Abstract Developing ways to use educational technology is not evident for mathematics ... According to this approach, the use of a technological tool involves a.
TEACHERS USING TECHNOLOGY: ORCHESTRATIONS AND PROFILES Paul Drijvers, Michiel Doorman, Peter Boon, Sjef van Gisbergen, & Helen Reed Freudenthal Institute, Utrecht University, The Netherlands Abstract Developing ways to use educational technology is not evident for mathematics teachers. With the theory of instrumental orchestration as a framework, we investigate the types of orchestrations three teachers use in a lesson series for grade 8 on the function concept, employing an applet embedded in a digital learning environment. The results show six types of orchestrations that differ in their ICT specificity. Interview data suggest that teachers’ preferences for types of orchestrations can be related to their views on mathematics learning and teaching.
1. INTRODUCTION It is generally acknowledged that the integration of technological tools into mathematics education is a complex issue. A theoretical framework that acknowledges this complexity is the instrumental approach (Artigue, 2002). According to this approach, the use of a technological tool involves a process of instrumental genesis, in which mental schemes and techniques for using the tool co-emerge. Many studies focus on students’ instrumental genesis and its possible benefits for learning (e.g., Kieran & Drijvers, 2006). Students’ instrumental geneses need to be guided, monitored and orchestrated by the teacher. To describe this, Trouche (2004) introduced the metaphorical notion of instrumental orchestration. Until today, however, the number of elaborated examples of instrumental orchestrations described in the research literature is limited (Drijvers & Trouche, 2008; Drijvers et al., in press; Trouche, 2004). Therefore, we wanted to investigate the different types of orchestrations teachers use, as well as the relationships between these types of orchestrations and teachers’ views on mathematical learning and teaching. We address these issues from meta-perspective, i.e. without detailed reference to specific mathematical, didactical and technological aspects. For a comprehensive analysis of these issues, the reader is referred to Doorman et al. (in press).
2. THEORETICAL FRAMEWORK The notion of instrumental orchestration (Trouche, 2004) addresses the question of how the teacher can fine-tune the students’ individual instruments and compose coherent sets of instruments, thus enhancing both individual and collective instrumental genesis. An instrumental orchestration is defined as the teacher’s intentional and systematic organisation and use of the various artefacts available in a – in our case computerised – learning environment in a given mathematical task situation, in order to guide students’ instrumental genesis. An instrumental orchestration consists of three elements: a didactic configuration, an exploitation mode and a didactical performance. The first two elements are described by Trouche (2004). As an instrumental orchestration is partially prepared beforehand and partially created ‘on the spot’ while teaching, we felt the need to add a third component, reflecting the actual performance. As such, the threefold model has a more explicit time dimension. A didactical configuration is an arrangement of artefacts in the environment, or, in other words, a configuration of the teaching setting and the artefacts involved in it. These artefacts can be technological tools, but the tasks students work on are important artefacts as well. In the musical metaphor of orchestration, setting up the didactical configuration can be compared with choosing musical instruments to be included in the orchestra, and arranging them in space so that the different sounds result in the most beautiful harmony. An exploitation mode of a didactical configuration is the way the teacher decides to exploit it for the benefit of her didactical intentions. This includes decisions on the way a task is introduced and worked, on the possible roles of the artefacts to be played, and on the schemes and techniques to be developed and established by the students. Decisions on the exploitation mode can be seen as part of the design of a Hypothetical Learning Trajectory (Simon, 1995). In the musical metaphor of orchestration, setting up the exploitation mode can be compared with determining the partition for each of the musical instruments involved, bearing in mind the anticipated harmonies to emerge. 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
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to set aside) any particular student input, how to deal with an unexpected aspect of the mathematical task or the technological tool? In the musical metaphor of orchestration, the didactical performance can be compared with a musical performance, in which the actual inspiration and the interplay between conductor and musicians reveal the feasibility of the intentions and the success of their realization. The instrumental orchestration model brings about a double-layered view on instrumental genesis. At the first level, instrumental orchestration aims at enhancing the students’ instrumental genesis. At the second level, the orchestration is instrumented by artefacts for the teachers, which may not necessarily be the same artefacts as the students use. As such, the teacher herself is also engaged in a process of instrumental genesis for accomplishing her teaching tasks (Bueno-Ravel & Gueudet, 2007), which include the development of operational invariants: the implicit knowledge contained in the schemes that is believed to be true.
3. METHODS To investigate the different types of orchestrations teachers use, we analysed video tapes of 38 50-minute lessons taught by three experienced female mathematics teachers. Their grade 8 classes participated in an innovative technology-rich learning arrangement 1 for the concept of function (Drijvers et al., 2007). The learning arrangement, which was developed over three research cycles, came with a teacher guide for the teachers including a planning scheme. The main technological artefact was an applet called AlgebraArrows embedded in an electronic learning environment called Digital Mathematics Environment (DME). The applet allows for the construction and use of chains of operations, and options for creating tables, graphs and formulae and for scrolling and tracing. The DME allows the student to access the work in any location, and the teacher to access student work to check progress, and to monitor the learning process. Data analysis focused in particular on whole-class episodes in which technology was used. The unit of analysis is an episode, which concerns the whole-class treatment of one task. If this treatment consists of different orchestration types, the episode was cut into sub-episodes. This way, a corpus of 83 episodes was identified. These data were organized and analysed with the help of software for qualitative data analysis 2 . The 1 2
For more information on the project see http://www.fi.uu.nl/tooluse/en/ . We use Atlas ti software, www.atlasti.com
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analysis combined a deductive, theory-driven approach with an inductive, bottom-up analysis. Six orchestration types were identified and the corpus was coded according to this categorization. A second coding of 29% of the episodes led to a good inter-rater-reliability (Cohen’s kappa) of .72. To investigate the relationships between these types of orchestrations and teachers’ views on mathematical learning and teaching, the researchers drew up a profile for each of the three teachers’ orchestrations, possible operational invariants and views on mathematical learning and teaching. These profiles were validated through semi-structured post-experiment interviews with the three teachers concerned.
4. RESULTS Different types of orchestrations The data analysis, partially based on a priori, theory-driven codes, led to the definition of six orchestration types: Technical-demo, Explain-the-screen, Link-screen-board, Discuss-the screen, Spot-and-show, and Sherpa-at-work. The first three are predominantly teacher-centred, whereas the last three are more interactive. The Technical-demo orchestration concerns the demonstration of techniques by the teacher. A didactical configuration for this orchestration includes access to the applet and the DME, facilities for projecting the computer screen, and a classroom arrangement that allows the students to follow the demonstration. As exploitation modes, we observed teachers treating a technique in a new situation or task, as well as adding new techniques to students’ work, anticipating what will follow. The Explain-the-screen orchestration concerns whole-class explanation by the teacher, guided by what happens on the screen. The explanation goes beyond techniques, and involves mathematical content. Didactical configurations can be similar to the Technical-demo ones. As exploitation modes, teachers sometimes took student work as a point of departure for the explanation, or started with a solution suggested by the teacher herself. In the Link-screen-board orchestration, the teacher stresses the relationship between what happens in the technological environment and how this is represented in conventional mathematics of paper, book and school board. In addition to DME access and projecting facilities, a didactical configuration includes a school board and a classroom setting such that both screen and board are visible. Similarly to the previously mentioned orchestration types,
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teachers’ exploitation modes take student work as a point of departure or start with a task or problem situation set by the teacher herself. The Discuss-the-screen orchestration concerns a whole-class discussion on what happens on the screen. The goal is to enhance collective instrumental genesis. A didactical configuration includes DME access and projecting facilities, preferably access to student work, and a classroom setting favourable for discussion. As exploitation modes, we once more see student work as point of departure, and having students react to it, as well as teachers setting a task, problem or approach as input for discussion. In the Spot-and-show orchestration, student reasoning is brought afore through the identification of interesting digital student work during preparation of the lesson, and its deliberate use in classroom discussion. Besides previously mentioned features, a didactical configuration includes access to the DME during the preparation of the teaching. As exploitation modes, teachers may have the students whose work is shown explain their reasoning, and ask other students for reactions, or herself provide feedback to the student work. In the Sherpa-at-work orchestration the student uses the technology, either to present his/her work, or to carry out operations the teacher requests. Didactical configurations are similar to the Discuss-the-screen orchestration type. The classroom setting should be such that one student can easily manage the technology, with both Sherpa-student and teacher easy to follow by all students. As exploitation modes, teachers may have student work presented or explained by the student using the technology, or may pose questions to the Sherpa student and ask him/her to carry out specific operations in the technological environment. Orchestration type Technical-demo Explain-the-screen Link-screen-board Discuss-the-screen Spot-and-show Sherpa-at-work Totaal
TeacherA TeacherA TeacherB TeacherC TeacherA cycle 1 cycle 2 cycle 2 cycle 3 cycle 3 5 3 2 7 5 0 0 0 7 1 3 0 6 0 3 4 4 3 1 2 0 1 12 2 2 2 7 0 0 1 14 15 23 17 14
Total 22 8 12 14 17 10 83
Table 1: Orchestration type frequencies by teachers and research cycles Table 1 shows the frequencies of each of the orchestrations for each of the teachers and, for teacher A, for each of the three times she taught the lesson series. Technical-demo and Spot-and-show are the most frequently used 5
orchestration types. Concerning the former, the post intervention interviews suggest that the three teachers felt the need to familiarize students with basic techniques, in order to prevent technical obstacles from hindering the mathematical activities. The Spot-and-show orchestration is also quite frequent, but is unevenly spread over the three teachers, even if they all appreciated the opportunity to browse through the digital student work while preparing the next lesson. This brings us to the issue of differences between teachers, and the underlying views they have on the learning and teaching of mathematics. Orchestrations reflecting teachers’ views Table 1 shows that Teacher A has high frequencies for Technical-demo, for Discuss-the-screen and for Sherpa-at-work. She spends most time on the three interactive orchestration types. In the post-intervention interview, she confirmed that she finds interaction in the classroom very important, and sees ICT as a means to stimulate this: “…so you could discuss it with the students using the images that you say on the screen, […] it makes it more lively…”
Teacher A mentioned time constraints as the main reason not to use Sherpaat-work in the final cycle. Technical constraints (e.g. slow internet connections during the first cycle or inappropriate classroom settings in the computer lab) also drive her choices for orchestrations. Teacher B has high frequencies for Link-screen-board, and particularly for Spot-and-show. She said she used the board to “take distance from the specific ICT-environment, otherwise the experience remains too much linked to the ICT”
Establishing the links between the ICT-work and paper-and-pencil mathematics is important to her, as she sees the use of ICT as a means to achieve her mathematical teaching goals. She appreciates Spot-and-show for discussing students’ common mistakes or original approaches, as she considers this as fruitful for learning. Teacher C has high frequencies for Technical-demo and Explain-the-screen, which are more teacher-centered orchestration types. Two reasons seem to explain her preferences. She described herself as a “typical teacher for midability students” who strongly believes that these students benefit from clear demonstrations and explanations. Furthermore, she wanted to be in control of what is happening in the classroom, and believed that more teachercentered orchestrations would support this. 6
5. CONCLUSION AND DISCUSSION One of the goals of this study was to investigate the different types of orchestrations teachers use and to extend the repertoire of exemplary orchestrations. The analysis revealed six orchestration types, which are very different in their specificity. Discuss-the-screen and Explain-the-screen can be seen as ICT-variants of regular teaching practices most teachers are familiar with. More specific for the use of ICT are Technical-demo, Sherpaat-work and Link-screen-board. Specific for ICT-tools that provide the teacher with online consultancy of digital student work is the Spot-and-show orchestration. We conclude that the repertoire of orchestrations is diverse, and that the orchestrations have different degrees of ICT-specificity. Technological and time constraints may influence the choice and exploitation of the orchestrations. A second goal was to investigate the relationships between these types of orchestrations and teachers’ views on mathematical learning and teaching. The matching of observed orchestrations and interview data reveals clear links between the three teachers’ preferences for orchestrations and their ideas on what is important to achieve during the teaching. We conclude that these three teachers’ choices for orchestrations and their exploitation are strongly related to their views on mathematics learning and teaching. While considering these conclusions, we should note that the three teachers, who participated on a voluntary basis, were confronted with a complex and innovative learning arrangement, which they were not familiar with. Dealing with that for the first time is not easy, and it is possible that a lack of familiarity partially guided their orchestration choices. Also, the time schedule for the learning arrangement was very tight, which as well may have affected the didactical performance and the decisions involved. At the end of section 2, we mentioned the instrumental genesis that teachers themselves are involved in. As developed schemes include operational invariants, the question arises as to what the operational invariants of these teachers are. Although the data are too limited to fully answer this question, it suggests that these operational invariants are at least partially determined by the teachers’ views on mathematics learning and teaching. A more finegrained analysis of the orchestrations, which goes beyond the categorization we presented here, is needed to shed more light on this. A next enterprise to be undertaken!
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ACKNOWLEDGEMENT We thank the teachers and students of the experimental sites for their collaboration, as well as the Netherlands Organisation for Scientific Research (NWO) for supporting the research study (project no 411-04-123).
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