Journal of Mathematics and the Arts,

Vol. 1, No. 3, September 2007, 203–207

Report

Innovations in mathematics education via the arts

BIRS Workshop 07w5062

22–26 January 2007

Banff, Alberta, Canada

M. ALAGIC*

Wichita State University, Wichita, Kansas, USA

(Received 15 April 2007; in final form 11 June 2007)

‘There are no excuses.’

Jim Oliver,Banff Centre Service Director

The Banff International Research Station for

Mathematical Innovation and Discovery (BIRS)

Workshop, ‘Innovations in mathematics education via

the arts,’ provided an outstanding opportunity for

developing a deeper understanding of the needs,

challenges, and future prospects of an emerging educa-

tional perspective that focuses on the interplay between

mathematics and the arts. The primary objective of the

Workshop was to bring together a group of profes-

sionals who incorporate the arts in their educational

activities, in order to discuss strategy about possible

ways of promoting and implementing mathematics

education via the arts. The intention was to develop a

range of projects in which the participants themselves

could engage and to plan for further development and

dissemination that would transfer these project ideas to

educators, students, and the public. This report

describes how the group worked productively and

what we accomplished during our five-day residency.Gerda de Vries (University of Alberta, Canada),

George Hart (Stony Brook University, USA), and Reza

Sarhangi (Towson University, USA) were the organizers

and principal facilitators of the Workshop. The parti-

cipants (see figure 1) were mathematicians, mathematics

educators and artists, many of whom owe their

connections to this group through their involvement in

Bridges conferences (http://www.BridgesMathArt.org).

The proposal for the BIRS Workshop and other

Workshop-related documents, including daily notes

and presentation materials, are posted at the web sitehttp://www.georgehart.com/birs.

Banff offered a unique environment for creativedialogue and the cross-fertilization of ideas. It wasintellectually energizing to be part of such a diversegroup, comprised of participants from specialized areasof mathematics and art within higher education, teachereducation and K-12 schools. The facilities were ideallysuited to the group’s needs and the BIRS staff provideda supportive and nurturing environment. The surround-ings were spectacular and inspiring; mountains, deer,birds, and a charming village all contributed to anenvironment where learning and creativity flourished.Excursions such as a hike up Tunnel Mountain or a visitto the hot springs enlivened the senses and fosteredcreative and constructive collaboration. There are fewother venues that offer such inspiring ambience forintense intellectual engagement. To quote Banff CentreService Director Jim Oliver, at BIRS ‘there are noexcuses’ to delay the work that you must do.

Mathematics education needs

BIRS workshop participants united in the belief that theintegration of mathematics and art can play a significantrole in teaching and learning mathematics, improvingpublic perception of the value of mathematics, andchallenging cultural norms that make mathematicaldeficiencies acceptable. They believe projects integratingmathematics and the arts can engage learners acrossmultiple modalities in a way that teaching mathematicsin isolation never could. Furthermore, K-12 level

Journal of Mathematics and the ArtsISSN 1751–3472 print/ISSN 1751–3480 online � 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/17513470701585977

*Email: Mara.alagic@wichita.edu

mathematics education should be broadening to life-long learning, both formally and informally. One ofthe most important aspects of integrating art into theteaching of mathematics is the realization that themathematical content must not be assumed to be self-evident. Mathematics and art projects that teachers,mathematicians, and artists choose to adopt as class-room activities for students must be carefully chosen tofit within the curriculum guidelines and requirements.Note that this need not be the case with mathematics/arts projects that take place outside the classroom

environment or in a more informal setting. TheWorkshop participants devoted considerable timetrying to identify the most significant challenges andfinding constructive ways in which at least some of thesechallenges might be met.

It soon became clear that the most important needsfor teaching mathematics by means of the arts were:(1) easily accessible and ready-to-use quality materialswhich integrate mathematics and arts in substantive andpedagogically sound ways, and (2) assessment tools forevaluating whether the integration of mathematics and

Figure 1. Workshop participants (photo courtesy of Glyn Rimmington). From back row, top left: Glyn Rimmington, MaraAlagic, Gene Klotz, George Hart, Godfried Toussaint k Reza Sarhangi, Robert Bosch, Susan Gerofsky, Gerda de Vries, GaryGreenfield, Pau Atela, David Richter k Craig Kaplan, Dirk Huylebrouck, Blake Mellor, Paco Gomez k Doris Schattschneider,Carolyn Yackel k Bill Higginson, Phil Wagner, Carlo Sequin, Gwen Fisher, David Rappaport and Carol Bier, Doug Burkholder kKevin Hartshorn, Stewart Craven, Daina Taimina, Nat Friedman.

204 M. Alagic

art is successful in improving mathematical conceptual

understanding and process skills. Furthermore,although it is tempting to assume that arts integration

improves students’ attitudes towards mathematics, moreresearch in that direction would be beneficial. At the

elementary, middle and secondary level, teachers oftenfeel that there is no opportunity to go beyond the scopeof the text, or integrate other material. Therefore

mentoring these teachers and involving them in appro-priate professional development activities to help learn

about integrating arts and mathematics is another needrecognized by BIRS participants. As one participant

emphasized, ‘We must improve test scores. That is thereality.’

From an educational perspective, mathematically

inspired art must be the launching point for a clearrepresentation of the underlying mathematical concepts,

or an end result of an educational process, and neveran end in itself. The challenge of bridging from the

specialized arts areas to making a measurable differencein learning mathematics is significant. It involves theongoing cultivation of multiple perspectives and con-cept-based representations.

Activities: What we shared and learned

BIRS workshop participants explored existing refer-ences, books and websites. They participated in a seriesof inspirational, creative, and hands-on activities thatallowed them to consider many constructive ideas forstudent and classroom projects. Example activitiesincluded:

. Rhythmic clapping exercises that were related tolearning about n-gons and periodic motions ona circle.

. Making models of manifold surfaces by connectingCDs with cable ties (see figure 2).

Figure 2. Assembling a truncated icosahedron CD sculpture (photo by Carlo Sequin).

Report on BIRS Workshop 07w5062 205

. Defining sets of points or line segments in 3-space thatyield different semantically meaningful shadows whenprojected in different directions.

. Studying the creative applications of L-systems tofacilitate student learning of a range of concepts, suchas 2D/3D visualization, recursion, iteration, branch-ing and evolving structures.

. Constructing truncated icosahedrons and six-point oreight-point stars through paper folding.

. Exploring the geometry of the circle through foldingand cutting to form patterns found in Islamic art andarchitecture.

This list of activities continues to be expanded throughnew partnerships and the planning of many excitingprojects.

Partnerships: Emerging projects

The workshop was dynamically structured; the weekwas orchestrated in such a way that the participants hadopportunities to group and regroup in various combina-tions. This promoted discussion of a wider range oftopics from numerous perspectives. Participants sharedtheir own work and their beliefs about the effectivenessof mathematics and arts integration for conceptualizingboth content and process skills inherent in mathematicslearning, understanding and applications. They sharedideas for enhancing mathematics education by suggest-ing a variety of new ways to incorporate arts intomathematics teaching and learning. As a result, manysynergistic partnerships, creative activities, and shortand long term goals emerged during the week. It maywell be that in the future the BIRS workshop would behailed as a landmark in the evolution of an emergingbranch of mathematics with important implications foreducation.

One of the key results of the BIRS workshop was tomake new contacts in addition to nurturing existingcollaborative partnerships. The dynamic structure of theworkshop allowed participants to be involved withmultiple research groups, and to form partnershipsbased on common interests outside the main flow of theworkshop. There was an opportunity for everyone toshare something about themselves and their specific areaof interest related to mathematics education via the arts,mathematics and/or arts. The common interest sharedby the participants brought many learning opportunitiesto the forefront, and provided for the design of short-and long-term individual and collaborative researchgoals. After discussing many possible outcomes, thegroup as a whole converged on the goal of developingpedagogical materials at various levels. In addition,there were several smaller groups of participants whocommitted their energies to projects that were of

a different nature. These smaller groups coalescedaccording to common interests and the clarity of theirvision. Thus the tangible outcomes can be clusteredaround two types of proposals: edited books about theintegration of mathematics and the arts intended forvarious levels and audiences, and creative projects of amore distinctive nature.

Edited books

1. A K-12 collaborative group intends to develop aframework for introducing mathematics and thearts into the elementary, middle and high schoolclassrooms. The group will take into account suchissues as prior learning and life experiences ofstudents and teachers. An important part of thisinitiative is the cross-indexing of arts with mathe-matics resources and vice versa. Such a frameworkmust include information for K-12 teachers onhow to effectively integrate the resources into theirclasses.

2. Two edited collections of activities are being devel-oped; one is intended for college-level liberal artsstudents and the other for post-calculus students.Each collection is expected to be published in theform of a book, probably with an electronicsupplement on CD or a web site. Representativeactivities for each collection were discussed at theWorkshop and a format is being developed thatprospective contributors can use when proposingactivities to the editors.

The teams of editors for these three collections arepreparing some example activities using standardizedformats and will soon announce the call for additionalsolicitations. Most of the workshop participants areexpected to be contributors.

Sample of other creative ideas

1. A survey article about the use of innovative circularrepresentations of rhythmic patterns in music tosupport mathematical concepts from topics arising innumber theory, geometry, abstract algebra, andcombinatorics. The possibility of extending this toa book encouraging thoughtful implementation ofmathematics teaching through music is beingexplored.

2. A collection of examples of mathematics projectspromoting artistic expression. Underdevelopment are projects based on graph embeddingquestions and on surfaces arising as knotcomplements.

3. A variety of collections of materials for broader(public) use ranging from Web-Based collections of

206 M. Alagic

Mathematics and Art with a well-supported galleryof mathematical images, CDs with collections of‘iconic’ images for mathematics and art, theme basedinstallations focused on, for example, L-systemsor phyllotaxis, or possibly even travelling museumexhibitions.

Long range impact

There was a real hope among participants that whatbegan at the BIRS workshop will have long-rangeimpact. Everyone left BIRS with new ideas, andeveryone made a commitment to carry out theirvarious proposed projects. It seems clear that anongoing and concerted effort will have to be nurturedin order to promote and sustain mathematics educa-tion via arts initiatives. A co-ordinated resourcematerials series offering integrated mathematics andarts activities with connections aimed at differentlevels of schooling and different artistic media appearsto be a backbone for the future work in this arena.This will promote a practical implementation ofenriched mathematics teaching via the arts with thepotential to reach many more students througha lively, humanistic grasp of abstract mathematicalconcepts.

The vision of improving learning outcomes in theK-12 mathematics classrooms can only be accomplishedthrough an ongoing dialogue between those with newideas in mathematics and the arts and the classroomteachers, instructional leaders and mathematics educa-tors. The proposed framework for the integration ofresources will help spread these ideas more broadly andsustain the efforts of teaching mathematics by means ofthe arts. Furthermore, BIRS workshop participants feltsustaining and expanding efforts should includea strategy for involving the burgeoning number ofmathematics and art organizations more closely.Participants shared ideas about systematic efforts thatwould bring together the Bridges, ISAMA, ISIS,NEXUS, Katachi, and Math and Design Conferenceconstituencies plus other interested groups for anInternational Congress. The Bridges 2009 conferencein Banff was mentioned as a practical stepping stone for

finalizing plans for this effort, with a Congress for allgroups possibly taking place in 2010.

The participants’ final reflections

Since the specific project proposals to work on needed tobe generated by the participants themselves after theirarrival, at the start of the workshop most peoplethought more in terms of their own forthcomingpublications, projects and interests, and how to connectthem to the interests of others within the general contextof the Mathematics and Art theme rather than collectiveefforts. The diverse backgrounds and experiences of theparticipants required a long period of grouping andregrouping, sharing ideas and then brainstorming inorder to identify what their common interests were andwhich tangible proposals deserved their attention. Thebreadth and intensity of the interests and experiences ofthe workshop participants was positively exploited bythe ongoing presentation of a rich set of examples to thegroup. These same characteristics presented somechallenges as the workshop progressed and participantstried to find a common vision and sense of purpose. Forthe workshop leaders, managing this focused creativityand enthusiasm required an ongoing balancing act offacilitating both the participants’ ideas for activitiessharing, and collaborating on envisioning largerprojects.

In their final reflection exit statements, participantsdescribed their feelings about revitalizing the teaching ofmathematics through the arts using phrases such as‘intellectually invigorating’, ‘enriching experience’, and‘rejuvenated my enthusiasm for teaching’. Furthermore,BIRS workshop participants emphasized that producinghigh quality interdisciplinary resource materials inmathematics education integrating the arts in areassuch as textbooks, workshop activity books, andinternet resources will increase awareness among tea-chers at all levels that mathematics can be taught ina more constructive and engaging manner for bothstudents and teachers. The implementation and realiza-tion of BIRS workshop goals will help accelerate themovement in education relating mathematics and art atall levels from pre-kindergarten to college as well aslife-long learning.

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