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Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations

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Abstract

Digital resources offer opportunities to improve mathematics teaching and learning, but meanwhile may question teachers’ practices. This process of changing teaching practices is challenging for teachers who are not familiar with digital resources. The issue, therefore, is what teaching practices such so-called ‘mid-adopting’ mathematics teachers develop in their teaching with digital resources, and what skills and knowledge they need for this. To address this question, a theoretical framework including notions of instrumental orchestration and the TPACK model for teachers’ technological pedagogical content knowledge underpins the setting-up of a project with twelve mathematics teachers, novice in the field of integrating technology in teaching. Technology-rich teaching resources are provided, as well as support through face-to-face group meetings and virtual communication. Data include lesson observations and questionnaires. The results include a taxonomy of orchestrations, an inventory of skills and knowledge needed, and an overview of the relationships between them. During the project, teachers do change their orchestrations and acquire skills. On a theoretical level, the articulation of the instrumental orchestration model and the TPACK model seems promising.

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Notes

  1. See http://www.fi.uu.nl/dwo/en.

  2. We used Atlas ti; see http://www.atlasti.com.

References

  • Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224.

    Article  Google Scholar 

  • Bokhove, C., & Drijvers, P. (2010). Digital tools for algebra education: Criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45–62.

    Article  Google Scholar 

  • Boon, P. (2009). A designer speaks: Designing educational software for 3D geometry. Educational Designer, 1(2). http://www.educationaldesigner.org/ed/volume1/issue2/article7 (Accessed 31 July 2013).

  • Borba, M., & Llinares, S. (Eds.) (2012). Online mathematics education. Special issue. ZDM—The International Journal on Mathematics Education, 44(6).

  • Brewer, M. B. (2000). Research design and issues of validity. In H. T. Reis & C. M. Judd (Eds.), Handbook of research methods in social and personality psychology (pp. 3–16). Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Cox, S., & Graham, C. R. (2009). Diagramming TPACK in practice: using an elaborated model of the TPACK framework to analyze and depict teacher knowledge. TechTrends, 53(5), 60–69.

    Article  Google Scholar 

  • Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41, 143–163.

    Article  Google Scholar 

  • Doorman, M., Drijvers, P., Gravemeijer, K., Boon, P., & Reed, H. (2012). Tool use and the development of the function concept: from repeated calculations to functional thinking. International Journal of Science and Mathematics Education, 10(6), 1243–1267.

    Article  Google Scholar 

  • Drijvers, P. (2012). Teachers transforming resources into orchestrations. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources: mathematics curriculum materials and teacher development (pp. 265–281). New York/Berlin: Springer.

    Google Scholar 

  • Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool; instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75(2), 213–234.

    Article  Google Scholar 

  • Drijvers, P., Tacoma, S., Besamusca, A., Van den Heuvel, C., Doorman, M., & Boon, P. (2013). Digital technology and mid-adopting teachers’ professional development: a case study. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), The mathematics teacher in the digital era. New York/Berlin: Springer (in press).

  • Even, R., & Ball, D. L. (Eds.) (2009). The professional education and development of teachers of mathematics. The 15th ICMI Study. New ICMI Study Series, Vol. 11. New York/Berlin: Springer.

  • Graham, C. R. (2011). Theoretical considerations for understanding technological pedagogical content knowledge (TPACK). Computers & Education, 57, 1953–1960.

    Article  Google Scholar 

  • Gueudet, G., Pepin, B., & Trouche, L. (Eds.). (2012). From text to ‘lived’ resources: Mathematics curriculum materials and teacher development. New York/Berlin: Springer.

    Google Scholar 

  • Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71, 199–218.

    Article  Google Scholar 

  • Gueudet, G., & Trouche, L. (2011). Mathematics teacher education advanced methods: an example in dynamic geometry. ZDM—The International Journal on Mathematics Education, 43(3), 399–411.

    Article  Google Scholar 

  • Gueudet, G., & Trouche, L. (2012). Communities, documents and professional genesis: interrelated stories. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 305–322). New York/Berlin: Springer.

    Chapter  Google Scholar 

  • Jaworski, B. (2007). Learning communities in mathematics: research and development in mathematics teaching and learning. In C. Bergsten, B. Grevholm, H.S. Masoval & F. Ronning (Eds.), In relating practice and research in mathematics education. Proceedings of Norma05, Fourth Nordic Conference on Mathematics Education (pp. 71–96). Trondheim: Tapir Akademisk Forlag.

  • Koehler, M. J., Mishra, P., & Yahya, K. (2007). Tracing the development of teacher knowledge in a design seminar: integrating content, pedagogy and technology. Computers & Education, 49, 740–762.

    Article  Google Scholar 

  • Lagrange, J.-B., & Ozdemir Erdogan, E. (2009). Teachers’ emergent goals in spreadsheet-based lessons: analyzing the complexity of technology integration. Educational Studies in Mathematics, 71(1), 65–84.

    Article  Google Scholar 

  • Llinares, S., & Valls, J. (2009). The building of pre-service primary teachers’ knowledge of mathematics teaching: interaction and online video case studies. Instructional Science, 37(3), 247–271.

    Article  Google Scholar 

  • Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: a framework for teacher knowledge. Teachers College Record, 108(6), 1017–1054.

    Article  Google Scholar 

  • National Council of Teachers of Mathematics (2008). The role of technology in the teaching and learning of mathematics. http://www.nctm.org/about/content.aspx?id=14233 (Accessed 31 July 2013).

  • Reed, H., Drijvers, P., & Kirschner, P. (2010). Effects of attitudes and behaviours on learning mathematics with computer tools. Computers & Education, 55(1), 1–15.

    Google Scholar 

  • Remillard, J. T. (2005). Examining key concepts of research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.

    Article  Google Scholar 

  • Roesken, B. (2011). Mathematics teacher professional development. In B. Roesken (Ed.), Hidden dimensions in the professional development of mathematics teachers (pp. 1–28). Rotterdam: Sense Publishers.

    Chapter  Google Scholar 

  • Ruthven, K. (2007). Teachers, technologies and the structures of schooling. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the V Congress of the European Society for Research in Mathematics Education CERME5 (pp. 52–67). Larnaca, Cyprus: University of Cyprus.

    Google Scholar 

  • Ruthven, K. (2013). Frameworks for analysing the expertise that underpins successful integration of digital technologies into everyday teaching practice. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), The mathematics teacher in the digital era. New York/Berlin: Springer (in press).

  • Sabra, H., & Trouche, L. (2013). Designing digital resources in communities of practice: a way to develop mathematics teachers’ knowledge. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), The mathematics teacher in the digital era. New York/Berlin: Springer (in press).

  • Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

    Article  Google Scholar 

  • Tabach, M. (2011). A mathematics teacher’s practice in a technological environment: a case study analysis using two complementary theories. Technology, Knowledge and Learning, 16, 247–265.

    Google Scholar 

  • Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments: guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307.

    Article  Google Scholar 

  • Voogt, J., Fisser, P., Pareja Roblin, N., Tondeur, J., & Van Braak, J. (2013). Technological pedagogical content knowledge—a review of the literature. Journal of Computer Assisted Learning, 29(2), 109–121.

    Article  Google Scholar 

  • Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. New York: Cambridge University Press.

    Book  Google Scholar 

Download references

Acknowledgments

We thank the participating teachers and their students for the collaboration, and Kennisnet for supporting the research study (Project No. C/OZK/2131).

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Authors and Affiliations

  1. Freudenthal Institute for Science and Mathematics Education, Utrecht University, Utrecht, The Netherlands

    Paul Drijvers, Sietske Tacoma, Amy Besamusca, Michiel Doorman & Peter Boon

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  1. Paul Drijvers
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  2. Sietske Tacoma
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  3. Amy Besamusca
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  4. Michiel Doorman
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  5. Peter Boon
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Corresponding author

Correspondence to Paul Drijvers.

Appendix: Orchestration overview and descriptions

Appendix: Orchestration overview and descriptions

Figure 5 provides an overview of whole-class and individual orchestrations identified in this study, as well as the correspondences between the two. The orchestrations are described below.

Fig. 5
figure 5

Overview of whole-class and individual orchestrations

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1.1 Whole-class orchestrations

Based on Drijvers et al. (2010), the following whole-class orchestrations are identified:

  • The Technical-demo orchestration concerns the demonstration of tool techniques by the teacher, which is recognized as an important aspect of technology-rich teaching. A didactical configuration for this orchestration includes access to the technology, facilities for projecting the computer screen, and a classroom arrangement that allows the students to follow the demonstration. As exploitation modes, teachers can demonstrate a technique in a new situation or task, or use student work to show new techniques in anticipation of what will follow.

  • The Guide-and-explain orchestration shares with Explain-the-screen and Discuss-the-screen a didactical configuration of access to the technology and projecting facilities, preferably access to student work, and a classroom setting favourable for students to follow the explanation. The exploitation mode, however, holds the middle between Explain-the-screen and Discuss-the-screen: on the one hand, a somewhat closed explanation based on what is on the screen is provided by the teacher. On the other hand there are some, often closed, questions for students, but this interaction is so limited and guided that it cannot be considered as an open discussion.

  • In the Link-screen-board orchestration, the teacher stresses the relationship between what happens in the technological environment and how this is represented in the conventional mathematics of paper, book and board. In addition to access to the technology and projection facilities, the didactical configuration includes a board and a classroom setting so that both screen and board are visible. The teachers’ exploitation modes may take student work as a point of departure or start with a task or problem situation they set themselves.

  • The Discuss-the-screen orchestration concerns a whole-class discussion about what happens on the computer screen. The goal is to enhance collective instrumental genesis. A didactical configuration once more includes access to the technology and projecting facilities, preferably access to student work, and a classroom setting favourable for discussion. As exploitation modes, student work, a task, or a problem or approach set by the teacher can serve as the point of departure for student reactions.

  • The Explain-the-screen orchestration concerns whole-class explanation by the teacher, guided by what happens on the computer screen. The explanation goes beyond techniques, and involves mathematical content. Didactical configurations can be similar to the Technical-demo ones. As exploitation modes, teachers may take student work as a point of departure for the explanation, or start with their own solution for a task.

  • In the Spot-and-show orchestration, student reasoning is brought to the fore through the identification of interesting student work during preparation of the lesson, and its deliberate use in a classroom discussion. Besides previously mentioned features, a didactical configuration includes access to the students’ work in the technological environment during lesson preparation. As exploitation modes, teachers may have the students whose work is shown explain their reasoning, and ask other students for reactions, or may provide feedback on the student work.

  • In the Sherpa-at-work orchestration, a so-called Sherpa student (Trouche 2004) uses the technology to present his or her work, or to carry out actions the teacher requests. A didactical configuration includes access to the technology and projecting facilities, preferably access to student work, and a classroom setting favourable for interaction. The classroom setting should be such that the Sherpa student can be in control of using the technology, with all students able to follow the actions of both Sherpa student and teacher easily. As exploitation modes, teachers may have work presented or explained by the Sherpa student, or may pose questions to the Sherpa student and ask him/her to carry out specific actions in the technological environment.

  • The Board-instruction orchestration is the traditional one of a teacher in whole-class teaching in front of the board. The board can be a chalk board, a whiteboard or an interactive whiteboard, but in any case it is just used for writing. No connections are made to the use of digital technology. The didactical configuration is the classical one of the teacher in front of the class working with the board. Different exploitation modes are possible, with different degrees of student involvement and interaction; however, no use of or reference to digital technology is made. We added this orchestration as we felt the need to also include the regular teaching in our analysis.

1.2 Individual orchestrations

The individual orchestrations all share the didactical configuration, that is, the students sitting individually or in pairs in front of their technological devices that provide access to their online work, and the teacher walking by in the classroom, but do differ in exploitation modes. Within this setting, the following individual orchestrations are identified and if appropriate named according to corresponding whole-class orchestrations:

  • In the individual Technical-demo orchestration, the didactical configuration is exploited for the individual demonstration of techniques for using the digital content by the teacher. The goal is to avoid obstacles that emerge from the student’s technical inexperience in using the digital environment.

  • The exploitation of the individual Guide-and-explain orchestration involves an individual exchange between teacher and (a pair of) student(s) in which the teacher takes the position of the instructor through providing guidance and instruction to the student, explains mathematical concepts or methods based on what happens on the screen, or raises questions to make the student reflect on his actions and results.

  • In the student–teacher interaction that characterizes the Link-screen-paper orchestration, the didactical configuration is exploited by the teacher for connecting the representations and techniques encountered in the digital environment and their conventional paper-and-pencil and textbook counterparts. The goal is to link the mathematics on the screen and the mathematics of the regular paper-and-pencil. As an extra requirement for the didactical configuration, the setting should allow switching between screen, notebook and textbook. This is not self-evident in often (too) full computer labs.

  • In the individual Discuss-the-screen orchestration, the phenomena on the screen lead to a discussion between teacher and student(s). This discussion may start by a question from the student, or by a remark made by the teacher. The goal of the discussion may not be clear beforehand and the student has considerable impact on the direction and the content of the talk, for example by expressing his difficulties.

  • In the individual Technical-support orchestration, in which technical issues play a central role, the teacher supports the student in technical problems that go beyond the DME technology, such as login difficulties, software bugs or hardware issues.

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Drijvers, P., Tacoma, S., Besamusca, A. et al. Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations. ZDM Mathematics Education 45, 987–1001 (2013). https://doi.org/10.1007/s11858-013-0535-1

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