The design of a mixed media learning arrangement for the

Report
The design of a mixed media
learning arrangement for the
concept of function in grade 8
Paul Drijvers
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
1 The research project: fact sheet
 “Tool Use in Innovative Learning Arrangements




for Mathematics“
Granted by the Netherlands Organisation for
Scientific Research
Timeline: 2006 – 2008
Research team:
• Peter Boon, programmer / researcher
0.2 fte
• Michiel Doorman, researcher
0.4 fte
• Paul Drijvers, PI / researcher
0.4 fte
• Sjef van Gisbergen, teacher / researcher 0.2 fte
• Koeno Gravemeijer, supervisor
PM
www.fi.uu.nl/nl/tooluse
Research Questions
1.
How can applets be integrated in an
instructional sequence for algebra, so
that their use fosters the learning?
2.
How can teachers orchestrate tool use
in the classroom community?
Applets
For the design products see:
 www.fi.uu.nl/wisweb/en/

(primary)
www.fi.uu.nl/rekenweb/en/
(secondary)
 So far: a lot of design / development of games /
applets than research on their use
Theoretical framework
 Realistic Mathematics Education
 Vygotsky and others on tool use
 Instrumental approach
to tool use
 Cultural-historical
activity theory
Research setup and methodology
 Design research (Gravemeijer): design of an
educational arrangement including
• Student worksheets (paper)
• Applet with tasks embedded in an electronic learning
environment
• Teacher guide with scenario’s of use
 Three cycles:
• Pilot
• First cycle
• Second cycle
One class, 2006
Three classes, 2007
Scaling up, 2008
 Data: both qualitative (video, screenvideo) and
quantitive (student work saved in ELE)
The project: Concretisation
 Mathematical subject: the notion of fonction
 Tools: an applet embedded in an electronic
learning environment
 Target group: mid – high achieving students in
grade 8 (14 year olds)
 Teaching sequence: 7 lessons of 50 minutes
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
2 The concept of function
 The function:
• Lower secondary level (13 – 15 year olds): a way to
describe a calculation process, a relation, a ‘machine’
• Upper secondary level (16 – 18 year olds): a
mathematical object, with several faces, which one can
consider as membre of a family, or that can be
submitted to a higher level procedure such as
differentiation.
 Intention: bridge the gap between the two,
facilitate the transition and promote a rich
conception of the notion of function.
2 The concept of function
 The function is a special kind of dependence,
that is, between variables which are
distinguished as dependent and independent.
(...) This - old fashioned - definition stresses
the phenomenologically important element: the
directedness from something that varies freely
to something that varies under constraint.
(Freudenthal, 1983, p. 496).
2 The concept of function
Three aspects of the notion of function:
 A directed relation from input to output
 A dynamical process of co-variation
 A mathematical object with several
representations
Mathematical phenomenology, but how
about the didactical phenomenology?
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
3 The tools
 The applet AlgebraPijlen (“AlgebraArrows”):
chains of operations, connected by arrows, with
tables, graphs and formulas.
3 The tools
 The Digital Mathematics
Environment (EDM) :
• Author: design tasks and
activities, ‘Digital textbook’
• Student: work, look back,
improve, continue, ‘Digital
worksheet’
• Teacher: prepare, comment,
assess, ‘Collection of digital
worksheets’
• Researcher: observe, analyse
the digital results,
‘Digital database’
The tools and the function concept
 The function as a directed relation from input to
output: construct and use chains
The tools and the function concept
 The function as a dynamical process of co-
variation: change input values to study the
effect, use trace (graph) and scroll (input/table)
The tools and the function concept
• The function as a mathematical object with
several representations: compose chains,
construct inverse chains, link representations
and study families of functions
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
4 The educational arrangement
Main ideas:
 Mixture of working formats: group work,
individual work, work in pairs with the
computer, plenary teaching and discussion
 Mixture of tools: paper – pencil, posters, cards,
applet, DME, both in school and at home
 First step: a hypothetical learning trajectory
The educational arrangement: lesson 1
 Group work on three central problems
The educational arrangement: lesson 2
 Posters, presentations and ‘living chains’
The educational arrangement: lesson 3
 First work in pairs with the applet after introduction
The educational arrangement: lesson 4
 Second work in pairs with the applet after plenary
homework review
The educational arrangement: lesson 5
 Group work on the ‘matching’ of representations
The educational arrangement: lesson 6
 Third applet session in pairs after plenary discussion
The educational arrangement: lesson 7
 Final work with the applet and reflections on the
concept of function and its notation
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
5 The data and analysis
1. Data on the use of applets and learning:
• Videos of the group work du travail en groupe
• Screen video of pairs of students
• Students’ work (digital, paper, posters)
2. Data on the use of applets and teaching:
• Videos of plenaries in the classroom
• Videos of student - teacher interactions
 Data analysis ongoing
Digital student work
Coding in Atlas ti
CODES
13 15 16 17
---------------------------------------docent-huishoudelijk 10 11 11
7 39
docent-inhoudelijkeh
5 10
8
8 31
docent-proceshulp
17
8
3
3 31
docent-reflectiehulp
4 10 10
6 30
docent-technischehul
8
4
3
4 19
functie-dynamiek-cov
5
1
2
4 12
functie-globale-in-u
0
6
7
7 20
functie-locale-in-ui
3
2
2
1
8
functie-object-repre
1
3
7
2 13
functie-variabele
2
4
4
2 12
techniek-in-uitzoome
3
5
5
1 14
techniek-ketting-mak
3
5
2
0 10
techniek-ketting-rec
1
0
0
0
1
techniek-scrollen-ta
1
3
1
3
8
Some prelimenary results
A. Difficulties to phrase the reasoning
B. Interaction paper-pencil / applet fruitful
C. Some lasting conceptual difficulties
D. Interaction teacher – student crucial
A Difficulties to phrase the reasoning
 Goes up sidewards
 Straigt line
 Further and further away





from 0
All equally steep
With the same jumps
The point is always
moving
It goes up steeper and
steeper
It gets higher and higher
B Interaction paper-pencil / applet fruitful
C Some lasting conceptual difficulties
• The work of Juliette, who insists on representing
a function with several chains.
 Atlas
 Online
D Interaction teacher – student crucial
 Either in individual contact and in plenary
discussions
 The teacher plays an important role while
orchestrating the discussion on, the reflection
on, and the convergence of techniques and
thinking within the class
Outline
1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
6 Provisional conclusions
1. How can applets be integrated in
an instructional sequence for
algebra, so that their use fosters
the learning?
 Global learning trajectory OK, but which


problem does the function concept solve for
the students?
Dynamic did not work -> change applet
More insight needed into the relation
between applet technique and concept
development
6 Provisional conclusions
2. How can teachers orchestrate tool





use in the classroom community?
Technical class management time
consuming and not self-evident!
Mixture of scenario’s and media
(ICT and P&P) led to a fruitful
arrangement that fosters transfer
between media
Demonstration/presentation/class
discussion important for reflection and
collective instrumental genesis
DME offers means to monitor the learning
Teacher interactions that focus both on
concept and technique are effective.
6 Provisional conclusions
Theoretical questions:
 Is the framework of instrumental genesis, with
its stress on the relation between technical and
conceptual development, is useful in case the
tool is as ‘simple’ as an applet?
 How to observe and to analyse the
orchestration by the teacher?

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