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1
Lived resources and mathematics teachers
professional development
  • Ghislaine Gueudet
  • (IUFM de Bretagne-UBO, CREAD)

2
Outline
  • Digital resources for mathematics teachers
  • A documentational approach
  • Teachers documentation a case study
  • Collaborative lesson design and professional
    development
  • Conclusion

3
Digital resources for mathematics teachers
  • A context of generalised availability of digital
    resources
  • Software
  • Digital textbooks
  • Online exercises
  • Forum
  • E-mail
  • Interactive Whiteboards
  • ...
  • Official / personal websites

4
Digital resources for mathematics teachers
  • Development of new forms of collaboration, the
    GeoGebra community a world-wide network,
    communication between users, sharing resources
    supporting the use of GeoGebra...

5
Digital resources for mathematics teachers
  • Intergeo European research project about
    interactive geometry. A platform offering
    resources, tested and evaluated by users. A
    central questioning about the quality of
    resources (Soury-Lavergne et al. 2010).

6
Digital resources for mathematics teachers
  • Online teacher associations
  • In France, Sesamath develops free online
    resources
  • Online exercises, lesson plans, a dynamic
    geometry environment (Tracenpoche), digital
    textbooks...

7
Digital resources for mathematics teachers
  • The issue of technology integration
  • A gap between the institutional expectations and
    the actual use of software a gap between the
    extensive use of digital means, of some online
    resources, and a sparse use of software
  • Development of new, holistic perspectives on
    technology integration, considering the
    structuring features of the classroom context
    (Ruthven 2007)
  • New aspects of digital resources
  • Design and use are strongly intertwined
  • Teachers do not work alone, they interact with
    various groups with teachers of the same
    school, with distant teachers on a forum, with
    their students ...
  • Evolutions, and new perspectives on old facts,
    requiring new approaches.

8
A documentational approach, origins
  • a conceptualisation of resources as anything
    re-sourcing the teachers practice (Adler 2000)
    material and socio-cultural resources
  • studies of curriculum material focusing on the
    interactions between the teacher and the
    curriculum material, central in the professional
    development (Remillard 2005)

9
A documentational approach, origins
  • Research about students working with technology,
    the instrumental approach (Rabardel 1995, Guin
    et al. 2005)
  • An artefact an outcome of human activity,
    designed for a specific aim.
  • An instrument developed by a subject from the
    artefact, in a goal-oriented activity.
  • Instrument artefact scheme of utilisation
  • Scheme a cognitive construct, comprising rules
    of action and structured by operational
    invariants
  • Extension of the approach development of a
    documentational approach (Gueudet Trouche 2009)

10
A documentational approach
  • Documentational genesis
  • a teacher develops a document from a set of
    resources
  • the document associates resources, and a scheme
    of utilization, in particular professional
    knowkledge
  • a double instrumentalization/instrumentation
    movement the teacher shapes the resources, and
    the resources frame the teachers choices and
    knowledge

11
A documentational approach
  • Geneses develop across different contexts for
    the same objective they are ongoing processes a
    given document yields resources that can be
    engaged in further documentation work.
  • Teachers develop coherent and structured
    resources systems and documentation systems
  • Documentational geneses are central in teachers
    professional development

12
Teacher documentation, a case study
  • Reflexive investigation (Gueudet Trouche 2010)
    a methodology to follow teacher documentation.
  • Long-term follow-up, collection of resources
  • Following the teacher in-class and out-of-class
  • The teacher as partner in the research logbook,
    representation of the resource system (SRRS)...

13
Teacher documentation, a case study
  • (A focus on evolutions, and on digital resources)
  • Myriam (aged 51), teaching grade 6 to 9. Followed
    in 2008-2009 and 2009-2010.
  • Material resources
  • Non digital textbooks, overhead projector,
    professional papers, students productions...
  • Digital laptop, software, online exercises,
    websites, e-mail...
  • Involvement in collectives
  • Local reflection group, In-service teacher
    training
  • Personal environment
  • Discussions with her husband (physics teacher)
    about the articulation between maths and physics
  • Her daughter in grade 10 works with the
    calculator, with GeoGebra

14
Myriam Schematic Representation of the Resource
System
15
Myriam, distant work with the students
  • Myriams grade 9 students are equiped with
    laptops her school has a virtual learning
    environment.
  • In 2009-2010, the school is blocked by the snow
    during two weeks. Myriam uses a mailing list of
    the class to send homework an exercise about
    functions, using a spreadsheet to build a
    graphic. (Instrumentalisation)
  • Some students send back a spreadsheet file
    others copy the graph in a word processing,
    Myriam can not see how it has been built.
  • She decides to go on sending distant homework,
    but retains to precise carefully the form of the
    work to send.
  • (development of teacher knowledge,
    instrumentation)

16
Myriam, introduction of functions
  • Where on AB should be M to obtain Perimeter
    (CNMP)9?
  • Objective introducing p(x)x6
  • Use of a diagram (Laborde et al. 2001),
    displaying the length AM and the perimeter. The
    teacher pilots the computer, the students
    observe.
  • The students formulate the conjecture, but can
    not find the proof...

17
Myriam, introduction of functions
  • Introduction of functions a new theme in the
    grade 9 curriculum (2008-2009)
  • First uses of GeoGebra other colleagues in the
    reflection group use it her daughters math
    teacher uses GeoGebra
  • Myriam professional knowledge, developed along
    her professional practice, and her use of a DGE
    (Geoplan) The students must be actively
    involved in the building of new knowledge
    Using a DGE diagram supports the formulation of
    conjectures by students guides her choices for
    this new topic and software.
  • The proof is too difficult the students consider
    that the function is not necessary...
  • Myriam decides to propose a more concrete
    situation the following year building
    rectangular boxes with maximum volume.
  • Development of new knowledge observing the
    joint evolution of two measures is not enough to
    justify the introduction of functions...

18
Individual and collective documentation
  • Teachers belong to many 'collectives' (Gueudet
    Trouche 2009), where documentation work takes
    place
  • Under specific conditions, communities of
    practice (Wenger 1998) emerge in these
    collectives.
  • A community of practice develops a resource
    system (development of the community and
    development of the resource system are
    simultaneous).
  • A complex articulation between individual and
    collective geneses.

19
Teachers collective work and professional
development
Teachers  groups  and in-service training
(Krainer Wood 2008) teams (purposely
designed), communities (self-selected), networks
(informal). Possible evolutions, from teams to
communities, from networks to communities (Wenger
1998)? Inquiry communities, gathering teachers
and didacticians (Jaworski 2004, 2008 Fuglestad
2007). Professional development, sustainable
evolutions Use of networking possibilities, to
develop distant collective work (Goos Bennison
2008, Borba Gadanidis 2008 ), to permit the
up-scaling of training programs (Cobb Smith
2008)
20
Collaborative lesson design and professional
development
Pairform_at_nce, a French national project set up by
the Ministry of Education
  • All disciplinary fields, primary and secondary
    school
  • Integration of ICT following the German project
    Intel Lehren
  • Design of training paths, providing the structure
    of training device to be carried out across the
    country
  • These training device are blended, using a
    distant platform they are grounded in
    collaborative lessons design.

21
Collaborative lesson design and professional
development
A research and development project (Gueudet et
al. 2009, project coordinated by Luc Trouche at
INRP) Designing training paths and investigating
the collective documentation work - of training
path designers - of trainers - of
trainees. Focus on the trainees, use of elements
of the 'reflexive investigation' methodology
(logbooks, collection of resources, classroom
videos, interviews) complemented by elements
collected on the platform teachers discussions,
successive versions of files...
22
Example of a training path inquiry in
mathematics with a DGE
Seven stages
23
Example of a training path inquiry in
mathematics with a DGE
  • Seven stages (like all the Pairform_at_nce paths)
    introduction, choice of a theme, self and
    co-training, design of the lesson, test of the
    lesson, reflection on the lesson, evaluation of
    the training.
  • A training over thirteen weeks, with three
    face-to-face days introduction, constitution of
    teams, work on the software (day 1) discussion
    on inquiry in maths, preparation of the lesson
    (day 2) presentation and discussion of the
    lessons (day 3).
  • Teams with 4 teachers 2 in one school, 2 in
    another. The lesson is tested and observed at
    least one time.
  • Resources on the platform lessons examples
    (studied in presence), description grid,
    observation grid, software guides, research
    articles
  • Communication via the platform forums, folders

24
Example of a training path inquiry in
mathematics with a DGE
25
A black-box example, reaction of the trainees
  • Find the point R on D (the river) such that PRRG
    is minimal.
  • (The solution is given by the point L)
  • A black box example, presented to foster the
    debate between trainees
  • A trainee  Maths are not physics! When I
    want them to investigate in geometry, I want
    deductions. 

26
Lessons designed by the trainees
  • The 9 teams (in 2008-2009) designed lessons where
    the students worked on the DGE.
  • For 2 teams the diagram has been built by the
    teacher, the students drag and observe
  • For 5 teams the diagram must at least be
    completed
  • For 2 teams, the diagram intervenes in the
    proof.
  • A shared operational invariant  the dragging
    of a dynamic diagram is helpful to formulate
    conjectures .
  • The teams appreciate the cross observation, and
    the observation grid provided by the trainers.

27
Lessons designed by the trainees
Example of a lesson functions and
optimizationGilda Lauren, teachers in the same
school, a lesson for grade 9
  • A lifeguard uses a rope and two buoys (B and C)
    to form the boundary of a swimming zone. He forms
    this way a rectangular zone. The length of the
    rope is 160 m 16 dam. He wonders where to place
    the buoys B and C to obtain a swimming zone with
    the largest possible area. The point A is fixed.

28
Lessons designed by the trainees
  • A paper-and-pencil work
  • A geometrical modelling with GeoGebra, the area
    of a rectangle ABCD
  • A geometrical modelling with GeoGebra the trace
    of a point M with coordinates (AB, area(ABCD))
  • A proof on paper

29
Gilda and Lauren, documentation work and teacher
knowledge
The students responsibility with the computer,
with the mathematics, stay limited. Professional
knowledge intervening in the teachers choices
(intrumentalisation)  connecting different
dynamic representations supports the learning of
the concept of functions by the
students . Professional knowledge developed
during the training (beginning of a genesis) 
 observing the dynamic evolution of values,
while dragging a figure on a DGE helps to
conjecture properties of the figure measures 
30
Conclusion
  • From a new context of generalized availability of
    online resources to a new perspective on the
    interactions between teachers and resources
  • - The features of the resources influence
    teachers work (instrumentation), but teachers are
    not passive resources users, aligning with the
    designers intentions.
  • - Teachers develop their own documentation work,
    they are the designers of their teaching
    (instrumentalisation).
  • Along these interactions, teachers develop new
    professional knowledge (geneses).
  • Collective lessons design a productive mode of
    in-service teacher training, likely to contribute
    to sustainable evolutions.
  • Online material to support lessons design
    methodological assistance (Gueudet et al. 2008)

31
Conclusion
Mathematics Curriculum Material and Teacher
Development from text to lived resources G.
Gueudet, B. Pepin L. Trouche (eds.) Springer
2011 Mathematics Education Library Table of
contents http//educmath.inrp.fr/Educmath/recherch
e/approche_documentaire/lived-resources
Need for further theoretical and methodological
developments, in the documentational perspective,
an ongoing work
32
References
Adler, J. (2000). Conceptualising resources as a
theme for teacher education. Journal of
Mathematics Teacher Education, 3(3),
205224. Borba, M.C., Gadanidis, G. (2008).
Virtual communities and networks of practising
mathematics teachers, in K., Krainer T. Wood,
T. (Eds.) Participants in Mathematics Teachers
Education Individuals, Teams, Communities and
Networks (Vol. 3, pp. 181-206). Rotterdam/Taipei
Sense Publishers. Cobb, C., Confrey, J., diSessa,
A., Lehrer, R., Schauble, L. (2003). Design
Experiments in Educational Research, Educational
Researcher, 32-1, 9-13. Fugelstad, A.B. (2007).
Developing tasks and teaching with ICT in
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Pitta-Pantazi G. Philippou (Eds.), Proceedings
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References
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