1. Mathematics, pedagogy and ICT David Wright

2. How ICT helps learners learn mathematics (National Council for Educational Technology (NCET) 1995) Learn from feedback Observe patterns See connections Work with dynamic images Explore data ‘Teach’ the computer

3. Learning from Feedback Effective feedback allows:  A context of exploration  A contingent claim to knowledge (it’s ok to be wrong) through  non-judgemental and impartial messages  possibility of privacy

4. Observing patterns The rapid production of results generates opportunities for:  explanation  justification  proof Develops skills of enquiry and communication

5. Seeing connections Multiple, dynamic representations (for example, linking formulae, tables of numbers and graphs) give opportunities for:  students making their own understandings  gives the pupil a sense of power and control over the mathematics

6. Working with Dynamic Images Manipulation of diagrams dynamically:  Unlocks the power of visual imagery  gives a sense of authorship to the pupil  fosters a sense of confidence in one’s ability to visualise mathematics and hence to think mathematically

7. Exploring data ICT allows pupils access to real data and its representation, interpretation and modelling  ownership of this process enhances the pupils’ sense of authority  access to different models encourages reflection and critique about models used in other situations  access to multiple representations encourages reflection and critique of other representations

8. “Teaching the computer” In order to make a computer achieve a result pupils must express themselves unambiguously and in correct order  They make their thinking explicit as they refine their ideas Pupils are able to pursue their own goals  they develop a sense of authorship and personal authority

9. Evolution of ICT in education Type 1 The learner and the computer Type 2 The learner, the teacher and the computer Type 3 (emphasising the ‘C’ in ICT) Mathematics Learner Mathematics Learner Teacher Classroom Learner Teacher Mathematics

10. Curricular specificityLow High Mathematical Expressivity Narrow Broad Spreadsheets Computer Algebra System Dynamic Geometry Graphical Calculator Autograph Microworlds Johnston- Wilder and Pimm (2005)

11. Current research Small software on handheld technology networks  With Pam Woolner and teachers at St Thomas More High School, North Shields

12. Equipment Two class sets of TI84+ calculators have been supplied to the school. One class have been given personal ownership, the other set is used by the department with a range of classes. The school has also been supplied with a range of software, including the TI Smartview emulator and a range of small software programs for the calculators. TI has supplied its Navigator system which will allow the calculators to be networked wirelessly with the teacher’s computer and the projector.

13. Small software

14. ‘Navigator’ network

15. Research focus: A socio-cultural analysis of the integration of ICT into the mathematics classroom An analysis of the mathematical meaning of the GC as an instrument in relation to a problem-solving task

16. Two theoretical frameworks Valsiner’s zone theory (Valsiner, Goos) Instrumentation theory (Verillon & Rabardel, Artigue, Trouche & Guin)

17. Valsiner’s zone theory ZFM – zone of free movement ZPA – zone of promoted action ZPD – zone of proximal development

18. The zones ZFM – environmental constraints  Resources  Access to learners  Technical support ZPA – activities which promote new skills and understanding ZPD – the possibilities for learning

19. Instrumentation theory (Guin & Trouche)

20. The instrumented activity system model (Verillon & Rabardel) Instruments emerge through a dialectical interplay between the technical demands of mastering a device and the conceptual work of making that device meaningful in the context of a task (Artigue, 2002) Instrumentalisation Instrumentation Utilisation scheme (theorems-in-action)

21. Instrumentation theory “Instrumental genesis thus makes artifacts meaningful in the context of activity, and provides a means by which users make meaning of that activity” (White, 2008) An instrument is more than object/artifact – it is a psychological construct consisting of a dialectical process of:  Instrumentalisation – Oriented towards the artifact – this is the process by which an artifact becomes the means of achieving an objective, solving a problem etc.  Instrumentation- Oriented towards the user - the user develops the schemes and techniques through which the artifact can be implemented in purposive action. “Instrumentation is precisely the process by which the artifact prints its mark on the subject …” (Trouche, 2004)

22. Utilisation schemes Comprise both the rules and heuristics for applying an artifact to a task and the understanding of the task in the form of ‘theorems in action’. “Theorems-in-action take shape as the domain-specific propositions on which learners rely as they interpret the capabilities [affordances] and constraints of a tool in relation to the features of a problem- solving task” (White, 2008) Hence the possibility of research focused on ‘theorems-in-action’ as a mechanism for linking the learner’s instrumented activity with learning goals and curricular content.

23. References Artigue, M (2002) Learning mathematics in a CAS environment: The genesis of a reflection about Instrumentation and the dialectics between technical and conceptual work International Journal of Computers for Mathematical Learning 7:245-247 Goos, M (2005) A sociocultural analysis of the development of pre-service and beginning teachers’ pedagogical identities as users of technology Journal of Mathematics Teacher Education 8:35-59 Guin,D and Trouche,L (1999) The complex process of converting tools into mathematical instruments: The case of calculators International Journal of Computers for Mathematical Learning 3: 195-227 Johnston-Wilder,S and Pimm,D (2005) Teaching Secondary Mathematics with ICT. Maidenhead:Open University Press National Council for Educational Technology (1995) Mathematics and IT: A pupil’s entitlement NCET Coventry Trouche,L (2004) Managing the complexity of human/machine interactions in computerised learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning 9(3): 281-307 Valsiner,J (1997). Culture and the development of children’s action: A theory of human development. (2 nd Ed) New York: John Wiley and Sons Verillon,P and Rabardel,P. (1995) Cognition and artifact: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology in Education, 9(3): 77 – 101 White,T (2008) Debugging an Artifact, Instrumenting a Bug: Dialectics of Instrumentation and Design in Technology-Rich Learning Environments International Journal of Computers for Mathematical Learning 13:1-26