Research Interests in Mathematical Music Theory (in construction)y
My interests are deterministic modeling of music analysis and structure and computational music analysis. I am particularly interested in modeling motivic (melodic) structure and analysis of musical compositions through a topological approach.
The motivic analysis of a music composition consists of identifying the short melody, called a motif, that units the composition through its strict repetitions, the so-called imitations, and its variations and transformations which are heard throughout the whole composition. Mainly using group theory, linear algebra and general topology concepts, we construct a (T_0-) topological structure corresponding to the motivic hierarchy of a composition. Our (computationnal topology) implementation (JAVA) , called Melos , can analyze music compositions such as Schumann’s Träumerei from Kinderszenen .
My ongoing interdisciplinary research mainly concerns:
- Concrete applications to a music corpus (recent works include Brahms' Op.51 No.1 and Schumann's Kinderszenen)
- A categorical extension of our model including e.g. continuous functions between two motivic spaces, products of different spaces, natural transformations (gestalt spaces);
- Visualisation of Melos’ multiple outputs in order to show and hear, and to explore mathematics and music results: now in our OM-Melos tool (see Buteau & Vipperman (2008));
- Cognitive experiments on melodic similarity in order to fix some parameters in the topological model.
- Buteau, C., & C. Agnagnostopoulou (2012): Mathematical and Computational Modeling Within a Music Analysis Framework: Motivic Topologies as a Case Study. In Journal of Mathematics and Music, 6 (1),pp. 1-16.
- Buteau, C., with K. Adiloglu, O. Lartillot, C. Anagnostopoulou (2009):Computational Analysis Workshop: Comparing Four Approaches to Melodic Analysis. In Communications in Computer and Information Science Series (37), Klouche, T., Noll, T. (eds), Springer, 247-249.
- Buteau, C. & G. Mazzola (2008): Motivic Analysis Regarding Rudolph Réti: Formalization Within A Mathematical Model. In Journal of Mathematics and Music, 2 (3), pp. 117-134.
- Buteau, C., J. Vipperman (2008): Representations of Motivic Spaces of a Score in OpenMusic. Journal of Mathematics and Music for the special issue on Computations, Vol. 2 (2), pp. 61-79.
- Buteau, C. & Anagnostopoulou, C. (2008): Computational Analysis Workshop Introduction: First movement of Brahms' Op 51 No 1 and an Overview of the Proposed Computational Approaches. Paper presented at Séminaire MaMuX: Mathématiques, musique et relations avec d’autres disciplines for the session on Computational Music Analysis: Ircam, Paris (France), April 2008.
Key Words: Topology, Modeling, Topological Model, T_0-space, Computational Topology, Weight Functions, Modeling Music Concepts, Motivic Analysis of Music, Melodic Similarity, Contour Similarity, Mathematical Music Theory, Formalization of Music Concepts, Rudolph Réti, Automatic Analysis, Motivic Structure of Music, Motivic Evolution Tree, Gestalt Space.
Research Interests in Mathematics Education
My main interests in mathematics education are the use of technology in teaching and learning mathematics, and mathematics teacher education. I’m also interested in developing tools using music for the exploration of mathematics concepts.
I am currently involved in three research projects:
- Together with Dr. Daniel Jarvis (Faculty of Education, Nipissing University) and Zsolt Lavicza (Faculty of Education, University of Cambridge, UK), we have initiated (funded by SSHRC International Opportunities Fund Grants, 2007 - 2010) an international research project about the integration of Computer Algebra Systems (CAS)-based technology in undergraduate mathematics teaching. Our current research project includes three main parts: (1) a comprehensive literature review; (2) a national survey on Canadian mathematicians practices; and (3) two case studies of mathematics departments that have integrated and sustained over time the use of technology in its teaching. Related to our project, a Canadian Mathematical Society (CMS) meeting session was recently organized (CMS Winter 2008 Meeting, Ottawa, December 2008): Technology Use in Post-Secondary Mathematics Instruction.
Together with Dr. Joyce Mgombelo (Faculty of Education, Brock University), we have initiated (internally funded) a project aiming at investigating 'Prospective Secondary Mathematics Teachers Repositioning, with Respect to Mathematics and Mathematics Didactics, by Designing and Implementing Mathematics Learning Objects '. The project addresses the need for a better understanding of how prospective teachers of secondary school mathematics are shaped by their learning experiences during their undergraduate education. In traditional secondary school mathematics teacher education programs, students learn mathematics content in departments of mathematics and mathematics didactics in faculties of education. This division creates problems clearly identified by previous research (Adler & Davis, 2006), such as students missing connections between academic mathematics and mathematics didactics (Fernandez, 2006). In addition, in most programs students spend their first years of education focusing exclusively on acquiring subject matter knowledge and do not focus on teaching as a practice until their final years, which narrows the time devoted to didactics considerably and contributes to the disconnect between mathematics content and didactics.
Our project addresses the division between mathematics content and didactic practices, integrates didactics and mathematics content in prospective teachers' first years of education. It will create new tools to map changes in learning that future researchers can utilize, and will broaden the understanding of how departments of both mathematics and education can create programs that produce highly effective mathematics teachers.
The project reinforces the rich tradition of Brock Department of Mathematics of being actively involved with mathematics teacher education.
Together with Eric Muller (Department of Mathematics, Brock University), we have been mainly working on practitioner reflections about the impact of Brock's Mathematics Integrated with Computers and Applications (MICA) core undergraduate mathematics program on student mathematics learning. Our recent work concerns the student development process (task analysis) of the activity of designing, implementing, and using mathematics " Exploratory and Learning Objects" (done by our students in our 1st and 2nd year courses, MICA I and MICA II ):
" An Exploratory Object is an interactive and dynamic computer-based model or tool that capitalizes on visualization and is developed to explore a mathematical concept or conjecture, or a real-world situation
A Learning Object is an interactive and dynamic computer-based environment that engages a learner through a game or activity and that guides him/her in a stepwise development towards an understanding of a mathematical concept." (Muller, Buteau, Ralph, Mgombelo, in press, p.5)
See Original MICA Student Exploratory and Learning Objects
- Buteau, C., Jarvis, D. & Z. Lavicza (in press). On the Integration of Computer Algebra Systems (CAS) by Canadian Mathematicians: Results of a National Survey. Canadian Journal of Science, Mathematics and Technology Education.
- Buteau, C. & Muller, E. (accepted) Evolving faculty teaching roles in technology intensive undergrad-uate mathematics courses. Refereed book chapter in A. Clark-Wilson, O. Robutti, N. Sinclair (eds): The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development. In Mathematics Education in the Digital Era series.
- Martinovic, D., E. Muller & Buteau, C. (in press). Intelligent partnership with technology: Moving from a math school curriculum to an undergraduate program. In S. Abramovich (Ed.), Computers in K-20 Mathematics Education. Computers in the Schools.
- Buteau, C., Marshall, N., Jarvis, D. H, & Lavicza, Z. (2010). Integrating Computer Algebra Systems in post-secondary mathematics education: Preliminary results of a literature review. In International Journal for Technology in Mathematics Education, 17(2), 57-68.
- Buteau, C. & E. Muller (2010): Student Development Process of Designing and Implementing Exploratory and Learning Objects. In Proceedings of the Sixth Conference of European Research in Mathematics Education (CERME 6), Lyon (France), 2009, 1111-1120.
- Mgombelo, J. & Buteau, C. (2009): Prospective Secondary Mathematics Teachers Repositioning by Designing, Implementing and Testing Learning Objects: A Conceptual Framework. In International Journal of Mathematical Education in Science and Technology, 40 (8), 1051-1068
- Muller, E., Buteau, C., Ralph, B., Mgombelo, J. (2009): Learning mathematics through the design and implementation of Exploratory and Learning Objects. In International Journal for Technology in Mathematics Education , 16 (2), pp. 63-74.
- Muller, E., Buteau, C., Klincsik M., Perjési-Hámori I. & Sárvári C. (2009): Systemic integration of evolving technologies in undergraduate mathematics education and its impact on student retention. In International Journal of Mathematical Education in Science and Technology , 40 (1), pp. 139-155.
- Buteau, C., Etchecopar, P. & Gadanidis, G. (2009): New mathematical and information technology use in post-secondary education. Working group report in the Proceedings for the CMESG (Canadian Mathematics Education Study Group) 2008 annual meeting, Sherbrooke (Canada), May 2008, 65-74.
- Buteau, C., Camilleri, S., Fodil, K., Lacroix, M.-E., Mgombelo, J. (2008): Fantasy Fractions: When a Grade 5 Class Creates Computer Mathematics Games, in The Ontario Mathematics Gazette , Vol. 46 (#3), March 2008, pp. 26-30.
- Buteau, C. & Muller, E. (2006). Evolving technologies integrated into undergraduate mathematics education . In L. H. Son, N. Sinclair, J. B. Lagrange, & C. Hoyles (Eds.), Proceedings for the Seventeenth ICMI Study Conference: Digital Technologies and Mathematics Teaching and Learning: Revisiting the Terrain, Hanoi University of Technology, 3rd-8th December, 2006, Hanoi (Vietnam) (c42)[CD-ROM], 8 pp. See paper supporting website and presentation file
- Muller, E. and C. Buteau (2006): Un nouveau rôle de l'informatique dans la formation initiale des enseignants . In Bednarz, N., Mary, C. (Eds.), "L'enseignement des mathématiques face aux défis de l'école et des communautés", Actes du colloque EMF 2006, Sherbrooke: Éditions du CRP [CD-ROM], 17 pp.
Research Projects with Students
(see current job posting)
A Literature Review on the Use of Computer Algebra Systems in University Mathematics Instruction
Neil Marshall (ongoing since Summer 2008)
Explicit (Mathematical) Motivic Analyses through Melos
John Vipperman (Summer 2006)
Visualizing Melos Output in the software OpenMusic
John Vipperman (Fall 2005 – 2006)
Extending the program Melos (JAVA)
Krishnendu Goswami (Winter 2006)
Teodora Dobrila (Fall 2005)
Designing a “ Mathematics and Music ” Website
Kaan Ersan (Fall2006 - Winter 2007) & Peter Gomes* (Summer 2006 - Winter 2007)
Chris Roy (Winter 2006)
Teodora Dobrila (Fall 2005)
Pascal Comte (Summer 2005)
Amanjot Toor** (2007-08): Gender Equity in Undergraduate Mathematics Curricula, Honour's Thesis, Brock University (Canada).
Jennifer Corbett** (2006-07): Recreational Math Clubs for Elementary and Secondary Schools, Honour's Thesis, Brock University (Canada).
Sarah Camillieri ** (2006-07): Fantasy Fractions Learning Object: A Collaborative Grade 5 Class Project, Honour's Thesis, Brock University (Canada).
Denis Poulin*** (2004-05): Topologie musicale, Final BSc Project, Université Laval, Québec (Canada).
Footnotes: * under the supervision of Michael Laurence (Multimedia Production & Innovation Centre, Brock University) ** co-supervision with Dr. Joyce Mgombelo (Faculty of Education, Brock University) *** co-supervision with Dr. Charles Cassidy (Université Laval)
Department of Mathematics & Statistics