McSquared project Communities of Interest production in Cycle 1

Mathematical Creativity Squared
Review meeting
Nov 6, 2014
Utrecht, Netherlands (hosted by Utrecht University)
Evaluation: Cycle 1 and cycle 2
productions processes
Ch. Mercat, WP6, UCBL
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”

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 2

Four different Coi's and many CoP's
UK – Spain – Greece – France
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
But also
- Environmental, Engineering,
Computer Science education
- Research in online teacher
communication
- Research on Creativity

Cycle 1 productions CMT driven design
Collaborative 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.

Cycle 1 productions – Social Creativity
CoiCode and critical episodes
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 – c-book units

Cross-case analysis & cycle 2
UK – France & Spain – Greece
Alien c-book units:
- evaluate CMT potential
- reconstruct
New c-books units:
- Version 2 of Cycle 1
- Completely new ones

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)

The “Eva’s windmills” c-book unit
Greek CoI
Maria Daskolia, CTI

Profiles of the Greek Coi members involved in the
design of the c-book units
1. Researchers in
Mathematics
Education
2. Teacher trainers
4. National
InitiativeTrainees
3. Technology
developers
6. Researchers in
Math Teacher
Education
5. Master Course
students – Master
Holders
7. Researcher in
Environmental
education
8. Phd
Holders/Students
10. Researchers in
On-line teacher
communication
9. Researcher in
Vocational
Education

“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)

The “Eva’s windmills” c-book unit:
Turtleworlds
• Eva’s adventures in a Greek island…..

“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

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 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)

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 Geogebra
1.ggb (attachment)
R: Elissavet’s post
V: Popi’s first Geogebra widget
instance
30
Mar14
Dimitris Building on Popi’s
microworld
2. 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

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

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

Cycle 2 map

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

New c-book units – Cycle 2 (WP6)
 Finished:
• Don Quixote
• Cycling in the park
 Under Development:
• Staircases
• Castles
• Sustainable Cities

The “Social networks” c-book unit
Spanish CoI
Berta Barquero & Mario Barajas, UB

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

Lídia Serrano [D, Secondary School teacher &
researcher in ME]
Berta Barquero
[D, researcher in ME]
Andrea Richter
[R, research assistant]
Víctor Martínez de Albéniz [D,
experts outside ME, in Operations
Research]
Dina Gómez [D, Secondary
School teacher &
technology user]
Ramon Eixarch [D, Technology
developers in ME, WIRIS creator]
Daniel Ramos
[R, ‘Experts’ on the
dissemination of
knowledge]
Sílvia Carvajal
[R, Creative Industry]

Rationale and genesis of the c-book unit
From the ‘protodesign’ of the unit …
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.
(1) …reaching agreement on the
rationale and principles for its design

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.

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

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).

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] …

Evaluation of the c-book units’ potential for CMT
Spanish CoI c-book unit: Viral behaviour of Social Networks
Modelling as an essential tool to promote CMT
Problematisation
Analysis Synthesis
Creativity
Institutionalization /
Communication
Validation/Evaluation
Connections Representations
Barquero, Richter, Barajas & Font (2014)

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.

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

REPRESENTATIONS – Evolution on the tools that were considered on the study of the
relation and complementarity among different representations

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)

The “Numbers” c-book unit
English CoI
Christian Bokhove, LKL

1. Researchers
in Education &
Mathematics
Education
2. Teacher
educators
3. Game
developer
6. Computer
scientists
4. ‘Mathematicians
5. Teachers

Alison Clark-Wilson [main idea ‘owner’ and
commenting further on these ideas]
Eirini Geraniou [core CoI members
commenting on different versions of the
cBook and eliciting feedback from
members of the extended CoI]
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]
Tess Andrew [contributor
to the initial idea]

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

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

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.

Evolution of the c-book: critical episode
1. First idea: the number 36

2. Expanding the idea

3. First prototype

4. Parallel idea involved expressions

5. Developing the first prototype

6. Adding open expression element for
pupils

Collaborative design process

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

- Velocity UK
- Graph transformations V2
- Generalisations
- Linear equations
- Planets
- Statistical literacy
• Multiple representations
• Flexibility, fluency, originality as part of CMT

The “Velocity” c-book unit
French CoI
Christian Mercat & Pedro Lealdino Filho, UCBL

French CoI, groups and CoPs
- IREM Lyon
- S2HEP – IFÉ
- IREM Grenoble
- Aristod
- Pôle Édition
- House of Math &
CS
(MMI)
Common domains:
- Math education
- Math teacher training
in- & pre-service
- Research in math education
& math didactics
- Math software designer
- Design, art, culture, edition

Composition of the CoI
1. Researchers in
Education &
Mathematics
Education
2. School teachers
& Teacher trainers
3. Technology
developers and
users in Maths
Education
(Aristod)
IREM Lyon epsilonWriter
4. ‘Expert’ on the
dissemination of
knowledge
(Publisher)
6. ‘Experts’
outside Maths
Education
(man/machine
Interface)
IREM Grenoble
5. Creative
Industry
(Artist)
Museum
Tactileo

CMT representation

SC representation

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)

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.

Velocity, 4 activities
- Tunnel
- Chronotachygraph
- Understanding
average speed
- Particle dancing

Velocity, Tunnel activity, p1

Velocity, Tunnel activity, p2

Velocity, Chronotachygraph activity, p2

Velocity, Chronotachygraph activity, p3

Velocity, Average speed activity

Velocity, Dancing particle

Velocity, Critical episode
Didactical 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 oeuvre 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.

- Numbers fr
- Velocity V2
- Limits V2
- Road dynamics
- Algebraic Tetris
- Modelling
- Museum pieces

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