‘Lived resources’ and mathematics teachers
professional development
Ghislaine Gueudet
(IUFM de Bretagne-UBO, CREAD)
Salamanque, September 2010
Outline
• Digital resources for mathematics teachers
• A documentational approach
• Teacher’s documentation: a case study
• Collaborative lesson design and professional
development
• Conclusion
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
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...
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).
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...
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.
A documentational approach, origins
- a conceptualisation of resources as anything resourcing the teacher’s 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)
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 goaloriented 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)
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 teacher’s choices
and knowledge
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
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)...
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
Myriam:
Schematic Representation of the Resource System
Myriam, distant work with the students
Myriam’s 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)
Myriam, introduction of functions
Where on [AB] should be M to obtain Perimeter (CNMP)=9?
Objective: introducing p(x)=x+6
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...
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
daughter’s 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”...
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.
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)
Collaborative lesson design and professional
development
Pairform@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.
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...
Example of a training path: inquiry in mathematics with a DGE
Seven stages
Example of a training path: inquiry in mathematics with a DGE
• Seven stages (like all the Pairform@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…
Example of a training path: inquiry in mathematics with a DGE
A black-box example, reaction of the trainees
Find the point R on D (the river) such that PR+RG 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. »
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.
Lessons designed by the trainees
Example of a lesson: functions and optimization
Gilda & 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.
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
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 »
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)
Conclusion
Need for further theoretical
and methodological
developments, in the
documentational perspective,
an ongoing work
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/recherche/approche_d
ocumentaire/lived-resources
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