Transcript The design of a mixed media learning arrangement for the

Research and practice on tool
use for mathematics in
secondary education
Christian Bokhove
19/9/08
Inhoud
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Introduction
From a classroom perspective
From a research perspective
Conclusion
Questions/ discussion
Feel free to ask questions during the presentation!
Introduction
Context
 Teacher mathematics and computer studies,
also ICT coordinator
 St. Michael College, a secondary school in
Zaandam, near de Zaanse Schans
 A lot of “tool” experience in the classroom
(classroom perspective)
 Now doing research on the use of tools for
acquiring, practising and assessing algebraic
skills (research perspective)
• In the Dudoc programme
• Programme for “allowing” research in secondary school
classroom.
Practice
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Wisweb project
Galois project
Sage project
ACTIVEMATH-EU
Many projects rooted in practice. “These seem to
work, these don’t”
But….these are not scientifically grounded
Theory
 A lot of research in the field
 Yielding many tools
 Case studies often indicate successes
But….how do these tools do in actual classroom
situations
From a classroom perspective
…but putting it in the context of
research
Several projects
 Wisweb project
Aimed at creating applets for various topics in
the dutch curriculum
• Students enthousiastic, some applets “seem to work”
• But…no coherence
 WELP project
aimed at grouping applets together for a
“leerlijn” (core curriculum)
• But…no integrated environment
Link to wisweb
From there on: from project to project…
Galois project
 Geïntegreerde Algebraïsche Leer Omgeving In
School
 Aims
• Inventarisation of tools / “state of the art”
• Using tools in the classroom
• AUthoring
• SCORM
• And more…
 Tools we “dabbled” in:
Link to Galois project
Maple TA
 Approached CAN diensten
 In the USA Maplesoft was released
 I wanted to see whether it could supplement
our math education
 Students and teachers are quick to give their
opinion. A valuable source for the developer or
researcher.
Link to maple TA 4
STACK
 Classroom experiment: accessing the server crashed it
 Stability and performance was deemed most important of
all points (research)
 Hard to install, access to server
 But a novel approach to feedback and multi-step
exercises.
Link to STACK
(Invitation: on october 9th I have organized a lecture by
Chris Sangwin at the Freudenthal Institute)
The practice of computer aided assessment of mathematics
raises some fundamental questions. What do to the
conventions of traditional mathematical notation mean?
How can we enter a mathematical expression into a
machine in a simple but unambiguous way? If we
generate random problems in mathematics how to we
identify and preserve their "essence"? When assessing
problems, what mathematical properties are we trying to
establish and which can we establish at a technical level?
Having established properties of a student's answer, what
outcomes should we assign, and how do we generate
feedback which is helpful? The STACK project proposes
answers to some of these questions in limited contexts.
Webwork
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Wanted to use this at our school
Moodle integration
But…very hard to install
Now I reverted to the server of the maintainers
Link to WebWork
Activemath
 Classroom experiment with wiskunde D material on
differentiation
 Group of about 15 students worked with AM on the St.
Michael College server
 One paragraph on the product rule
 Use of registration
Link to Activemath site
This…
Becomes…
…
Plus other sources (animation)
applet
Outside AM for registration by Moodle
Wiris
 Integrated in our Virtual Learning Environment
 Input editor and CAS
 Demo in our
VLE moodle
But…we as teachers want to be able to edit course
material
Sage project
 Reinvested prize money from Galois project
 Aimed at improving the Scorm Applet Generator
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(See Peters talk)
And providing 20 SCORM packages (creative
commons) with algebra.
But also improving Digital Mathematical
Environment (developer Peter Boon)
 Demo of material
 Demo of student material (together with
Activemath material)
II. From a research perspective
Problem statement
 Bridging the gap Secondary-Higher education
Algebraic skills ▼ ?
Entry exams e.d.
(NKBW, 2007; Tempelaar, 2007; Vos, 2007)
 Tool use ▲
Vision of cTWO (2007):
(Commission future mathematics education)
“Use to learn” vs. “Learn to use”
Position statement NCTM (2008)
Research question
 In what way can ICT be used for acquiring,
practising and assessing algebraic skills?
 This involves three key topics
• 1. ICT tools
• Acquiring, practising and assessing > 2. Assessment
• 3. Algebraïc skills
1 ICT tools
 Anthropological approach
(Lagrange, 1999)
Task
Technology
Theory
 Instrumental approach
(Drijvers, 2007)
tool + mental scheme = instrument
 Focus on (online) software
(ICT) Tool use Examples
10th january 2008
26
2. Assessment
 Practising and testing...
 Assessment for learning
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(Formative)
More qualitative
and
Assessment of learning
(Summative)
More quantitative/performance
(Black and Wiliam, 1998)
Feedback: several types
(Hints!!)
Assessment framework
3. Algebraic skills
 Basic skills
 Symbol sense (Arcavi, 1994)
 Assessment framework > pyramid
Simple example
 Equation
(2 x  3)(x  3)  (3x  6)(2 x  3)
 Students could do this:
(conditioned to expand)
x 2  3x  9  6 x 2  21x  18
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etc…
5x 2  24x  27  0
Or the student “sees”
(Gestalt):
2x  3  0
or
But…are practice
and related?
x  3  3x  6
2 x  9
 Advocates of insightful learning are often
accused of being soft on training. Rather
than against training, my objection to drill
is that it endangers retention of insight.
There is, however a way of training —
including memorisation — where every
little step adds something to the treasure
of insight: training integrated with
insightful learning.
Freudenthal, H. (1991). Revisiting
mathematics education.
Design research
 Conceptual framework, grounded in existing
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research
Design cycles
Aimed at making an intervention, but also
improving the body of research on a certain
topic (e.g. here tool use, symbol sense etc.)
Qualitative analysis (first and second cycle),
what feedback, misconceptions etc.
Quantitative (second and third), including highstakes performance
Combining research and practice
 Use research to give a theoretical basis for, for
example:
• Criteria on which to rate tools. 27 criteria in 4
categories. Examples
 Stability. See example STACK
 Authoring. Hardcoding as in Webwork or
Wysiwyg
 Modes/strategies. (based on literature on
formative assessment and feedback)
 Process/Answer.
• Leading to “Tool choice”
• Focus of the research
Conclusion
 All tools have their particular strengths and
weaknesses
 Practice and theory should go hand in hand
 Sharing knowledge and more collaboration
between seondary education, higher education
and researchers could create a win-win
situation.
Questions and discussion
Selected references
Arcavi, A. (1994). Symbol Sense: Informal Sense-Making in Formal
Mathematics. For the Learning of Mathematics, 14(3), 24-35.
Black, P., & Wiliam, D. (1998). Assessment and classroom
learning. Assessment in Education: Principles, Policy & Practice,
5(1), 7-73.
cTWO. (2007). Rijk aan betekenis, visie op vernieuwend
wiskundeonderwijs.
Drijvers, P. (2007). Instrument, orkest en dirigent: een
theoretisch kader voor ICT-gebruik in het wiskundeonderwijs.
Pedagogische Studiën, 84(5), 358-374.
Lagrange, J.-B. (1999). Complex calculators in the classroom:
Theoretical and practical reflections on teaching pre-calculus.
International Journal of Computers for Mathematical Learning,
4, 51-81.
NKBW. (2007). Eindrapport Nationale Kennisbank
Basisvaardigheden Wiskunde.
Tempelaar, D. (2007). Onderwijzen of bijspijkeren? Nieuw
Archief voor Wiskunde, 8, 55-59.
Vos, P. (2007). Algebra-prestaties van tweedeklassers. Euclides,
82, 129-132.