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
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