Transcript The design of a mixed media learning arrangement for the

An ICT-rich learning arrangement
for the concept of function in
grade 8: student perspective and
teacher perspective
Paul Drijvers
Freudenthal Institute for Science and
Mathematics Education
Utrecht University
Universität Köln, 20.01.09
www.fi.uu.nl
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
1 The project
 Project name: Tool Use in Innovative Learning
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Arrangements for Mathematics
Granted by the Netherlands Organisation for Scientific
Research NWO
Timeline: 2006 – 2008
Research team:
• Peter Boon, programmer / researcher
• Michiel Doorman, researcher
• Paul Drijvers, PI / researcher
• Sjef van Gisbergen, teacher / researcher
• Koeno Gravemeijer, supervisor
• Helen Reed, master student
www.fi.uu.nl/tooluse/en
Project theme: math & technology
 Integrating technology in mathematics education seems
promising
 But optimistic claims are not always realized!
 Technology for ‘drill & practice’ or also for conceptual
development?
 If yes, how to achieve this?
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 collections of applets see:
 www.fi.uu.nl/wisweb/en/
(primary)
 www.fi.uu.nl/rekenweb/en/
(secondary)
 So far: rather much design / development of games /
applets than research on their use in the classroom
Project concretisation
 Mathematical subject: the concept 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: 7-8 lessons of 50 minutes
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
2 The function concept
Two quotes:
 “The very origin of function is stating and producing
dependence (or connection) between variables occurring
in the physical, social, mental world (i.e. in and between
these worlds).”
(Freudenthal, 1982)
 “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)
Function definitions
 "a quantity composed in any of [a] variable and constant"
(Bernoulli, 1718)
 an "analytic expression" (Euler, 1747)
 f is a function from a set A to a set B if f is a subset of the
Cartesian product of A (the domain) and B (the range),
so that for each a in A there exists exactly one b in B with
(a, b) in f. (Dirichlet-Bourbaki, 1934)
How useful are these definitions for lower secondary
mathematics education?
The ‘function gap’
 Lower secondary level (SI, 13 – 15 year olds):
a way to describe a calculation process, an input-output
‘machine’ for numerical values.
 Upper secondary level (SII, 16 – 18 year olds):
a mathematical object, with several representational
faces, which one can consider as membre of a family, or
that can be submitted to a higher level procedure such as
differentiation.
Intentions and didactical ideas
Intentions:
 To bridge the gap between the two, facilitate the
transition and promote a rich conception of the notion of
function including both the process and the object view.
Relevant ideas from mathematics didactics:
 Vinner (1983), Vinner & Dreyfus (1989):
Concept definition and concept image
 Janvier (1987):
Multiple representations – formula, graph, table
 Sfard (1991): Process – object duality
 Malle (2000): Function as assignment and as co-variation
Proces-object duality (Sfard, 1991):
 Operational conception:
processes
 Structural conception:
objects
 In the process of concept
formation, operational
conceptions precede the
structural
Three aspects of the notion of function:
a. Dependency relation from input to output
b. Dynamical process of co-variation
c. Mathematical object with several representations
Mathematical phenomenology or didactical
phenomenology?
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
3 The ICT tools (1)
 Freudenthal (1983) mentions activities with arrow chains
as one means to approach the function concept
 ICT tool: The applet AlgebraPijlen (“AlgebraArrows”):
chains of operations, connected by arrows, with tables,
graphs and formulas.
3 The ICT tools (2)
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The Digital Mathematics
Environment (DME) :
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
a. The function as a dependency relation from input to
output: construct and use chains
The tools and the function concept
b. 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
c. 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 function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
4 Learning 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
Learning arrangement: lesson 1
 Group work on three central problems
Learning arrangement: lesson 2
 Posters, presentations and ‘living chains’
Learning arrangement: lesson 3
 First work in pairs with the applet after introduction
Learning arrangement: lesson 4
 Second work in pairs with the applet after plenary
homework review
Learning arrangement: lesson 5
 Group work on the ‘matching’ of representations
Learning arrangement: lesson 6
 Third applet session in pairs after plenary discussion
Learning arrangement: lesson 7 (+8)
 Final work with the applet and reflections on the concept
of function and its notation
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
5 Some results on learning
A. Difficulties to express the reasoning
B. Mixed media approach fruitful (paper-pencil <-> applet)
C. Form-function shift as a model for describing conceptual
change in ICT-rich learning
A Difficulties to express the reasoning
Students explaining dynamic covariation:
 “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 Mixed media approach fruitful
C Form-function shift
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Form-function shift as
a model for describing
conceptual change in
ICT-rich learning
Example: task 1.6
The work of two girls
 Their work ‘real time’: Atlas (clip 59:9)
 Their final product:
Hypothesis: form-function shift (1)
A form-function shift (Saxe, 1991) takes place concerning
the functions that arrow chains have for the student:
 Initially, the arrow chain represents a calculation process,
and is a means to calculate the output value once the input
value is given. The arrow chain helps to organize the
calculation process.
 Evidence: students make new chains for the same
calculation:
Hypothesis: form-function shift (2)
 Later, the arrow chains become object-like entities that
represent functional relationships and can be compared
and reasoned with.
Verfication of the hypothesis: Task 4.1
The work of the two girls
Results of three classes
Theoretical interest
 Form-function shift here might be a suitable construct to
explain conceptual change when there is little technical
development in the use of the ICT tool.
 Instrumental genesis, which was one of the points of
departure of this study, seems to be more appropriate for
more versatile technological tools.
Chain of operations
Table of independent
input values
Formula
Table of dependent
output values
Graphic representation
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
6 Some results on teaching
A. Different whole-class orchestrations
B. Relations with teachers’ views on teaching and learning
C. Interaction teacher – student
A Different whole-class orchestrations
Main orchestrations observed:
1. Technical demo
2. Explain the screen
3. Link screen board
4. Discuss the screen
5. Spot and show: example
6. Sherpa at work: example
Orchestrations by teacher
Orchestration type
TeacherA
cycle1
TeacherA
cycle2
TeacherB
cycle2
TeacherC
cycle3
TeacherA
cycle3
Total
Technical-demo
5
3
2
7
5
21
Explain-the-screen
0
0
0
7
1
8
Link-screen-board
3
0
6
0
3
12
Discuss-the-screen
4
4
3
1
2
13
Spot-and-show
0
1
12
2
2
19
Sherpa-at-work
2
7
0
0
1
10
Total
14
15
23
17
14
83
B Relations with teachers’ views
on teaching and learning
 Teacher A:
“…so you could discuss it with the students using the
images that you say on the screen, […] it makes it more
lively…”
 Teacher B:
“I use the board to take distance from the specific ICTenvironment, otherwise the experience remains too much
linked to the ICT”
 Teacher C:
“I am a typical teacher for mid-ability students, and these
students need clear demonstrations and explanations”
C Interaction teacher – student
Different types of interactions:
 Content of interaction:
• Mathematical meaning
• Technical meaning
• Situational meaning
• Interaction-meaning-technical
 Form of interaction:
• Revoicing
• Questioning
• Answering
Outline
1. The project
2. The function concept
3. The ICT tools
4. Learning arrangement
5. Some results on learning
6. Some results on teaching
7. Conclusion
7 Conclusion on learning
1. How can applets be integrated in
an instructional sequence for
algebra, so that their use fosters
the learning?
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Global learning trajectory works, but which problem
does the function concept solve for the students?
Mixed media approach fruitful Subtle relation
between applet technique and concept development
(instrumentation, FFS)
Form-function shift as a model for describing
conceptual change in ICT-rich learning
7 Conclusion on teaching
2. How can teachers orchestrate tool
use in the classroom community?
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Technical class management not self-evident!
Mixture of whole-class orchestrations, related to teachers’
views
Demonstration/presentation/class discussion important for
reflection and collective instrumental genesis
DME offers means to monitor the learning
The teacher important for orchestrating discussion /
reflection / convergence of techniques and thinking
7 Conclusion on theory
Theoretical questions:
 Is the framework of instrumental genesis, with its stress
on the relation between technical and conceptual
development, useful in case the tool is as ‘simple’ as an
applet?