Information and communication technologies (ICT) become increasingly important in modern education. Teachers can use ICT as a tool to support students’ learning; or a tool for teachers’ personal productivity; or a medium for interacting and collaborating with colleagues (Ponte et al., 2002). In spite of its potential for teaching and learning, the integration of ICT into mathematics education lags behind the high expectations of researchers (Assude, 2007; Cayton et al., 2017; Drijvers et al., 2010). And how technology can be used effectively and the effects it brought into mathematics teaching and learning are important in mathematics education (Bretscher, 2014). Researchers think it is important for us to make empirical studies on the real use of technology in teaching and learning (Sinclair et al., 2016). However, in China, a few studies are about teacher’s teaching practices with technology. So this research wants to fill in the gap. We take dynamic geometry software (DGS here and after) as an example to analyze how teacher integrates DGS into mathematics lessons and their opinions towards using DGS.
TEACHING PRACTICE WITH DYNAMIC GEOMETRY SOFTWARE
DGS is developed for visualization and for checking properties by dragging geometrical objects (Laborde & Laborde, 2008; Sinclair et al., 2016). Its main feature: dragging mode, a kind of direct manipulation towards the software, means the simultaneity between students’ action and DGS feedback (Sinclair et al., 2016) which can solve the gap between experimental and theoretical mathematics (Leung, 2003). This mode is mainly used to: 1, check the correctness of the supposed (known) geometrical properties in the figure; 2, to look for new geometry properties through the perception of what remains invariant when dragging; and 3, to check whether the construction preserves its geometrical properties when dragging (Healy, 2000; Laborde, 2001). But the use of DGS by teachers is often limited to the first modality, students are expected to drag figures to confirm empirically the properties (Hölzl, 2001). Because teachers may find that students’ activities with DGS may lead them to the production which is not their expectation (Olivero, 2002).
In China, Shang (2008) investigated how teachers integrated DGS in mathematics teaching. She observed 8 mathematics lessons and found DGS played mainly as a servant for teachers rather than a patterner. In these lessons, students manipulated DGS mainly for constructing different objects not for exploring mathematics knowledge. The author concluded that what teachers did is not integrating DGS in mathematics lessons but for replacing traditional teaching methods. This indicates that although researchers think DGS can change their teaching methods (Hu, 2005), promote the effects of mathematics teaching (Zhao et al., 2012), and let students explore mathematical knowledge by themselves (Fan, 2003), small amounts of teachers choose to let students manipulate DGS in the classrooms (Hu, 2005). And in these lessons, students have no chance to exchange their findings of special questions and even do not know whether what they have is right or not (Liu, 2009).
One of the reasons why teachers do not use DGS as researchers want is that teachers need to make a transformation between old and new didactical practices (Assude & Gelis, 2002; Lagrange et al., 2003). Another important reason is that teachers lack enough support for making educational use of DGS (de Castell et al., 2002; Norris et al., 2003). Teachers lack enough strategies to effectively use DGS or other technologies for learning mathematics (Niess et al., 2009). Like Kortenkamp et al. (2009) point out: “still, the adoption of DGS at school is often difficult. … Many teachers do not seem to know about the new possibilities……” (p. 1,150). Other factors, like the change of room location and physical layout, change of class organization and classroom procedures (Jenson & Rose, 2006), and the resources they need to use in the class such as mathematics tasks (Hegedus et al., 2017; Pierce & Stacey, 2013; Author, 2009) also affect teachers practice with DGS.
Thus, how to use DGS to create new opportunities for students’ learning process (Angeli & Valanides, 2009) is a big problem for teachers. Some of the researchers began to think about how to help teachers to teach with DGS, such as building the repository which includes DGS to support their didactic process (Trgalová & Jahn, 2013). In order to help teachers to integrate technology into classrooms, Trouche (2004) introduces instrumental orchestration to describe how teachers orchestrate classrooms with technology. In this study, we want to make sense of teaching practice with DGS and answer the following questions: How teachers organize these nine mathematics lessons with DGS? And what characters can be identified in these mathematics lessons with DGS?
Instrumental orchestration comes from instrumental approach which acknowledge the complexity of using technologies in mathematics teaching and learning (Artigue, 2002). This approach focus on a psychological construction process called instrumental genesis during which an artifact comes into instrument. During this process the user develops a scheme to use the instrument for a specific task (Drijvers et al., 2010). In this scheme, users’ technological knowledge about the artifact and specific knowledge about the domain like mathematics are intertwined. So that instrumental genesis is useful for us to analyze how to use technology in mathematics education.
There are many studies focus on the students’ instrumental genesis in learning mathematics with technology. Researchers believe that students’ instrumental genesis is guided by teachers. For example, teachers chose to use certain technological tools in order to guide students’ learning process(Kendal & Stacey, 2002). So Trouche (2004) introduced instrumental orchestration to describe how the teacher can guide students’ instruments genesis within the classroom.
Instrumental orchestration tries to answer questions about what technological artifacts mathematics educators should introduce to learners and what guidance they should provide so that learners can appropriate and use artifacts as instruments to mediate their activities with various artifacts (Drijvers et al., 2010; Trouche, 2004). it is defined as how teachers want to organize the class teaching with different kinds of artefacts available in learning environment in order to help students learn mathematics (Trouche, 2004). We can distinguish instrumental orchestration into three different elements: a didactic configuration, an exploitation mode and a didactical performance (Drijvers et al., 2010).
In this study, instrumental orchestration is used to describe how teachers organize classroom teaching with DGS. Instrumental orchestration tries to answer questions about what technological artifacts mathematics educators should introduce to learners and what guidance they should provide so that learners can appropriate and use artifacts as instruments to mediate their activities with various artifacts (Trouche, 2004). We can distinguish instrumental orchestration into three different dimensions:
A didactic configuration which means an arrangement of artefacts in the didactical environment, or in other words, a configuration of the teaching setting and the artefacts involved in it. Like in the musical metaphor of orchestration, the didactical configuration can be seen as choosing what musical instruments need to be included, and how to arrange them in space so that the different sounds result in polyphonic music, which in the mathematics classroom might come down to a sound and converging mathematical discourse.
An exploitation mode means the way teachers decide to exploit a didactical configuration for the benefit of his/her didactical intentions. This includes decisions on the way a task is introduced and worked through, on the possible roles to be played by the artefacts, and on the schemes and techniques to be developed and established by the students.
A didactical performance involves the ad hoc decisions taken while teaching on how to actually perform in the chosen didactic configuration and exploitation mode: what question to pose now, how to do justice to (or to set aside) any particular student input, how to deal with an unexpected aspect of the mathematical task or the technological tool, or other emerging goals.
According to Drijvers (2010, 2013), the following types can be used to describe different orchestration of classroom with technology in mathematics teaching: Technical-demo, Explain-the-screen, Guide-and-explain, Link-screen-board, Discuss-the-screen, Spot-and-show, Sherpa-at-work, Board-instruction (see Appendix). In this study, we would analyze Chinese mathematics lessons to check if all these types can be identified or if there are some new types or new characteristics can be found in Chinese mathematics lessons.