McSquared project Communities of Interest production in Cycle 1

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


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


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.


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


Cycle 1 productions – c-book units


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


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)


Greek CoI

Maria Daskolia, CTI


The “Eva’s windmills” c-book unit


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


“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:


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)


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


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


Spanish CoI

Berta Barquero & Mario Barajas, UB


The “Social networks” c-book unit


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]

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]


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.


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

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)



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)


English CoI

Christian Bokhove, LKL


The “Numbers” c-book unit



1. Researchers in Education

& Mathematics Education

2. Teacher

educators

3. Game developer

4. ‘Mathematicians

6. Computer scientists

5. Teachers



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]


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 episode1. 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



• Multiple representations

• Flexibility, fluency, originality as part of CMT

- Velocity UK- Graph transformations

V2- Generalisations- Linear equations- Planets- Statistical literacy


French CoI

Christian Mercat & Pedro Lealdino Filho, UCBL


The “Velocity” c-book unit


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


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


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 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.



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