Supporting Creative Design Processes for the Support of Creative

Mathematical Thinking

Capitalising on Cultivating Synergies between Math Education and Environmental

Education

Chronis Kynigos

1,2

and Maria Daskolia

1,3

Computer Technology Institute and Press "Diophantus" (CTI), 26-28 Mitropoleos str, 10563 Athens, Greece

Educational Technology Lab, Department of Pedagogy, University of Athens, Athens, Greece

Environmental Educational Lab, Department of Pedagogy, University of Athens, Athens, Greece

Keywords: Mathematical Creativity, Social Creativity, Design of Educational Digital Resources, c-Book Technology,

Communities of Interest, Environmental Education, Education for Sustainable Development, MC

Project.

Abstract: Mathematical creativity is acknowledged as a backbone lifelong competence necessary to be fostered in all

students. However, this is not an easy task to accomplish, due not only to a lack of appropriate technologies

enabling the creative design of digital educational resources for creative mathematical thinking (CMT), but

also to the absence of insightful methodologies to support creative design processes of this kind among

professionals. The Mathematical Creativity Squared (MC

) project aims to address this twofold problem in

the following ways: (a) by designing and developing a new genre of technological environment for the

design of CMT resources, 'the c-book' environment, and (b) by adopting and further developing a

methodology based on the generation of Communities of Interest (CoI) as a social milieu which will

facilitate the creative design of CMT resources in collectives of educational designers stemming from

diverse professional and educational domains. Especially with regards to the latter, the rationale is that CoIs

will support synergies among designers with a math education background with others carrying a more

socially-relevant educational orientation, such as environmental educators and educators for sustainable

development, both on epistemological and pedagogical level, with the aim that their 'boundary crossing'

interactions will positively effect social creativity in the design process for digital resources for CMT.

1 INTRODUCTION

1.1 On Creativity and Mathematical

Creativity

Creativity is perceived as the backbone of the skills

required for new jobs and as an essential ability to

be developed in the context of lifelong learning (EC,

2008; 2011). It is recognised as a transversal skill

needed to foster each of the eight key lifelong

competences identified by the European

Commission (EC, 2008) as particularly necessary for

personal fulfilment and development, social

inclusion, active citizenship and employment.

Mathematical competence is among these core life-

long competences identified by the European

Commission, comprising the ability to develop and

apply mathematical thinking to solve a range of

problems in everyday situations and the application

of this knowledge and methodology in response to

perceived human wants or needs (EC, 2006).

Developing students’ creative potential in

mathematical thinking has been also ranked among

the most needed skills to develop for the 21

century

in the States (National Academies of Science, 2007).

Nevertheless, although creativity is deemed as a

tool for succeeding personal development and social

empowerment and as an impetus for achieving

professional innovation and economic change

(Banaji et al., 2006), it still remains an elusive target

to attain and a largely under-researched topic,

especially in terms of whether and how it can be

enhanced. The wide array of theories and

perspectives on 'creativity' (e.g., Runco, 2007),

coupled with the inherent vagueness of the concept

(e.g., Kampylis and Valtanen, 2010), its context-

specificity (Amabile, 1983; Csikszentmihalyi, 1996),

and the inadequacy of most traditional educational

342

Kynigos C. and Daskolia M..

Supporting Creative Design Processes for the Support of Creative Mathematical Thinking - Capitalising on Cultivating Synergies between Math

Education and Environmental Education.

DOI: 10.5220/0004965603420347

In Proceedings of the 6th International Conference on Computer Supported Education (CSEDU-2014), pages 342-347

ISBN: 978-989-758-021-5

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

systems and processes, set several burdens in the

fostering of creativity.

This is particularly true with mathematical

creativity. The lack of an accepted definition among

researchers and practitioners (Mann, 2006) on the

one hand, and educational reality which

systematically deemphasize creative math problem-

solving ability on the other cannot but hinder any

concerted activity in the field (Mann, 2009;

Sriraman, 2005). Even though almost any academic

perspective on math education has been promoting

mathematical creativity (Menghini et al., 2008),

well-meant intentions are not enough for developing

math learning through creativity and vice versa.

Endeavours of this kind are acknowledged to be

closely associated with putting forth changes in

educational systems, processes and outcomes; taking

full advantage of the potential of information and

communication technologies; and working towards

materialising creative ideas into concrete, new and

more effective, products and services (EC, 2008;

Ferrari et al., 2009).

1.2 Fostering Mathematical Creativity

through Digital Technologies

In terms of technologies, twenty or more years ago,

emergent approaches to how digital media could be

used to foster mathematical thinking were almost

exclusively visionary and about using technology as

an expressive medium for cutting edge pedagogy

involving engagement with mathematical thinking

and meaning making (Hoyles and Noss, 2003).

Despite the existence of some excellent digital tools

affording dynamic manipulation, interconnected

representations (including mathematical formalism),

simulations of phenomena and situations embedding

mathematical rules, visualisations of data

representations and handling of probability, their

uses in education were frequently instrumented to a

large extent towards contexts of traditional lecturing

and demonstration of exercise-type activities

(Hennessy et al., 2005; Ruthven, 2008). Some

technological advances of the past few years have

even further skewed attention back to uses of digital

media to support traditional routine learning of

mathematics through the assistance of learning

management systems, portals, video and intelligence

in tutoring systems to that effect (e.g., the Kahn

Academy).

However, especially during the last decade there

is a growing interest in Europe in the role and use of

appropriate digital technologies as the necessary

broad-based strategies to bring forth the envisioned

shifts in terms of creativity and innovation in

education (Ferrari et al., 2009). This rests to be

explored with regards to the teaching and learning of

any knowledge domain, including mathematics.

Nevertheless, whether digital technologies can act as

real enablers of creative mathematical thinking has

to be combined with other processes, mechanisms

and tools of school education, among which the

design and use of appropriate educational resources.

1.3 Creativity in the Design of

Educational Resources for CMT

This last point brings forth an inter-related and

equally challenging issue, that of the design of

appropriate educational resources for creative

mathematical thinking (CMT), and its relation to

creativity. Taking into consideration that any kind of

design process is inseparably connected to creativity

(Taura and Nagai, 2010), the task of designing

educational resources for CMT becomes thus a

'squared' creativity issue.

Design creativity is recognised as a

multidimensional concept, that can be identified,

among others, in the processes or the outcomes of

the design activity, or in the context within which

such an activity is embedded (Gero, 2010). This is

also the case with the design of educational

resources (and instructional design in general),

which although not overly acknowledged and

studied as a design discipline compared to other

domains (i.e., software design or architecture),

creativity has recently also being acknowledged as a

‘built-in’ dimension of it (Clinton and Hokanson,

2012).

The 'social' component in various collectives of

design professionals and its role in enhancing both

the individuals' creativity and the creative capacity

of the group in addressing complex design problems

is also a relatively new focus of research. 'Social

creativity' actually explores the social and technical

environment within which participatory design

processes take place (Fischer, 2001; 2011), when

specialists from different domains coordinate their

efforts to achieve a common design goal. Social

creativity arises from the synthesis and synergy of

the different perspectives towards a complex design

task of shared interest.

Designing for digital educational resources for

CMT can be therefore viewed as a squared creativity

challenge, since it requires not only to define

mathematical creativity but also to situate the design

process itself within a socio-technical environment

that can boost educational designers' creative

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potential. However, both challenges to be dealt with

appropriately need some paradigm shifts in terms of

thinking of, learning and designing pedagogical

materials for mathematics, already represented in

some debates in math education.

1.4 Boosting Math Creativity through

Synergy with Other Domains: the

Case of Environmental Education

For example, a key debate in the field is the one

questioning the traditional paradigm of focusing

exclusively on abstract mathematical concepts and

problems as isolated from broader phenomena and

social contexts in which they could become more

relevant, and without highlighting the relevance and

value of this knowledge to the students’ life and

society in general. Within this paradigm, math

education seems to reproduce the false myth of an

objective and value-free discipline, alienated from

current reality. Criticisms of this genre are supported

by evidence from many countries indicating that

school curricula give preponderant emphasis on

foundationalist approaches of learning about

mathematics and on pedagogies advancing the

transmission of dry knowledge, as if all students

were being prepared to become the big

mathematicians of tomorrow. However, there are

serious doubts about whether such curricula can

succeed to trigger in students any meaningfully

creative engagements with maths.

A suggestion so that math education explores

new creativity potentials and meaningfulness to a

wider range of students is to ‘bridge’ it with

educational domains which are more socially-

oriented and centred to real-life problems, such is

the case of Environmental Education (EE) and

Education for Sustainable Development (ESD)

(Kynigos et al., 2013). Although several scholars

have stressed the opportunities for developing a

beneficial relationship between science education

and EE/ESD (such as Gough 2002, 2007; Sjøberg

and Schreiner, 2005, etc), no relative bridging has

been proposed to motivate students to get more

actively involved with math concepts and processes

by identifying the 'mathematics' hidden inside some

of the most challenging current socio-scientific and

sustainability issues.

Especially mathematical problem-posing and

problem-solving which are indicated as appropriate

learning formats to allow math creativity to emerge,

are proposed to exemplify some characteristics with

respect to the 'problematic situation'. For example,

for a situation to be amenable to creative

(mathematical) problem-posing and -solving (Torp

and Sage, 2002) it:

• should be fairly ill-structured and messy

• may change with the addition of new

information

• is not solved easily or with a specific formula

• does not result in one right answer.

These characteristics are directly applicable to

most environmental and sustainability issues, which

are by nature ill-defined, complex, controversial,

value-laden and require the application of various

perspectives to grasp them more thoroughly

(Daskolia and Kynigos, 2012). However, it is this

messiness and complexity that make them 'good'

examples of creativity-triggering problems and

amenable to be treated in learning situations that

foster the students' creative thinking. It also turns

them into “boundary objects” (Star and Griesemer,

1989), that is entities that can interpreted and

employed by more that one groups or communities

in ways that make sense to them.

At the same time, dealing with issues of these

characteristics at a pedagogical level provides many

opportunities for teachers and learners to get

engaged in dialogical forms of meaning-construction

and perspective-sharing and to expand the

“boundaries” of their knowing of and being in the

world both inside and across the realms of their

discipline (Daskolia and Kynigos, ibid). This is what

some would also identify as a creative appropriation

of subject-matter and pedagogical knowledge.

2 THE PROJECT RATIONALE

For mathematical creativity to be fuelled new

designs are needed, new ways of thinking and

learning about mathematics through synergy with

other educational domains, such as those of EE and

ESD, and the support of learners' engagement with

creative mathematical thinking in collectives using

dynamic digital media. Designers of educational

digital resources (either professionals or teachers)

have therefore to look for and benefit from new

designs that would assist them to explore, identify

and creatively produce new educational resources

and tools with a potential to stimulate creative ways

of mathematical thinking.

This is not an easy task to accomplish, due not

only to a lack of appropriate technologies enabling

the creative design of digital educational resources

for creative mathematical thinking (CMT), but also

to the absence of insightful methodologies to support

creative design processes of this kind. The MC

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project aims to address this twofold problem in the

following ways: (a) by designing and developing a

new genre of technological environment for the

design of CMT resources, that is an authorable e-

book we call 'the c-book' (c for creative), and (b) by

adopting and further developing a particular

methodology based on the generation of

Communities of Interest (CoI) (Fischer, 2001;

2011) as a social environment that would facilitate

the creative design of CMT resources in collectives

of professionals. The rationale is that by providing

educational designers with an appropriate socio-

technical environment, they will have more

opportunities for richer creative design processes.

The objective of the MC

project is thus to build

both a technology and a social milieu that will

jointly and interactively enhance creativity among

educational designers in the design of digital

educational resources fostering creative

mathematical thinking in learners. Our approach

addresses social creativity in design through

specially formed Communities of Interest (CoIs)

which will act as agents of potential cultures of

participation among professional designers and

teachers from different/ diverse educational domains

(not only from the math education sector but also

from environmental education) and will lead to the

development of innovative frames of addressing the

design of educational resources for creative

mathematical thinking.

The MC

project places at the centre of its

concerted action the design and development of the

c-book, an integrated and beyond the state-of-the-art

digital system which will facilitate all foreseen

creative design processes and practices and will

provide support for creative mathematical thinking.

The main characteristics of this system will be:

 an authorable data-analytics engine and a

graphical interface providing professionals from

diverse creative industries involved in the design of

digital resources for CMT (publishers, developers,

researchers, school educators from diverse

educational domains) with the ability of customizing

the kinds of information they need for assessing end-

user creativity and the kinds of automated reactions

they wish the tool to provide to end-users' activity.

 an authorable dynamic e-book infrastructure

to be used by pedagogical design professionals for

collaborative design of CMT resources and by

students to create their personalized versions of such

resources.

 a set of dynamic, exploratory and

constructionist digital tools integrated in the e-book

infrastructure designed to foster creativity in

students' mathematical expression, investigation and

meaning generation.

The design of the c-book environment will

require advancements not only in terms of

technology but also in the design processes. MC

project will develop, operationalize and document

methods for the emergence of creativity in the

design process based on 'middle c' social creativity

in CoI embedding diverse actors and norms (Moran,

2010; Hämäläinen and Vähäsantanen, 2011;

Sonnenburg, 2004). In effect, it will address and

promote the creativity that emerges when

communities share common goals and well-designed

tools. The project partners have the capacity and

experience to act as catalysts for systemic change in

cultures characterizing current school and workplace

practices. By situating the CoI in large-scale

systemic national initiatives MC

will test how the

process and productions of collaborative creative

design for CMT can empower professionals (and

teachers) from diverse (disciplinary and/or

educational) domains to collaborate and create new

educational designs. At the scientific level, the

project will contribute to our knowledge of methods

and processes in using digital media to enhance

creativity in interdisciplinary CoIs. The emergence

of creativity is perceived as 'a system' involving

'collectives with tools' characterized as 'cultures of

participation' (Fischer, 2011). This project will

advance our knowledge by integrating two

constructs to generate and support such cultures for

professional design, 'documentational approach'

(Gueudet and Trouche, 2012) and the 'boundary

crossing with digital media as boundary objects'

approach (Akkerman and Bakker, 2011; Kynigos &

Kalogeria, 2012). The technology will play a

fundamental role by transforming an otherwise

costly endeavour to an efficient knowledge co-

construction process through both the collaborative

and authoring features of the c-book which will be

useable in the six languages of the partnership. This

will provide stakeholders with the means to both

advance and share creative intelligent mathematical

artifacts and by thus to enable a broader reach.

Accordingly, the impact of MC

will be long-lasting

as the malleability of the creative widgets and the

authorable intelligent feedback will create the

foundations for subsequent evidence-based creative

designs beyond the life of the project.

3 NEXT STEPS

Our position is that digital media designed to

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support practices and productions of added

pedagogical value fall short of target if they address

only the conceptual/ cognitive/ performance aspects

of the learning process. There is a need to consider

how to directly support future citizens in their social

and work capacity. Mathematical creativity is

viewed as a backbone ability and disposition for

students, which can be cultivated and taught as a

social phenomenon per se but also through teaching

and learning about more socially-related and focused

issues (i.e., current environmental and sustainability

issues) and be widened up to all students. Social

creativity amongst designers of educational

materials with an explicit aim to support such a

creative potential in learners is a core competence

which nevertheless needs some appropriate frames

to be identified, studied and boosted. Technologies

supporting the design and deployment of educational

materials and digital media for enhancing the

students' creative mathematical thinking have so far

been too much influenced by more traditional

paradigms of thinking where such media have the

role of unquestionably accredited tools to be

employed by the students. New genres of

technologies, such as the proposed c-book system,

will help the questioning and development of the

process of design and the kinds of media and

accompanying paradigms which may become

available to students in the future. Within this

rationale, a c-book unit can be a medium containing

questionable artifacts, statements, arguments for

students to change, improve, negotiate over, re-

construct; something for the students to re-draft, to

shape or make on their own. They are viewed as

objects for students to argue over and to collectively

're-write'. How does this will consequently change

the designers' perspectives, methodologies and ways

of thinking, collaborating and working? Our

proposed c-book technology and supporting

methodology aim to re-address how to design

educational resources for creative mathematical

thinking by turning into a creative learning asset the

crossing of various pre-established boundaries

across many inter-related or not until recently related

fields.

ACKNOWLEDGEMENTS

The research leading to these results has received

funding from the European Union Seventh

Framework Programme (FP7/2007-2013) under

grant agreement n° 610467 - project “M C Squared”.

This publication reflects only the authors’ views and

Union is not liable for any use that may be made of

the information contained therein.

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