mathematical content influences their changes in teaching strategy (Huang et al., 2021). Thus, some key points need to be considered in teachers’ preparation and development, such as making better subject-matter preparation for teachers and forming a powerful content-specific pedagogical preparation founded on significant and comprehensive subject-matter knowledge (Even, 1993). As many researchers have argued, developing teachers’ PCK and subject matter knowledge is essential for improving the effectiveness of teaching and affecting student learning (Kleickmann et al., 2013; Berry et al., 2016; Schiering et al., 2023b). Consequently, the proficiency levels of teachers’ subject matter knowledge and PCK should be given more attention. In this study, participants’ proficiency level of PCK has improved after participating in a professional development course, which is in line with Schiering et al.’s (2023a) finding that teacher education has the ability to promote transitions into higher proficiency of PCK. In this regard, this study also indicates a possible interrelation between instrumental orchestrations and the proficiency levels of PCK, which encourages a feasible future study networking these two frameworks, with the helpful contribution of four specific proficiency levels identified by Schiering et al. (2023a) in PCK.

At last, there is an unanticipated result that the communication and discussion with other participants changed T2’s attitude toward integrating technology into teaching, which is in line with that of Yuan et al. (2023), who found social influence had a great impact on teachers’ behavioral intentions towards dynamic mathematics software. Given that behavioral intention affects usage behavior, it is reasonable to assume that communicating and discussing with others may have an effect on some aspects of using instrumental orchestrations. Further studies mixing quantitative analysis with qualitative analysis need to be undertaken.

Conclusion

From the findings and discussion above, the conclusions corresponding to two research questions are summarized as follows: First, after participating in the professional development course, there were some preservice mathematics teachers whose final instrumental orchestrations were different from the initial, and the three selected cases were presented as evidence. Second, the three preservice mathematics teachers’ types of instrumental orchestrations did not change, but the frequencies of those increased significantly. Their changes in didactical configurations were obvious, mainly reflected in the layout of GGB; the changes in exploitation modes were more detailed, mainly manifested in the techniques teachers used and the management of mathematics tasks. But their changes in didactical performance were hardly observable. Third, the theoretical support (instrumental approach), technical support (knowledge about using GGB), and community support (“Same Content Different Designs” activity and communicating and discussing with others) were the useful course elements inviting changes in instrumental orchestrations. Fourth, when selecting cases that provided more information about changes, two group leaders were chosen unintentionally, which meant group leaders had an advantage in changing their instrumental orchestrations, as this identity increased their sense of responsibility and motivated them to behave more actively.

This study established an analytical framework of instrumental orchestrations within a GGB environment and applied it, which confirmed the feasibility of the framework. Additionally, some preservice mathematics teachers’ instrumental orchestrations changed after participating in the designed professional development course. Given the short duration of the course and the participants being less exposed to other interventions, it was determined that the professional development course was effective, which provided an intervention for reference to change teachers’ instrumental orchestrations for the better. Moreover, it provided an excellent chance for teachers to think about how to design student activities, which partly responds to the recommendation of Pittalis and Drijvers (2023). Thus, it is suggested that some content of the designed professional development course should be offered to in-service teachers, helping develop their competencies in integrating technology into teaching.

Finally, it should be pointed out that there were only 26 participants in this study, and because only three preservice mathematics teachers’ changes were present, some of the results may not be generalizable. In addition, it was difficult to recognize changes in didactical performance since there were no real students present, and the teaching implementation was just simulated teaching. Therefore, considerably more work will need to be done in a real classroom context to verify the results and determine the influence of the professional development course on teachers’ didactical performance. Moreover, the established analytical framework has not been designed to score the level of instrumental orchestrations with a quantitative approach, and the degree to which different factors influence instrumental orchestrations has also not been studied. In this regard, further studies about instrumental orchestrations would be a fruitful area.