The teacher and the tool
Paul Drijvers
Freudenthal Institute
Utrecht University
The Netherlands
cadgme, 30-06-2010
www.fi.uu.nl/~pauld
The Freudenthal Institute
 Aims at developing, investigating and
improving mathematics education at
primary, secondary and tertiary level
 Aims at intertwining research, design, professional
development and implementation
 Is the Dutch National Expertise Centre of Mathematics
Education
 Key concepts:
• Realistic Mathematics Education
• Design Research
 www.fi.uu.nl/en/
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Outline
1. What’s the problem?
2. Theoretical framework
3. The DME environment
4. Results
• Four examplary clips
• Orchestration types and tables
• Teacher profiles
5. Conclusion
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c o n c l u s i o n
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What’s the problem?
 High expectations concerning the integration of
technology into mathematics education have not yet
come true.
 Growing awareness that teachers are crucial in the
integration of technology into mathematics education:
• Gueudet & Trouche (2009)
• Lagrange & Monaghan (2009)
• Lagrange & Ozdemir Erdogan (2009)
• Maracci & Mariotti (2009)
• Pierce & Ball (2009)
• Ruthven, Deaney & Hennessy (2009)
 Issue at stake: what do teachers do when they integrate
technology in their teaching and why do they do so?
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Theoretical framework
 Instrumental orchestration as a model for analysing
teacher activity
(Trouche, 2004; Drijvers & Trouche, 2008)
 An instrumental orchestration: the teacher’s intentional
and systematic organisation and use of the various
artefacts available in a learning environment in a given
mathematical task situation, in order to guide students’
instrumental genesis.
 An instrumental orchestration:
• a didactical configuration
• an exploitation mode
• a didactical performance
Didactical performance
Exploitation mode
Didactical configuration
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Didactical configuration
 A didactical configuration is an arrangement of artefacts
in the environment, a configuration of the teaching
setting and the artefacts involved in it.
 Cf. the musical metaphor:
• choosing the musical instruments to include in the
orchestra
• arranging them in space and time to achieve harmony
(= convergence of mathematical discourse).
Didactical performance
Exploitation mode
Didactical configuration
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Exploitation mode
 An exploitation mode is the way the teacher exploits the
didactical configuration for her didactical intentions:
decisions on the way a task is introduced and worked, on
the possible roles of the artefacts, and on the schemes
and techniques to be developed.
 Cf. the musical metaphor:
• determining the partition for each of the musical
instruments involved.
Didactical performance
Exploitation mode
Didactical configuration
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Didactical performance
 A didactical performance involves the ad hoc decisions
taken while teaching, on how to actually perform in the
chosen didactic configuration and exploitation mode:
what question to pose now, how to do justice to (or to set
aside) student input, how to deal with an unexpected
aspect of the task or the tool?
 Cf. the musical metaphor:
• the actual musical performance
Didactical performance
Exploitation mode
Didactical configuration
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The orchestration model ...
 has a time dimension:
ranging from teachers’ preparation in advance (didactical
configuration) to ad hoc decisions while teaching
(didactical performance)
 has a structural – incidental dimension:
ranging from teachers’ operational invariants to incidental
actualizations
 Includes two levels of instrumental genesis, namely the
students’ level and the teachers’ level, the latter being
the focus of this talk
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A symphony orchestra conductor?
Rather a jazz band leader...
Trouche, L. & Drijvers, P. (in press). Handheld technology: Flashback into the future.
ZDM, The International Journal on Mathematics Education.
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Questions
1. Which types of instrumental orchestration emerge in
technology-rich classroom teaching?
2. To what extent are teachers’ repertoires of orchestrations
related to their views on mathematics education and the
role of technology therein?
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Two studies
I. ‘Tool Use in an Innovative Learning Arrangement for
Mathematics’:
• NWO-granted small-scale design and evaluation of
an instructional sequence on the concept of
function in grade 8, using an applet embedded in
the electronic learning environment DME
• See www.fi.uu.nl/tooluse/en
II. ‘EPN-pilot Algebraic Skills through ICT’:
• Publisher initiated bigger-scale pilot on algebraic
skills in grade 12, using applets in the DME, and
replacing a textbook chapter
• See http://www.fi.uu.nl/dwo/gr-pilot/ and
http://www.epn.nl/wps/portal/epn/getalenruimte/ictpilot#vwo
Both studies use the Digital Mathematics Environment
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3. The DME environment
http://www.fi.uu.nl/dwo/en/
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The DME environment
 FI’s Digital Mathematics Environment DME:
• Content (applets, SCORM)
• Authoring environment (including design of feedback)
• Administration and monitoring system (LMS, CMS,
including log facilities)
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Examplary feedback
 Cf yesterday’s presentation by Christian Bokhove
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DME qualities
 DME scores good on feedback facilities
Bokhove, C. & Drijvers, P. (2010). Assessing assessment tools for algebra: Design
and application of an instrument for evaluating tools for digital assessment of
algebraic skills. International Journal of Computers for Mathematical Learning,
15(1). Online First.
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4.
Results

Examplary clips ABC (Study I) and D (Study II)
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Orchestration types and tables
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Teacher profiles ABCD
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Case Teacher A: grade 8
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Case Teacher A
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Case Teacher A [clip]
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Case Teacher A
 Orchestration type: Sherpa-at-work (?)
• Didactical configuration:
Whole-class setting, laptop computer with projector,
students in ‘regular’ rows
• Exploitation mode:
Have a pair of students explain their work, one talking
to the class and the other one operating the computer
• Didactical performance:
Teacher in ‘marginal’ position, but taking care of the
process of the student at the laptop supporting the
words of his peer
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Case Teacher B: grade 8
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Case Teacher B
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Case Teacher B [clip]
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Case Teacher B
 Orchestration type: Link-screen-board
• Didactical configuration:
Teacher computer + data projector, positioned next to
the ‘regular’ blackboard
• Exploitation mode:
Show students’ work on screen; use the blackboard to
transform applet notation and technique into
conventional paper-and-pencil equivalents
• Didactical performance:
Interactively use students’ input
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Case Teacher C: grade 8
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Case Teacher C [clip]
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Case Teacher C
 Orchestration type: Technical-demo
• Didactical configuration:
Teacher computer connected to the interactive
whiteboard. Students in ‘traditional setting’
• Exploitation mode:
Students listen and watch the teacher’s demonstration
of applet technique
• Didactical performance:
Teachers remains seated and does not use the
facilities of the IAW
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Case Teacher D: grade 12
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Case Teacher D
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Case Teacher D [clip]
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Case Teacher D
 Orchestration type: Work-and-walk-by
• Didactical configuration:
Individual netbook computers in a wifi network,
digital module including textbook files
• Exploitation mode:
Students work on their own, teacher assists and
answers questions
• Didactical performance:
Taking neighbour’s netbook to compare findings; both
technical and mathematical problems; student and
teacher on different tracks
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Orchestration types
 Seven orchestration types identified through theory-
driven (1,3,6) as well as bottom-up analysis (2,4,5,7):
1. Technical-demo
2. Explain-the-screen
3. Link-screen-board
4. Discuss-the-screen
5. Spot-and-show
6. Sherpa-at-work
7. Work-and-walk-by
 1-2-3 teacher-centred:
teacher dominates communication
 4-5-6-7 student-centred:
interactive communication, student input
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Orchestration table from study I
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Orchestration table from study II
Drijvers, P. (submitted). Teachers transforming resources into orchestrations. In
Gueudet, G., Pepin, B., & Trouche, L. (Eds.), Mathematics Curriculum Material and
Teacher Development: from text to lived resources? (pp. - ). New York/Berlin: Springer.
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Teacher profiles
Teacher A:
 prioritizes student-centred orchestrations
 sees learning as an interactive process in which students
should have a voice
 sees technology as a means to achieve this
”…so you can have discussions with the students using the
images that you saw on the screen, … that makes it more
lively”
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Teacher profiles
Teacher B:
 prioritizes teacher-centred orchestrations
 finds mathematical content to be paramount
 sees technology as a means to teach this
 attention to the links between the DME-work and paperand-pencil or blackboard mathematics.
”… take distance from the specific ICT-environment;
otherwise the experience remains too much linked to the
ICT ”
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Teacher profiles
Teacher C:
 prioritizes teacher-centred orchestrations
 finds clear explanations and instructions important
 has a concern for student difficulties when learning
mathematics, and when using technology
”As a teacher, one has to tell students clearly what they
should do with ICT ”
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Teacher profiles
Teacher D:
 prioritizes student-centred orchestrations
 leaves much initiative to the students
 does not see a technology-rich lesson as a specific one to
prepare
“Well, I don’t know much about it myself. I did not invest
time in preparation. […] I prefer the kids act, and raise
questions based on their actions.”
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Conclusion
1. Which types of instrumental orchestration emerge in
technology-rich classroom teaching?
Seven orchestration types are identified, as well as their
frequencies.
2. To what extent are teachers’ repertoires of orchestrations
related to their views on mathematics education and the
role of technology therein?
The four teacher profiles suggest a clear relationship
between orchestrations and views.
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Reflections
 How generalizable are the orchestration types, to other
topics, to other types of technology, to other forms of
teaching, to other teachers … ?
 Why such a preference for individual orchestrations in
Study II? Do teachers feel that they should step back as
soon as technology enters the classroom?
 What are possible implications for teachers’ professional
development concerning ‘TPACK’?
 What does the model of instrumental orchestration offer?
A fruitful framework for analysing teachers’ practices
when teaching mathematics with technological tools?
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Literature & Links
 Drijvers, P., Boon, P., Doorman, M., Van Gisbergen, S.,
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Gravemeijer, K., & Reed, H. (in press). The teacher and the
tool: whole-class teaching behavior in the technology-rich
mathematics classroom. Educational Studies in Mathematics.
Drijvers, P., & Trouche, L. (2008). From artifacts to
instruments: A theoretical framework behind the orchestra
metaphor. In G. W. Blume & M. K. Heid (Eds.), Research on
technology and the teaching and learning of mathematics: Vol.
2. Cases and perspectives (pp. 363-392). Charlotte, NC:
Information Age.
http://www.fi.uu.nl/dwo/gr-pilot/
http://www.fi.uu.nl/dwo/en/
http://www.fi.uu.nl/tooluse/en/
http://www.epn.nl/wps/portal/epn/getalenruimte/ictpilot#vwo
http://ec.europa.eu/dgs/education_culture/eve/alfresco/n/browse/work
space/SpacesStore/1fe0b2bc-d45d-11de-b89b-09f36ec598be
Thank you!
Paul Drijvers
[email protected]
www.fi.uu.nl/tooluse/en/