Mathematics, pedagogy and ICT

David Wright

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

Learning from Feedback

Effective feedback allows:A context of explorationA contingent claim to knowledge (it’s ok to be

wrong) through

non-judgemental and impartial messagespossibility of privacy

Observing patterns

The rapid production of results generates opportunities for:explanation justificationproof

Develops skills of enquiry and communication

Seeing connections

Multiple, dynamic representations (for example, linking formulae, tables of numbers and graphs) give opportunities for:students making their own understandingsgives the pupil a sense of power and control

over the mathematics

Working with Dynamic Images

Manipulation of diagrams dynamically:Unlocks the power of visual imagerygives a sense of authorship to the pupil fosters a sense of confidence in one’s ability

to visualise mathematics and hence to think mathematically

Exploring dataICT allows pupils access to real data and its

representation, interpretation and modellingownership of this process enhances the pupils’

sense of authorityaccess to different models encourages reflection

and critique about models used in other situations

access to multiple representations encourages reflection and critique of other representations

“Teaching the computer”

In order to make a computer achieve a result pupils must express themselves unambiguously and in correct orderThey 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

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

LearnerTeacher

Mathematics

Curricular specificityLow High

Mathematical

Expressivity

Narrow

Broad

Spreadsheets

Computer Algebra System

Dynamic Geometry

Graphical Calculator

Autograph

Microworlds

Johnston- Wilder and Pimm (2005)

Current research

Small software on handheld technology networksWith Pam Woolner and teachers at St

Thomas More High School, North Shields

EquipmentTwo 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.

Small software

‘Navigator’ network

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

Two theoretical frameworks

Valsiner’s zone theory (Valsiner, Goos)

Instrumentation theory (Verillon & Rabardel, Artigue, Trouche & Guin)

Valsiner’s zone theory

The zones

ZFM – environmental constraintsResourcesAccess to learnersTechnical support

ZPA – activities which promote new skills and understanding

ZPD – the possibilities for learning

Instrumentation theory (Guin & Trouche)

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)

InstrumentalisationInstrumentalisation

InstrumentationInstrumentationUtilisation scheme (theorems-in-action)Utilisation scheme (theorems-in-action)

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)

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.

ReferencesArtigue, 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. (2nd 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