mcsquared project communities of interest production in cycle 1
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MCSquared (http://mc2-project.eu) is funded by the European Commission under FP7 (Project no. 610467), Strategic Objective ICT-2013.8.1 “Technologies and scientific foundations in the field of creativity”
Mathematical Creativity Squared
Nov 6, 2014Utrecht, Netherlands (hosted by Utrecht University)
Review meeting
Ch. Mercat, WP6, UCBL
Evaluation: Cycle 1 and cycle 2 productions processes
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2
CoIs, c-books and conceptions
- The different Communities of Interest (Coi)
- The design process and Cycle 1 products (c-book units)
- Cross-case analysis and Cycle 2
- Detailed presentation of each Coi and a selected c-book unit
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3
Four different Coi's and many CoP's
Common domains:- Math education- Research in Math education- Math teacher training
in & pre-service- Software designers
math, education, games
- Design, art, culture, edition,creative industry
UK – Spain – Greece – France
But also- Environmental, Engineering,Computer Science education- Research in online teacher
communication- Research on Creativity
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 4
Cycle 1 productions CMT driven designCollaborative design process – Coi Conceptions
Each partner surveyed conceptions about Social Creativity and Creative Mathematical Thinking among CoI members.It boiled down to CMT potential evaluation criteria that led the design process.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 5
Unexpected Ideas Dimitri, when you used the word baker in your files, I imagined a windmill worker and unexpected ideas came to mind.
I attach them without having thought of them really well...In the 1st file the student observes the windmill’s motion and knowing that in three turns it produced two sacks of flour, builds the function and draws it.
In the 2nd file the mill works in a different rate each time and the student will need to discover a function with three branches, its formula and create its graphical representation to explain why the mill didn’t have the expected production.
In the 3rd and 4th file I put there trigonometry. In 5th rate of change
All the files are available and if you like them, we can discuss about them, you may evolve them and modify them so as to use them well didactically
Popi_2.zip [5 Geogebra files] (attachment)
Cycle 1 productions – Social Creativity
CoiCode and critical episodes
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 6
Cycle 1 productions – c-book units
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 7
Cross-case analysis & cycle 2
Alien c-book units:- evaluate CMT potential - reconstruct
UK – France & Spain – Greece
New c-books units:- Version 2 of Cycle 1 - Completely new ones
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 8
The “Eva’s windmills” c-book unit:Design rationale – cycle 1
• Theme: Windmill• Math topics: 2d shapes, cartesian coordinates,
rate of change, multi-branch functions, proportional amounts, trigonometric functions, rotational symmetry
• CoI CMT conceptions: “mathematics in use”• Audience: “low threshold” – “high ceiling”
students up to 11th grade• Widgets used: o DME widgets, o Geogebra and o E-Slate Turteworlds (half-baked microworlds)
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 9
Greek CoI
Maria Daskolia, CTI
Evaluation: Cycle 1 and cycle 2 productions processes
The “Eva’s windmills” c-book unit
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 1010
Profiles of the Greek Coi members involved in the design of the c-book units
1. Researchers in Mathematics Education
2. Teacher trainers
3. Technology developers
4. National InitiativeTrainees
6. Researchers in Math Teacher
Education
5. Master Course students – Master
Holders
7. Researcher in Environmental
education
8. Phd Holders/Students
9. Researcher in Vocational Education
10. Researchers in On-line teacher communication
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 11
“Eva’s windmills”: Design rationale – cycle 1
• Theme: Windmills
• Math topics: 2d shapes, angles, cartesian coordinates, rate of change, multi-branch functions, proportional amounts, trigonometric functions, rotational symmetry
• CoI CMT conceptions: “Mathematics in use”
• Audience: “low threshold” – “high ceiling” students up to 11th grade
• 12 widget instances from: o DME widgets, o Geogebra and o E-Slate Turteworlds (half-baked microworlds)
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 12
The “Eva’s windmills” c-book unit:Turtleworlds
• Eva’s adventures in a Greek island…..
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 13
“Eva’s windmills”: Some Data
• CoI members: 10 members, 9 communities of practice
• Duration: 4 March – mid June
• CoIcode map: 1 (and 1 more for tech issues)
• Contributions: 115 in 11 trees
• Attachments: 24 (widget instances, ppts, docs, images, links)
• Overall: 14 pages - 12 widgets instances from 3 different widgets/widget factories
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 14
CoICoDe map cycle 1:
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 15
CoICoDe map cycle 1:Unexpected Ideas Dimitri, when you used the word baker in your files, I imagined a windmill worker and unexpected ideas came to mind.
I attach them without having thought of them really well...In the 1st file the student observes the windmill’s motion and knowing that in three turns it produced two sacks of flour, builds the function and draws it.
In the 2nd file the mill works in a different rate each time and the student will need to discover a function with three branches, its formula and create its graphical representation to explain why the mill didn’t have the expected production.
In the 3rd and 4th file I put there trigonometryIn 5th rate of change
All the files are available and if you like them, we can discuss about them, you may evolve them and modify them so as to use them well didactically
Popi_2.zip [5 Geogebra files] (attachment)
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 16
CoICoDe map cycle 1:
DATE AUTHOR POST RESOURCES/VERSIONING
28 Mar14
Dimitris Geogebra try out (discussion initiator) The sails (attachment)
R: Foteini’s attached image, Elissavet’s post
V: Dimitris’ first cbook link – geobebra instance included
30 Mar14
Popi 1st idea with Geogebra1.ggb (attachment)
R: Elissavet’s postV: Popi’s first Geogebra widget
instance30
Mar14Dimitris Building on Popi’s
microworld2. Rar (2 ggb attachments)
R: Giannis’ Ideas, Popi’s initial widget instance
V: 2nd widget instance by Dimitris (+ possible students’ constructions)
30 Mar14
Popi Unexpected Ideas Popi_2.zip (5 ggb
attachments)
R: Dimitris’ new widget instance with Geogebra
V: Five new widget instances by Popi
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 17
CoICode Map discussions
• Initially: Creating and exchanging widget instances
• Significantly later: story added
• Towards the end: Started viewing c-book technology as integrating narrative and widgets
• Discussions on how the story will flow along the pages – internal structure – distribution of math ideas – c-book unit as a whole
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 18
Between Cycle 1 and Cycle 2: f2f meeting
• Critical discussion and reflection on the design process and products of Cycle 1
• Demonstration of the products of other Coi
• Main Points:
o How integrated were the widget instances and narrative
o How functional was the story with respect to the instances
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 19
Cycle 2 map
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 20
What changed in Cycle 2 design process:
• Two design strands:
o Activities around a specific mathematical concepts,
o Thematic activities to allow the emergence of new mathematical ideas
• Story: Make the storyline more realistic – Don Quixote
• Which Cycle 1 widget instances to keep?
o those more appropriate for the narrative
• Ideas for using new widgets - 3d software
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 21
New c-book units – Cycle 2 (WP6)
Finished:• Don Quixote
• Cycling in the park
Under Development:• Staircases
• Castles
• Sustainable Cities
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 22
Spanish CoI
Berta Barquero & Mario Barajas, UB
Evaluation: Cycle 1 and cycle 2 productions processes
The “Social networks” c-book unit
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2323
Profiles and roles of the CoI members involved in the design of the c-book unit
1. Researchers in Education & Mathematics Education
2. School teachers &
Teacher trainers
3. Technology developers and users in Maths
Education
4. ‘Experts’ on the
dissemination of knowledge
6. ‘Experts’ outside Maths
Education 5. Creative Industry
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2424
Profiles and roles of the CoI members involved in the design of the c-book unit
Lídia Serrano [D, Secondary School teacher & researcher in ME]
Berta Barquero [D, researcher in
ME]
Víctor Martínez de Albéniz [D, experts outside ME, in
Operations Research]
Ramon Eixarch [D, Technology developers in ME, WIRIS creator]
Dina Gómez [D, Secondary School teacher & technology user]
Andrea Richter [R, research assistant]
Daniel Ramos[R, ‘Experts’ on the
dissemination of knowledge]
Sílvia Carvajal[R, Creative
Industry]
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2525
Rationale and genesis of the c-book unitFrom the ‘protodesign’ of the unit
…
(1) …reaching agreement on
the rationale and principles for its
design
Introduce students to the study of:
Real, functional and live questions and problems,
Through an interdisciplinary study, where mathematics could ‘help’ to act providing answers,
Where mathematical modelling will adopt a central role.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2626
The critical episode in the collaborative design process
(2) …a turning point that made
the CoI members come down to
‘school’
Making decisions about the:
Theme/Topic focused on the use of the exponential growth properties to describe and model the behaviour and evolution of the number of users in a social network.
Learning goals focused on introducing students into the study of ‘real and live’ questions from which a process of mathematical modelling would begin.
Targeted audience and context of use: Secondary Education, 14-15 years old.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2727
Structure of the unit: different interconnected phases
First phase [6 pages, DME, Geogebra]
Description and understanding of the viral
behaviour of Social Networks
Second phase [6 pages, Cinderela,
DME and Geogebra]
The six degree of separation: Can SN
help to reduce the degree of
separation between people?
Third phase [2 pages, all]
You in Facebook and your connections
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2828
The critical episode in the collaborative design process
(3) Going further: the internal
review process of the c-book unit
Since the beginning, we could distinguish some ‘criteria’ to foster CMT, used as …
Indicators related to the conception of CMT of this CoI working group,
Criteria or principles to integrate in the design of the c-book unit,
Tools to evaluate the creative potential of the c-book unit (finally produced).
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 2929
Evaluation of CMT in the design process of the c-book unit and on the final product
Spanish CoI c-book unit: Viral Behaviour of Social Networks
1. Facing ‘real’ and ‘true’ problems [PROBLEMATISATION, CONNECTIONS]
2. Generating the need of looking for answers [PROBLEMATISATION, EXPLORATION]
3. Split the initial question into sub(questions) [ANALYSIS-SYNTHESIS]
4. Need for tools of validation and evaluation of partial and final answers [EVALUATION-VALIDATION]
5. Looking for external validator outside [EVALUATION-VALIDATION]
6. Mathematical modelling as a continuous process [MODELLING, evolution of the questions and of the models to work with]
7. Integration, use and combination of technological tools for the exploration, contrast, evaluation of the use of models with data, etc. [EXPLORATION, CONTRAST AND EVALUATION] …
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3030
Evaluation of the c-book units’ potential for CMT
Creativity
Problematisation
Analysis Synthesis
Connections Representations
Institutionalization / Communication
Modelling as an essential tool to promote CMT
Validation/Evaluation
Spanish CoI c-book unit: Viral behaviour of Social Networks
Barquero, Richter, Barajas & Font (2014)
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3131
Evaluation of the c-book units’ potential for CMT
PROBLEMATISATION – Posing situations and open questions at the beginning of the c-
book unit. They have to be ‘partially’ answered along each phase of the unit.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3232
Evaluation of the c-book units’ potential for CMT
ANALYSES – Break the initial ‘open-question’ as a chain of questions more concrete and easy (sub)questions
SYNTHESIS – Include moments where students will have to elaborate their own answers
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3333
Evaluation of the c-book units’ potential for CMT
REPRESENTATIONS – Evolution on the tools that were considered on the study of the
relation and complementarity among different representations
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3434
WP6 - cycle 2 & WP7 - cross-case redesign
Forecasting FACEBOOK users
The musical plane
Dominos, Tetris, Pentominos and other 2D figures
Viral Behaviour of Social Networks (v2)
Coordinates and points (v2)
Eva’s windmills (WP7 – Cross-CoI design)
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 35
English CoI
Christian Bokhove, LKL
Evaluation: Cycle 1 and cycle 2 productions processes
The “Numbers” c-book unit
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 36
Profiles and roles of the CoI members involved in the design of the c-book unit
1. Researchers in Education
& Mathematics Education
2. Teacher
educators
3. Game developer
4. ‘Mathematicians
6. Computer scientists
5. Teachers
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 3737
Profiles and roles of the CoI members involved in the design of the c-book unit
Alison Clark-Wilson [main idea ‘owner’ and commenting further on these ideas]
Christian Bokhove [buddy author of this cBook, capitalising on authoring expertise, taking feedback and
implementing it]
Manolis Mavrikis [core CoI member commenting on different versions of the cBook and eliciting
feedback from members of the extended CoI]
Eirini Geraniou [core CoI members commenting on different versions of
the cBook and eliciting feedback from members of the extended CoI]
Tess Andrew [contributor to the initial
idea]
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 38
The “Number” c-book unit
• Theme: Numbers• Math topics:
• Knowing multiple numerical expressions for the same ‘object’;
• Understanding equivalence of expressions;• Rewriting numbers and expressions into
equivalent numbers and expressions;• Audience: 6th and 7th grade
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 39
The “Number” c-book unit
• “Numbers can be written in different ways”
• First c-book made, so also used to elucidate thoughts of CoI on Social Creativity and Creative Mathematical Thinking
• Embed in classroom practices, for example as ICE breaker
• Unique features:• Randomization• Equivalence checking and feedback
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 40
Structure of the c-book unit • Introductory page;
• Pages that have 10 boxes that ask students to construct expressions equivalent to (random) number;
• Same question for (randomized) algebraic expression;
• A page with drawing and open textbox widget to pose/create their own task;
• Page to share their idea with a fellow class member.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 41
Evolution of the c-book: critical episode1. First idea: the number 36
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 42
2. Expanding the idea
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 43
3. First prototype
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4. Parallel idea involved expressions
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5. Developing the first prototype
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 46
6. Adding open expression element for pupils
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Collaborative design process
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 48
Creative mathematical thinking
• Open character; not giving question with fixed answer but leaving ‘correct’ answer open;
• Problem posing as creative process
• For example gauge:• Fluency• Flexibility• Originality
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 49
Cross-case analysis & cycle 2
• Multiple representations
• Flexibility, fluency, originality as part of CMT
- Velocity UK- Graph transformations
V2- Generalisations- Linear equations- Planets- Statistical literacy
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 50
French CoI
Christian Mercat & Pedro Lealdino Filho, UCBL
Evaluation: Cycle 1 and cycle 2 productions processes
The “Velocity” c-book unit
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 51
French CoI, groups and CoPs
- Aristod- IREM Grenoble- S2HEP – IFÉ
- Pôle Édition- House of Math & CS
(MMI)
- IREM LyonCommon domains:- Math education- Math teacher training
in- & pre-service
- Research in math education& math didactics
- Math software designer- Design, art, culture, edition
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 5252
Composition of the CoI
1. Researchers in Education & Mathematics Education
2. School teachers & Teacher trainers
3. Technology developers and users in Maths
Education(Aristod)
4. ‘Expert’ on the dissemination of
knowledge(Publisher)
6. ‘Experts’ outside Maths
Education(man/machine
Interface)
5. Creative Industry(Artist)
IREM Lyon epsilonWriter
IREM Grenoble
Museum
Tactileo
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 53
CMT representation
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SC representation
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The TraAM CoP, Velocity
Travaux Académiques Mutualisés: an IREM of Lyon group (3 secondary math teachers, 2 ESPÉ math teacher trainers, 1 academic) summoned by the French ministry of Education for a national program on open ended problems and ICT. Members of this CoP belonging to the CoI:
Jean-Louis Bonnafet and Christian Mercat
Main CoI inputs from Nataly Essonier (S2HEP), Michèle Gandit
(IREM Grenoble) and Pedro Lealdino Filho (S2HEP)
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 56
The TraAM CoP, Velocity
Main Objectives: -Installing the relationship between speed, distance and time in a robust way; -taking decisions and reaching conclusions in spite of uncertainty; -understanding the graphs of position over time, of speed over time, and their relationship.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 57
Velocity, 4 activities
- Tunnel
- Chronotachygraph
- Understanding average
speed
- Particle dancing
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 58
Velocity, Tunnel activity, p1
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 59
Velocity, Tunnel activity, p2
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Velocity, Chronotachygraph activity, p2
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Velocity, Chronotachygraph activity, p3
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Velocity, Average speed activity
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 63
Velocity, Dancing particle
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 64
Velocity, Critical episodeDidactical contract clash
When tested in two classes, the most successful class in math rebelled against the activity while the lower achieving students complied and enjoyed it. We had to add a table of trained competencies taken from official curriculum to restore peace in the classroom.
- Je suis capable de comprendre le problème, de commencer les recherches et de formuler des questions, de modéliser ;- Je suis capable d’émettre des hypothèses, des conjectures ;- Je suis capable de proposer un protocole et de mettre en œuvre une expérience ;- Je suis capable de mener un raisonnement cohérent, de faire une preuve ou de valider une hypothèse ;- Je sais communiquer ma démarche par écrit ou par oral ;- Je fais preuve d’initiative, d’originalité, d’autonomie ;- Je suis capable de rester concentré, de travailler dans le calme.
MCSquared FP7 Project 610467, Review meeting, Nov 6, 2014, Utrecht 65
Cross-case analysis & cycle 2
- Numbers fr- Velocity V2- Limits V2- Road dynamics- Algebraic Tetris- Modelling- Museum pieces
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