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Music as a "language" of expression for understanding multiplication in grade three Moriarty, Corry Suzanne 2010

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MUSIC AS A “LANGUAGE” OF EXPRESSION FOR UNDERSTANDING MULTIPLICATION IN GRADE THREE by Corry Suzanne Moriarty B.A., Augustana University College, 1994 Music Diploma, Selkirk College, 1997 B.Ed., University of Calgary, 2001  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in The Faculty of Graduate Studies (Music Education) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2010 © Corry Suzanne Moriarty, 2010  Abstract This study explored how music could be used as a language of expression for understanding multiplication for grade three children. Using a/r/tography as a research methodology, a class of grade three students, their teacher, and I worked together on a coemergent inquiry project to create musical compositions that conveyed meanings about multiplication to the listener. The design of this a/r/tographic inquiry involves the components of the cyclical Inquiry Process used by the International Baccalaureate program, as well as the Reggio Emilia’s approach, known as Progettazione, which involves emergent, child centered project work. Through my research I offer credence to the area of interdisciplinary studies at the early childhood level. Such studies support the development of new notions and forms of music instruction—created by and for children—that advance both music and related learning. As is evident in my account, I demonstrated how music (1) can be taught in and of itself, (2) can be thought of as a medium (i.e., a “language,” as in the Reggio definition) for the expression of concepts in multiplication, (3) is instrumental in fostering knowledge of both musical and mathematical concepts, and (4) when linked with mathematics, can show learning transfer and access related learning between the two disciplines. This study contributes to on-going scholarly conversations concerning the present structure and role of the music teacher (and other “specialists”) in our schools.  ii  Table of Contents ABSTRACT........................................................................................................................ ii TABLE OF CONTENTS .................................................................................................. iii LIST OF TABLES ..............................................................................................................v LIST OF FIGURES .......................................................................................................... vi ACKNOWLEDGEMENTS ............................................................................................. vii DEDICATION ................................................................................................................ viii 1 INTRODUCTION............................................................................................................1 1.1 CO-EMERGENCE .........................................................................................................2 1.2 WHY MUSIC AND MATH?............................................................................................8 1.3 THEORETICAL FRAMEWORK ........................................................................................9 1.4 THE REGGIO APPROACH ...........................................................................................10 1.5 INTERNATIONAL BACCALAUREATE PRIMARY YEARS PROGRAM (IBOPYP) ..............14 1.6 THE PAST/THE PROJECT METHOD/A BRIEF OVERVIEW OF THE HISTORY OF MUSIC EDUCATION .............................................................................................................17 1.7 WHY THIS MATTERS TO ME – THE MOTIVATION BEHIND THIS STUDY ..........................20 2 LITERATURE REVIEW ..............................................................................................24 2.1 2.2  THEORY AND QUALITATIVE RESEARCH IN REGGIO AND THE IBO...............................24 TWO, FOUR, SIX, EIGHT, WE JUST WANT TO INTEGRATE! INTEGRATED CURRICULUM = TRANS/CROSS/INTER/MULTI-MODAL- DISCIPLINARY EDUCATION .........................................26 2.3 QUANTITATIVE RESEARCH STUDIES SUPPORTING OR REFUTING COGNITIVE TRANSFER IN MUSIC RELATED TRANS-DISCIPLINARY PROJECT WORK/INTEGRATED CURRICULUM .............28 2.3.1 Music listening and social sciences = supportive research ................................29 2.3.2 Music listening and memory = supportive research...........................................31 2.3.3 Music listening (and instruction) and spatial-temporal reasoning = supportive research ......................................................................................................................32 2.3.4 Music and general math ability = some support ...............................................38 2.3.5 Music and high school biology and math = some support.................................39 2.3.6 Music and reading ability = no support............................................................40 2.4 QUALITATIVE RESEARCH ON MUSIC RELATED INTEGRATED CURRICULUM ...................41 2.5 SUMMARY OF THE RELATED RESEARCH AND HOW IT RELATES TO THE PRESENT INQUIRY 46 3 METHODOLOGICAL CONSIDERATIONS, DESIGN, AND TEACHING AND LEARNING PROCESSES ........................................................................................49 3.1 OUR CONTEXT ..........................................................................................................49 3.2 IT’S MESSY! A/R/TOGRAPHY = LIVING INQUIRY + PERFORMATIVE INQUIRY ................52 3.2.1 Living inquiry .....................................................................................................55 3.2.2 Performative inquiry...........................................................................................56 3.3 PROCEDURE .............................................................................................................59 iii  3.4 FRONT LOADING/PRE-ASSESSMENT “WHAT DO WE KNOW?”.......................................62 4 THE INQUIRY CYCLE—THE NARRATIVE UNFOLDS ........................................72 4.1 ASK .........................................................................................................................73 4.2 INVESTIGATE ...........................................................................................................78 4.3 CREATE ...................................................................................................................91 4.4 DISCUSS ................................................................................................................ 125 4.5 REFLECT ............................................................................................................... 140 4.6 THE STUDENT’S COMPOSITIONS.............................................................................. 155 5 SYNTHESIS ................................................................................................................. 175 5.1 THE ART FORM INFLUENCES THE RESEARCH INFLUENCES THE ART FORM ................. 175 5.2 THE PUDDIN’ ......................................................................................................... 177 5.2 REFLECTIONS ON THE INQUIRY CYCLE ................................................................... 180 5.2.1 Front Loading/Ask: What did we know (about the research question)?........... 180 5.2.2 Investigate: What did we want to know (about the research question)? ........... 181 5.2.3 Create: How can we express what we know (about the research question)? .... 183 5.2.4 Discuss: What did we learn (about the research question)?............................. 188 5.2.5 Reflect: How did we learn what we learned (about the research question)?..... 190 5.4 VISION FOR THE FUTURE ........................................................................................ 194 REFERENCES................................................................................................................ 197 ENDNOTES .................................................................................................................... 206 APPENDIX A ETHICS REVIEW BOARD PERMISSION ........................................ 208  iv  List of Tables Table 3.1 Comparative Model of the Inquiry Cycle and the Reggio Project Approach......... 61 Table 4.1 Common Multiples of 12 Comparison in number groups ..................................... 87 Table 4.2 Music Communication Strategies for Multiplication ............................................ 95 Table 4.3 Music Communication Strategies for Multiplication (a review for the reader)..................................................................................... 156 Table 4.4 Summary of application: Music Communication Strategies ............................... 156 Table 5.1 Music Communication Strategies for Multiplication (a review for the reader)...................................................................................... 183  v  List of Figures Figure 1.1 Body (from “enactivism” from Figure 1.3) ........................................................... 4 Figure 1.2 Mind (from “enactivism” from Figure 1.3) ........................................................... 4 Figure 1.3 “enactivism” (acrylic, pen and ink, gravel) ........................................................... 6 Figure 1.4 Theoretical framework ......................................................................................... 9 Figure 1.5 Noticed by the shark - from the series: Little Kelp (acrylic, pencil crayon, construction paper, pen and ink) ........................................................................ 23 Figure 3.1 My “living system” of this research project ....................................................... 52 Figure 3.2 The Inquiry Cycle .............................................................................................. 60 Figure 3.3 Music Vocabulary Self-assessment..................................................................... 70 Figure 4.1 Session Summary in Inquiry Cycle Design ......................................................... 72 Figure 4.2 Subdivision of a Whole Note.............................................................................. 84 Figure 4.3 Inquiry Cycles in Inquiry Cycles ........................................................................ 92 Figure 4.4 Subdivision of a Whole Note............................................................................ 114 Figure 5.1 mind/body reunion – [a portion of “enactivism” from Figure 1.3]..................... 175 Figure 5.2 The Inquiry Cycle deconstructed ...................................................................... 180  vi  Acknowledgements You can’t really tattoo an acknowledgement section on your newborn baby. I suppose that’s what birth announcements are for. And although I would never presume to suggest that undertaking this Master’s degree was as momentous or arduous as giving birth, it certainly has been the most strenuous achievement of my life. I owe the successful completion of this thesis to the loving kindness of my family and friends, my advisor, and my thesis committee: My Mom - who is my greatest enthusiast - a never ending, unconditional source of emotional, social, spiritual, psychological, aesthetic, fashion, home-organizing, editorial, and intellectual support. I live this path because of her. My Dad - who loves, spoils, and protects me no matter how eccentric I become. He knows who I am, and I only hope I can say the same about him. I admire his thoughtfulness: often unspoken and inwardly insightful. Jil - who practiced while I theorized. She mirrored and shared my academic experience in her lived experience of home-schooling her boys. In addition to endless conversations about the meaning of life, from her I learned that spaghetti fried in butter and garlic with a glass of wine can cure almost anything. Ken - who makes academia “normal,” and combines it with nature. Over an open fire he told me I was best teacher in the world. Tracy - who reminds me to be humble and clear, and whose life, although the opposite of my own, is a constant source of motivation. Artie - in whose footsteps I walk. His gentle humour, perceived simplicity, and lack of judgment ease me. Brenna, Jordan, Graeme, Finley, Lila, Ben, and one more - who are the children I hope most to impassion with the joy of living, the love of learning, the need for sharing, the zest of experiencing, and a respect for diversity - my little loves. Alan - who shares love, comfort, and balance with me. Anne-Marie - whose intellect and depth inspire me. Beth and Dianna - who walked a while with me. Lynn – who, with flare, gave me performative inquiry and encouraged academic precision. Scott – who taught me to appreciate rigor. And Peter - whose academic versatility and personable style offered me independence, freedom of expression, and advancement of ideas. He expanded my “vocabulary,” listened to my singing, and made me fluent in a more than one language. Thanks everybody!   vii  Dedication  For those guys  ⬆  viii  1  Introduction “We could have one song times another song,” suggests Katy. “But one times one is one, so that wouldn’t make two songs. One PLUS one is two,”  replies Kalen. “How about two songs times two songs is four songs?” offers Lara. “How can we do that?” Katy replies. I stand in the middle of the noisy music room zooming in on this conversation with the video camera, while trying to attend to the other children moving about the class. “Can we go in the closet?” asks Paige. “We’re trying to make it harder.” Chan tell me. Distracted by the video camera, in one breath I manage to answer Chan and Paige with, “Sure. Okay,” and then holler to the rest of the class over the rising din, “There’s paper on the table if you need it!” “Ms. Moriarty, it’s so loud in here I can’t even hear my own music,” complains Christian, looking over at Sarah and Carly whacking away on a djembe and a xylophone. “Maybe you boys should head outside to find a quieter space of your own?” As the boys gather up as many hand drums as they can carry out the door, I hear Ethan’s favorite piano piece, the “Mission Impossible” theme song, thundering out above the clutter of chimes, percussion instruments, and recorders. Just then Kira and Ginny run by. They are practicing their “ending exit” for their performance, and I wonder, “do they even have a beginning or any part of a song yet?”  1  Joshua wanders by with his recorder in his mouth, playing a series of notes going from a whole note, to two half notes, to four quarter notes, to eight eighth notes, and back up again. “What part of multiplication are you trying to express?” I ask him. Joshua gazes out the window. He doesn’t answer me for a long moment. Finally, he says, “Well, actually it’s not really multiplication that I am playing, it’s … equivalent fractions, I think.” Jenny, the class teacher, walks in just in time to hear Joshua’s pronouncement. We look at each other and grin.  1.1  Co-emergence The typography of the borderlands is simultaneously the suturing space of multiple oppressions and the potentially liberatory space through which to migrate toward a new subject position. The geographical trope is at once psychological, physical, metaphysical, and spiritual, since it functions as a space where cultures conflict, contest, and reconstitute one another. Smith, 1993, p. 169 A/r/tographers re-represent their questions, practices, emergent understandings, and creative analytic texts and compositions as they integrate knowing, doing, and making through texts and/or performances that convey meaning rather than facts. Springgay, S., Irwin, R.L., & Wilson Kind, S., 2005, p. 903 In the educational context, co-emergence is the process through which curriculum  content, methods, and conclusions emerge as project participants and the learning environment contribute to the generated learning process (Davis, Sumara, & Kieren, 1996; Maturana, 1987; Reid, 2010). In what I call co-emergent inquiry, a project begins with group discussions between students and adults (teachers, researcher, parents, volunteers) that determine the direction or focus of the project. As these discussions and interactions unfold,  2  more, different, and elaborated ideas are often generated. In the case of the inquiry in this thesis, we attempted to use music elements to communicate/express grade three students’ understandings of the content in a multiplication-music integrated curriculum project. The grade three students, their classroom teacher, and I were active participants in this artmaking, or “music-ing” (Elliott, 1995) process, thus we cooperated as the inquiry emerged. Davis, Sumara, and Kieren in their article Cognition, co-emergence, curriculum (1996) described emergence, and co-emergence, as the creating or unfolding of worlds. It can be seen as the emergence of community and the emergence of understanding. In the case of this research study, through co-emergent inquiry we were creating a place where participants and their environments could co-evolve and were co-implicated in the context of the inquiry (Davis et al., 1996). In this way, Aristotelian knowing/researching (theoria), doing/teaching (praxis), and making/music-ing (poesis) were honored as in keeping with a/r/tographic living inquiry and performative inquiry (Gouzouasis, 2007; Irwin & de Cossen, 2004; Fels & Belliveau, 2008). In a/r/tographic living inquiry, participants are invited to the borderlands, to the edge of chaos (Fels & Stothers, 1996), to experience together re/creating, re/searching, and re/learning “ways of understanding, appreciating, and representing the world … in an elegance of flow between intellect, feeling, and practice” (Irwin & de Cossen, 2004, p. 29). For a/r/tographers who dwell in the borderlands, in the theoria, praxis and poesis of art-making/music-ing, a third aesthetic space is created that allows for expression outside of Cartesian dualism and into a pluralistic, holistic, evolving dance (Gouzouasis, 2006; Irwin & de Cossen, 2004; Fels, 2008). Just as trans-disciplinary study breaks borders of either/or, so  3  too does a/r/tography consider both/and in encompassing co-emergent inquiry, living inquiry, and performative inquiry (Palmer, 1998; Tarr, 2001; Fels, 1997).  Figure 1.1 Body (from “enactivism” from Figure 1.3) Figure 1.2 Mind (from “enactivism” from Figure 1.3) © Corry Moriarty 2008  © Corry Moriarty 2008  In these interlingual acts, there is at once an acceptance of playing with particular categories and a refusal to be aligned with any one category. Where two would be inclined to dialogic opposition, a third space offers a point of convergence – yet respect for divergence – where differences and similarities are woven together. (Irwin & de Cossen, 2004, p. 29) A/r/tography, co-emergent inquiry, and performative inquiry are at the heart of a wider education philosophy referred to as enactivism (Maturana, 1987; Reid, 2010; Sumara & Davis, 1997; Davis et al., 1996), Davis, 2004; Doll, 2010). Enactivists are interested in events of adaptation and emergence within a complex system “in those instances when coherent collectives arise through the co-specifying activities of individuals” (Davis & Simmt, 2003, p. 140). On this, Davis, Sumara, and Kieren (1996) said, “enactivism suggests that the 4  teacher and the students are working on a common project – the simultaneous bringing forth of themselves and the world – even if their respective interpretations of their actions and experiences differ…the mere act of teaching contributes to the dynamic, unpredictable, complex unfolding of an as-yet unrealized world” (p. 4). The present research project was an exercise in co-emergence, when applied to the teaching/learning context, in that when the initial question was posed to the class, there was no predetermined inquiry conclusion. It was not guided inquiry using a constructivist methodology. This research project asked the people involved to work together with the environment and each other to explore multiplication with musical tools. The only prescriptive element of this project was the thesis question itself: Can music be used as a language of expression for understanding multiplication in a grade three class? If so, how? If not, why not? What came out of the students’ and teachers’ time together was completely open and evolved as the weeks of the project went by. While this project was not student-generated in its inception, it was fully co-emergent throughout its formation and summation. The key with co-emergent curricula is that all participants are considered equal throughout the project; therefore, it was imperative that the adults refrained from dominating or manipulating the creative process, yet remaining engaged as facilitators in the inquiry process (Davies et al, 1996). In this way, all participants in the inquiry were considered artists-teachers-researchers (see section 3.2).  5  Figure 1.3 “enactivism” (acrylic, pen and ink, gravel) © Corry Moriarty 2008  If I have a living system ... then this living system is in a medium with which it interacts. Its dynamics of state result in interactions with the medium, and the dynamics of state within the medium result in interactions with the living system. What happens in interaction? Since this is a structure determined system ... the medium triggers a change of state in the system, and the system triggers a change of state in the medium. What change of state? One of those, which is permitted by the structure of the system. Maturana, 1987, p. 75  6  The perceived messiness of the theoretical and methodological background of this thesis are accounted for in the very nature of a/r/tography itself. The painting (figure 1.3) is my visual re-representation of enactivism, but can also be seen to re-represent the complex nature of this thesis. It is enactivism, it is emergence, it is multi-dimensional, it is embodiment, it is organicism, it is the borderlands, it is the space created when the work of artists, researchers, and teachers collide, and the design of a/r/tographic processes unfold. This research project as a whole, with its many parts, could be seen as a complex living system. And like Maturana (1987), Fels (1996), Davis et al. (1996), and others (Reid, 2010; Doll, 2010) suggest, my “living system,” which is this study, is inside a “medium” of a/r/t/ography, my “dynamics of state” are the research components (i.e., the theoretical background, the research methodology, the research question, and the research design). The dynamics of state interact with the medium, and vice versa, which affect the system itself, so that we end up with an evolving emergent system with interwoven, rhizomatic connections, that create openings for learning, interpretation, tension, action, and inquiry. When applying these scientific thoughts on living systems to an educational context, especially an a/r/tographic one, I find Lynn Fels words about the edge of chaos in her work with performative inquiry help to explain the flowing, interactive nature of my complex a/r/tographic experience. “The edge of chaos is where life dances to action. And it is in this space, on the edge of chaos, we suggest, that Academic Performance executes its glorious tangos” (Fels & Stothers, 1996, p. 257). My study explored the ways in which children attempted to communicate their understanding of multiplication using musical devices; e.g., rhythm, melody, singing and song lyrics, instruments, and other music elements. Since the students’ musical “language”  7  differed from one another and only some common music “vocabulary” previously existed within the class, for assessment purposes oral artist statements/reflections accompanied the summative musical compositions that the children created about multiplication. This interdisciplinary study between music and mathematics explored the conditions under which music learning processes and music, itself, served to communicate knowledge of non-musical concepts with or without the use of verbal language to support the musical communication. My research explored how learning processes transfer between music and mathematics (and vice versa). It also gained some insights into indicators that demonstrated the learning transfer (i.e., images, discourse, language, interaction). In chapter two of this thesis, I describe the existing structures and functions of the disciplines of music education, math education, and interdisciplinary education. In the third and fourth chapters of this thesis I describe my research methodology and findings in detail. In the final chapter, I discuss the implications for music inclusion in the interdisciplinary context and scholarly significance in terms of what it offers teachers and students.  1.2  Why music and math? In trans-disciplinary projects there are many combinations for subject matter that  seems to go together well. Some obvious links are: Language Arts and Music, Language Arts and Drama, Art and Music, Art and Science, Dance and Physical Education, Dance and Music, Social Studies and Drama, Music and Mathematics. In actuality there are ways to link almost every subject together in meaningful or superficial (i.e., surface) ways. I chose to combine mathematics with music in this research study because the two have been linked previously in a number of ways, but never in this way. I clearly indicate in chapter two of this thesis, music listening has been used in a number of psychological studies  8  to influence/enhance mathematical areas of the brain with particular emphasis on spatial awareness (Ivnov & Geake, 2003; Cockerton, Moore, & Norman, 1997; Rauscher, Shaw, & Ky, 1993; Rauscher, Shaw, & Ky, 1995; Rauscher, Shaw, Levine, Wright, Dennis, & Newcomb, 1997; Catterall, Chapleau, & Iwanaga, 1999). Also, music and math have joined forces in a number of educational methodology books for fraction work or math fact memorization with steady beat (Caudle & Caudle, 2010; Banfill, 2010; Nolan, 2009). However, not one book, article, or research study has examined how music--the elements of music, the vocabulary of music--can be used to express understanding, how it can serve as a language for conceptual understanding, as opposed to merely enhancing the learning of another subject.  1.3  Theoretical framework The theoretical framework for this study drew from two main philosophical  frameworks.  Reggio Emilia Educational Philosophy (The Reggio Approach)  ⇔  International Baccalaureate Primary Years Program (IBOPYP)  Figure 1.4 Theoretical framework To support my position, in chapter two I included a review of research studies that examined how music and other areas of human interest have been used together to demonstrate that music can enhance or influence learning and development in the social sciences. In this literature review, what the research on music in interdisciplinary studies failed to demonstrate is how music can be used to express learning in other subjects, or how  9  music as a discipline can be experienced in ways other than listening when working with material from other subjects. The two educational philosophies/approaches that influenced my study--Reggio Approach and International Baccalaureate Primary Years Program (IBOPYP)--both hold in common the premise that music, and all of the arts, can be used as a means of expression, a “language,” for conceptual understanding rather than for merely emotive expression in transdisciplinary projects/units of inquiry. I considered the philosophy and organizational structure of both the Reggio Approach and the International Baccalaureate (IB) curriculum that puts great importance on finding authentic links between subjects and emphasizes trans-disciplinary study.  1.4  The Reggio approach No way. The hundred is there. The child is made of one hundred. The child has a hundred languages a hundred hands a hundred thoughts a hundred ways of thinking of playing, of speaking. A hundred always a hundred ways of listening of marveling of loving a hundred joys for singing and understanding a hundred worlds to discover a hundred worlds to invent a hundred worlds to dream. The child has a hundred languages (and a hundred hundred hundred more)  10  but they steal ninety-nine. The school and the culture Separate the head from the body. They tell the child: to think without hands to do without head to listen and not to speak to understand without joy to love and to marvel only at Easter and Christmas. They tell the child: to discover the world already there and of the hundred they steal ninety-nine. They tell the child: that work and play reality and fantasy science and imagination sky and earth reason and dream are things that do not belong together. And thus they tell the child that the hundred is not there. The child says: No way. The hundred is there. Loris Malaguzzi (in Edwards, Gandini, & Forman, 1998, p. 3) In the Reggio Emilia philosophy of education, music is considered one of the hundred languages of children (Edwards et al., 1998). Rather than subscribing to traditional frameworks of understanding on the meaning of language, Reggio educators have embraced a new definition of “language” to mean the many ways in which small children communicate their understanding of a concept.i This non-verbal, non-discursive communication (e.g., sculpture, painting, wire work, collage, drama play, use of metaphor) might not use a vocabulary that is interpretable to anyone other than the child who is “speaking” and perhaps those who know them well. Rather it serves as a method of “speaking” (i.e., communication  11  and representation) that clearly demonstrates implicit meaning and articulation of conceptual understanding of a given topic, regardless of whom else speaks their language (Edwards et al., 1998). In the early childhood centers of Reggio Emilia and in Reggio inspired schools around the world, the visual arts are emphasized in the hundred languages for children’s expression (Edwards et al., 1998). There are many articles and books written about the atelierista (art teacher) and the atelier (art studio) in these schools, but there is very little mention of music or music use in the research or available documentation. While Reggio schools have music rooms, and have engaged in sound projects, music is not broken down into a separate “vocabulary” like the elements of visual art are delineated. Although such programs exist, there is a hole in educational research supporting the implementation of Reggio inspired music programs and music as a language of expression similar to the extent to which visual art is used for communication. This thesis will serve to provide research in this area. Similar to co-emergent inquiry, another important aspect of the Reggio Emilia Approach is its use of student-generated projects as the main body of the curriculum. This flexible curriculum is called progettazione. Their philosophy is similar to The Project Method spearheaded by Dewey at the turn of the 20th century (Costa, Grasselli & Morrow, 2001). The translators in the book Making Learning Visible define progettazione clearly: The use of the noun progettazione in the educational context, at least in Reggio Emilia, is in opposition to programmione, which implies predefined curricula, programs, stages, and so on. The concept of progettazzione thus implies a more global and flexible approach in which initial hypotheses are made about classroom work (as well as about staff development and relationships with parents), but are subject to modifications and changes of direction as the actual work progresses. (p. 17)  12  Traditionally, music instruction in the school context has not allowed much room for student choice or opinion. Music teachers teach the skills necessary to master objectives that pertain directly to music theory and instrumental or vocal technique. More modern music teachers are implementing creative strategies and tactics, as well as music games, in order to teach musical proficiency. While this is more fun for children, rarely are music teachers looking at music as a means for children to express their understandings of the world around them. Tiaziana Filippini’s metaphor (as cited in Rankin, 1998) of the ball is a good way to describe the present co-emergent research project. Our expectations of the child must be very flexible and varied. We must be able to be amazed and to enjoy – like the children often do. We must be able to catch the ball that children throw at us, and toss it back to them in a way that makes the children want to continue the game with us, developing, perhaps, other games as we go along. (p. 217) Lella Gandini (2004) summarizes some of the main Reggio principles as: • • • • • • • • • • •  The Image of the Child (as capable, independent and trustworthy) The Three Subjects of Education: Children, Parents, and Teachers The Role of Parents (involved, interested, welcomed) The Role of Space: An Amiable School The Value of Relationships and Interactions of Children in Small Groups The Role of Time and the Importance of Continuity Cooperation and Collaboration as the Backbone of the System Teachers and Children as Partners in Learning Flexible Planning vs. Curriculum and projects (Progettazione) The Power of Documentation The Many Languages of Children: Atelierista (Artist) and Atelier (Studio) (p.15-23)  Forman and Fyfe (1998), contributors to the book the hundred languages of children, explain that in Reggio schools “a language is more than a set of symbols” (p. 249). They elaborate that if syntax is created—an order to the symbols that convey meaning—then a language is born.ii “It is the nature of the relation among the symbols that converts the medium into a message; and it is the presence of an intended message that motivates children  13  to negotiate shared meaning and to co-construct knowledge” (Foreman & Fyfe, 1998, p. 249).iii  1.5  International Baccalaureate Primary Years Program (IBOPYP) The International Baccalaureate (IB) aims to develop inquiring, knowledgeable and caring young people who help to create a better and more peaceful world through intercultural understanding and respect. To this end the organization works with schools, governments and international organizations to develop challenging programmes of international education and rigorous assessment. These programmes encourage students across the world to become active, compassionate and lifelong learners who understand that other people, with their differences, can also be right. (International Baccalaureate Organization, 2010) There are three programs in the International Baccalaureate (IB). The Primary Years  Program (PYP) for students age three to twelve, the Middle Years Program (MYP) for students age eleven to sixteen, and the Diploma Program (DP) for students age sixteen to nineteen. For the purposes of this thesis, only the PYP will be discussed since I worked with a grade three class. Schools in North America consider grade three to be the last year of early childhood, with grade four moving into middle years. IBPYP, however, uses developmentally appropriate primary years programming and educational philosophy from Junior Kindergarten all the way to grade five or six. To be truly educated, a student must also make connections across the disciplines, discover ways to integrate the separate subjects, and ultimately relate what they learn to life. Boyer, 1995, p. 82 The IB Primary Years Program (IBPYP) introduces six trans-disciplinary themes in each consecutive year of the program that are taught through Units of Inquiry. These globally significance themes “create a trans-disciplinary framework that allows students to ‘step up’ 14  beyond the confines of learning within subject areas” (International Baccalaureate Organization, 2010). The six trans-disciplinary themes are: - Who we are - Where we are in place and time - How we express ourselves - How the world works - How we organize ourselves - Sharing the planet Around these themes, PYP classroom teachers and specialists collaborate in using the PYP unit planner template to build guided inquiry projects. These units are detailed investigations guided by the teachers’ research and planning with students critically thinking in active participation. They differ from co-emergent inquiry in that they are guided inquiries, as opposed to open-ended inquiries. Teachers in the PYP are deliberately constructivist in their teaching, rather than enactivist, meaning that they have benchmarks for the students to scaffold to during units of inquiry. The summative assessment piece, along with the central idea and lines of inquiry are well established before the students begin the inquiry. “Inquiry, as the leading but not exclusive pedagogical approach of the PYP, is recognized as being intimately connected with the development of students’ comprehension of the world” (Primary Years Programme, 2009, p. 15). According to the PYP Arts Scope and Sequence (2009) document, there are three main ways that specialist teachers can engage in collaboration with class teachers, and other specialist teachers, in trans-disciplinary units of inquiry. 1. They can directly develop a Unit of Inquiry with their subject serving as a main area for inquiry. 2. They can do preliminary or follow-up work for a Unit of Inquiry that supports a unit 15  of inquiry. 3. They can do stand-alone inquiry that is not a part of the class Units of Inquiry. Another fundamental feature of the IBPYP program is the Learner profile. The Learner Profile is a set of guiding ethics that informs the ethos and human interaction in an IB school. The vocabulary of the learner profile is used in daily in lessons, posted on the wall in every classroom, and often referred to during discipline issues and classroom management. The learner profile permeates all areas of the lived curriculum as teachers and students engage in holistic inquiry based learning experiences. Inquirers: They develop their natural curiosity. They acquire the skills necessary to conduct inquiry and research and show independence in learning. They actively enjoy learning and this love of learning will be sustained throughout their lives. Knowledgeable: They explore concepts, ideas and issues that have local and global significance. In so doing, they acquire in-depth knowledge and develop understanding across a broad and balanced range of disciplines. Thinkers: They exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems, and make reasoned, ethical decisions. Communicators: They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication. They work effectively and willingly in collaboration with others. Principled: They act with integrity and honesty, with a strong sense of fairness, justice and respect for the dignity of the individual, groups and communities. They take responsibility for their own actions and the consequences that accompany them. Open-minded: They understand and appreciate their own cultures and personal histories, and are open to the perspectives, values and traditions of other individuals and communities. They are accustomed to seeking and evaluating a range of points of view, and are willing to grow from the experience.  16  Caring: They show empathy, compassion and respect towards the needs and feelings of others. They have a personal commitment to service, and act to make a positive difference to the lives of others and to the environment. Risk-takers: They approach unfamiliar situations and uncertainty with courage and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are brave and articulate in defending their beliefs. Balanced: They understand the importance of intellectual, physical and emotional balance to achieve personal well-being for themselves and others. Reflective: They give thoughtful consideration to their own learning and experience. They are able to assess and understand their strengths and limitations in order to support their learning and personal development. Primary years Programme, 2009, p. 9  1.6 The past/the project method/a brief overview of the history of music education I tiptoe over the well-substantiated arguments, words of scholars more respected than myself. I reach down and pick one little idea the ripest strawberry to pluck and hold to the light.  In 1840 Ralph Waldo Emerson wrote: Our education, like other institutions and formularies of the present age is poor. It has no breadth. It speaks in a dialect. As we construe it, education refers to a narrow circle of experiences, powers, and literature of its own. Education should be as broad as man and demonstrating whatever elements are in him … Today’s education does none of this. We confront the vanity of our education when we look at its result: society. (as cited in Young, 1990, p. 5)  17  Educational philosophers, academics, and critics such as John Dewey (1916), William H. Kilpatrick (1918), and now Eliot Eisner (2002), Madeleine Grumet (1996), Howard Gardner (1993), and Charles Ungerleider (2003) reiterate similar complaints. In the 19th century, Horace Mann had a strong vision of the “common school’s” social function in building an orderly and productive society and was the driving force behind the factory model school (Young, 1990). John Dewey fundamentally disagreed with this approach to education. Consequently in 1896, he began to experiment with progressive education and opened a “laboratory school” through the University of Chicago as a response to Mann’s factory model education that industrialized Americans were quickly being seduced toward (Ungerleider, 2003). Dewey’s approach was entirely child centered with focus on individual student interest and learning needs. William H. Kilpatrick (1918) then elaborated on Dewey’s philosophies and devised a curriculum that was called “The Project Method.” Despite the initial acceptance of “The Project Method” in the early part of the 20th century by educators and educationalists, its application petered out after WWII as Americans put societal concerns ahead of individualistic needs due to the Cold War and competition with Russia. This meant educational reform toward academic achievement and excellence and less emphasis on meaning, relevance, and student interest (Young, 1990). It was during this time that “aesthetic education” developed as a result of this shift in educational values and philosophy (McCarthy & Goble, 2002). Referring to education in the arts, “aesthetic education” is a “branch of philosophy that is concerned with the analysis of concepts and the solutions of problem that arise when one contemplates aesthetic objects” (Elliott, 1991, p. 50). In the case of music education in particular, Music Education as Aesthetic Education (MEAE), spearheaded by Bennett Reimer, has been the sole theoretical  18  reference for practitioners in the field of music education until the 1990’s, despite obvious problems with the theory (Elliott, 1991). Philosopher Susanne Langer influenced many of Reimer’s theories on arts education (Langer, 1951; 1957). Critics in music education, including David Elliott (1991) and Wayne Bowman (2006) assert that the modern concept of aesthetics arose within the 18th century European nobility and advocate elitist views on what constitutes “fine, high art.” Elliott reasons that the main flaw of Reimer’s “aesthetic education” is that it fails to allow music to be “a rich and complex human phenomenon” (Elliott, 1991, p. 51). He says that the concept behind the MEAE is an aesthetic philosophy rather than an artistic philosophy and that it excludes musical styles and forms of musical practice that do not conform to the MEAE’s ideas of fine, high art. Paradoxically, Reimer’s influence on music as I have come to know and teach it today have stemmed from a double sided philosophy, that of philosophy versus advocacy for music education as aesthetic education. In a well-written critique of Reimer’s latest edition of A Philosophy of Music Education – Advancing the Vision, Wayne Bowman (2006) makes the following point. Regardless of their philosophical validity, points of view, which do not lend readily apparent support to such non-philosophical functions as advocacy as instructional method, are expendable. Since, for instance, the existential perspective – despite ‘powerful insights’ – does not lend itself ‘directly or abundantly to problems of mass education,’ it is not a dependable base for a philosophy of music education. Philosophical perspectives, then, are not to be examined on their philosophical merits when ‘selecting’ one upon which to build a music education philosophy. Rather, the criterion is their ease of applicability to mass education. (p. 79). Thus, in an effort toward mass education and the factory model of education, there was a shift away from aesthetic music education and toward praxial music education. The present research suggests that times have changed again as we enter an age where praxial  19  experiences and aesthetic experiences are of great concern to today’s students. For too long we have made music-ing a scientific experience rather than something meaningful – something not only to be created, but to also be felt and understood. Although valid critiques of Music Education as Aesthetic Education (MEAE) philosophy exist and varied and alternative approaches to music pedagogy also exist (e.g., Orff, Kodaly, Delcoze, Gordon, the Manhattanville Music Curriculum Project, Kindermusic, Music for Children), the organizational structure of music education in North American public schools is still based on aesthetic music principles and remains largely unchanged from what is was in the early 20th century. Despite the solid curricular models (e.g., Reggio approach, the International Baccalaureate curriculum, and even the Project Method itself), extensive educational theory (Dewey, 1916; Kilpatrick, 1918; Eisner, 2002; Grumet, 1996; Gardner, 1993), and research studies (Youm, 2007; Parsons, 2004; Gouzouasis, Guhn, & Kishor, 2007; Schmithorst & Holland, 2004; Johnson & Memmott, 2006) that clearly demonstrate methods for and the benefits of alternative practices in music and art education (e.g., trans/cross/inter-disciplinary project work) in elementary schools, the fact is that educators in schools are not using what we know (Perkins, 1992).  1.7  Why this matters to me – the motivation behind this study Eleven years ago, at the beginning of my journey as a teacher, I believed, as I still do,  that children are capable, thoughtful, trustworthy, creative, fun, adventurous, trusting, and excited about life and learning. As a musician and future fine arts teacher my personality and past music training and artistic experiences allowed for a liberal, non-traditional view of human potential for learning and discovery. “Artistic training at least gave me an approach to  20  teaching that wasn’t overly structured – perhaps freer with more potential for irony, humour, or pleasure” (Vecchi, 1998, p.140). Fine arts teachers often possess this liberal, creative view of education. As a new teacher, I was excited about the joyful and meaningful work I was about to embark on with children. To me, the possibilities for discovery were endless. Becoming Stepping out and looking in, intent exposed from deep within. We clearly see from close away the wonders of our everyday. Inside ourselves, growth can start; Once ego free change becomes art. Pondering the what, the why and the how develops the past to enhance the now. Consideration of the where, when and who, may mean taking the old and making it new. Digging out pieces from a well packed jar Uncovers assumptions and reflects who we are. While working toward my Bachelor of Education degree in 1999, my beliefs about children—my “image of the child” (Malaguzzi, 1998)—were formed more thoroughly as I began studying the teaching approach used in early childhood education in Reggio Emilia, Italy. The corner stone of our experience, based on practice, theory and research, is the image of the child as rich in resources, strong and competent. The emphasis is placed on seeing the children as unique individuals with rights rather than simply needs. They have potential, plasticity, openness, the desire to grow, curiosity, a sense of wonder, and the desire to communicate and interact with other people and to communicate. Rinaldi, 1998, p. 114  21  In my sixth year of teaching, after experiencing numerous challenges relating to my liberal views on teaching and learning, I considered leaving the profession. With my potent image of the child I sought to articulate and explore new curricular ideas and teaching strategies in a system that was often resistant to change; an institution caught in habits of engagement that precluded opportunities for collaborative work and/or exploration. Rather than give up on teaching all together, I decided to start a Master’s program to learn how to influence change from within. I have come to see that tension— pedagogical/political/philosophical variance between colleagues and students—can be an important part of the learning process on the road to change. “Conflicts and the recognition of differences are essential … opposition, negotiation, listening to others point of view and deciding whether or not to adopt it, and reforming an initial premise – are part of the processes of assimilation and accommodation into the group” (Rinaldi, 1998, p. 115-117). Hence, I undertook this a/r/tographic study exploring how music can serve as a language of expression for understanding multiplication in order to research my personal philosophies about co-emergent teaching and learning. Through this study I desired to (1) contribute supportive research to the field of integrated curriculum, so that children might have greater opportunities to communicate using the “languages” necessary to fully express their understandings about the world, and (2) offer Reggio inspired, inquiry-based creative arts teachers validation that they have a place within the “institution” of education.  22  Figure 1.5 Noticed by the shark - from my story: Little Kelp (acrylic, pencil crayon, construction paper, pen and ink)  Hope has two beautiful daughters. Their names are anger and courage; anger at the way things are, and courage to see that they do not remain the way they are. Augustine of Hippo  23  2  Literature Review  2.1  Theory and qualitative research in Reggio and the IBO Chapter one of this thesis provided a brief overview of the Reggio Emilia approach  and the International Baccalaureate Primary Years Program curricular models that provide the theoretical foundation for my study. An extensive body of literature of a philosophical nature (e.g., the many works of John Dewey, Lev Vygotsky, Howard Gardner, Jerome Bruner, and Elliot Eisner) informs both the Reggio Approach and IB programs (International Baccalaureate , 2007; Costa et al, 2001; Edwards et al, 1998). While the theoretical frameworks of those two curricular models is comprehensive and the programs are widely practiced, it is only within the last two decades that an extensive body of qualitative and quantitative research has been undertaken that attempt to demonstrate how trans-disciplinary projects affect teaching and learning (Hetland & Winner, 2001). The Municipal infant-toddler centers and preschools of Reggio Emilia engage in constant action research that digs deeply into the factors influencing learning in their contexts. In The hundred languages of children (Edwards et al., 1998), there are numerous documented examples of project work what use the well-theorized Reggio philosophy to guide the student inquiry. (e.g., The Amusement Park for Birds, a Shadow study, the Dinosaur project, and how to organize a long jump competition). Reggio Emilia’s art focused environments have teachers act as researchers through on-going data collection via photo, video, and anecdotal record documentation (a major component of the Reggio philosophy) of student projects and of teacher collaborative planning sessions. I believe that it is a fair to say that The hundred languages of children (Edwards et al., 1998), is an example of a/r/tographic writing supporting project work in early childhood education since 24  the authors are artists, researchers, and teachers who have written about their experiences working with children on these projects. This research is qualitative in methodology; my study uses similar lived inquiry/project work as research methodology. The book Making learning visible, children as individual and group learners (Giudici, Rinaldi, & Krechevsky, 2001) is a collaborative research documentary between Project Zero, the independent research group from Harvard Graduate School of Education, and Municipal infant-toddler center and preschools of Reggio Emilia. This book outlines the research and practice between these two institutions, which propose to: 1. Put forth a conceptual framework that we hope will inform future research and practice relative to group learning. 2. Identify seven sets of propositions about how learning groups in early childhood forms, function, and demonstrate understanding. 3. Provide examples of documentation of individuals learning in groups. 4. Take a closer look at adults (teachers and parents) in learning groups as documenters of children’s learning processes and as learners themselves. 5. Draw on research from Reggio Emilia and Project Zero to reflect further on such issues as the relationship between context and group learning, the role of research and documentation in teaching and learning, and connections among learning groups in diverse settings and age groups. 6. Examine the cultural contexts that supported the development of these ideas in Reggio Emilia and the United States and identify what we call cultural knots – assumptions values, and beliefs that frame out understanding and images of individuals learning in groups – which can become barriers to the creation of learning groups. (Edwards et al., 1998, p. 19) This body of research supports my study through its consideration toward studentgenerated project work, group work, and the Reggio concept of teachers as learner (i.e., researcher). Inherently, of course, the Reggio concept of the hundred languages is the theoretical backbone of my thesis study. The research documented in these two books offers sound evidence for the positive affects of Reggio philosophy influencing teaching and learning.  25  We see the traditional isolation of teachers and school staff, and their isolation from families and the social environment, as a sort of longstanding existential imprisonment, an obstacle to the professional growth and knowledge of the individual which is constructed by means of the comparison of experiences and backgrounds…What we want to ensure, then, is that the traditional theory of separation gives way to the theory of participation. Loris Malaguzzi (From the catalogue of the exhibit “The Hundred Languages of Children”) Through discussions with the PYP coordinator in my school, I know that the IBO encourages action research projects within schools that implement IB curriculum. However, in the IB documents (e.g., Making the PYP Happen; The Primary Years Programme - A basis for practice; Arts Scope and Sequence) available on the On-line Curriculum Center, a curriculum support website accessible to practitioners of IB programs, no research studies are mentioned directly, although they are implicit in the books referenced in these documents.  2.2  Two, four, six, eight, we just want to integrate! Integrated curriculum = trans/cross/inter/multi-modal-disciplinary education A problematic aspect of the theoretical, research-based, and practical literature in the  field of trans/cross/inter/multi-modal-disciplinary education is the terminology (Parsons, 2004). Project work/inquiry of this nature goes by many names depending on geographical locations and curricular models. During my research of the literature in journals and databases specific to Education, Psychology, Social Science, Music, Arts, and Neurology I used the words music and art with search terms like: interdisciplinary study, multi-modal education, projects, Reggio, math, language, brain, inquiry, cross-disciplinary education, trans-disciplinary education, international baccalaureate education, inquiry. These search terms led me to some related studies, but it wasn’t until I came across the terms “integrated curriculum” and “arts integration” that I really started to find relevant research studies. While some studies exist that have examined various ways that music has been linked  26  to other subjects (Do Rosario Sousa, Neto, & Mullet, 2005; Hetland & Winner, 2001, Smithrim & Upitis, 2005), most of the articles retrieved about music integration were theoretical and/or methodological in nature (Parsons, 2004), rather than research specific. I found articles that discussed various methods for integrating music (and other arts) into subjects such as language acquisition (Dudley, Pecka, Lonich, Kersten, Hauser & Trimble, 1994), literacy (Prescott, 2005), and mathematics (Church, 2004; 2001; Nolan, 2009). While these types of activities are fine in some contexts when working with children, the suggestions were often superficial examples of subject integration and certainly were not examples of authentic trans-disciplinary links described in the Reggio or IBPYP programs (Snyder, 2001). Ideas presented in these articles include activities like: moving to music, counting to music, echoing sounds and notes, sorting sounds or instruments depending on timbre or material, pattern games, repetition songs for memorizing times tables or letters, using music symbols as metaphoric representations of fraction (e.g., with pies or oranges), adding music to characters in books/stories, playing household items to the beat, learning songs about thematic units, or using familiar melodies to memorize subject specific vocabulary (Church, 2004; 2001; Jensen, 2000; Dudley, Pecka, Lonich, Kersten, Hauser, & Trimble, 1994; Anderson & Lawrence, 2004; Prescott, 2005). There are books and other programs available that integrate music and mathematics. Various multiplication rap series are available to help memorize multiplication facts (Caudle & Caudle, 2010; Banfill, 2010). The book Musi-matics, music and arts integrated math enrichment lessons (Nolan, 2009) is full of ideas on how to teach basic math facts through music games and activities. These articles and books are examples of music being used as a vehicle through  27  which to enhance cognition of non-music subject matter, but they do not make reference to music as an expression of cognition - as a language for conceptual understanding - as in the case of the present study, and they do not consider learning objectives specific for the enhancement of music cognition. There are also studies undertaken by scholars in mathematics and music at from the Canadian Math Education Study Group (Reid, 2004) that demonstrate relationships between concepts (e.g., patterns and processes) of music and mathematics. Although these studies do not directly discuss these relationships in terms of teaching and learning, their findings helped to further my curiosity about how music can be used as a language of expression for the conceptual understanding of multiplication (Gómez, Talaskian, & Toussaint, 2009; Guastavino, Gómez, Toussaint, Marandola, & Gómez, 2009).  2.3  Quantitative research studies supporting or refuting cognitive transfer in music related trans-disciplinary project work/integrated curriculum The focus of this section is to review research that exemplifies the existing  connections between music and other disciplines in the fields of education and the social sciences. In this literature review I have found that the existing research on trans-disciplinary project work/integrated curriculum supports my study by demonstrating the impact on cognition when integrating (i.e., studying, making, or listening) music into non-art subject matter. Although it is beyond the scope of this thesis, another relevant area for review would be in the fields of dance, drama, and visual art education and the ways they have been integrated with the general curriculum and/or been used as a language to express cognition outside of the realm of arts education. For example, researchers Hetland & Winner (2001) demonstrated that there are significant causal relationships between classroom drama and the  28  development of verbal skills. Moreover, the research of Howard Gardner, Eliot Eisner, Rita Irwin, and Alex de Cossen could be considered in more detail in such an extended review. While the studies that follow are interesting in their own right, for my purpose they only serve to demonstrate the glaring gap in the research which affirms that music has, until now, been used only as a tool to enhance achievement in other disciplines, as in the case of the “Mozart Effect,” or to be studied or listened to in isolation without meaningful transfer in understanding being made between music and other disciplines being studied or vice versa.  2.3.1 Music listening and social sciences = supportive research One area where music has shown significant instrumental impact is with the social sciences. In this example, Do Rosario Sousa, Neto, & Mullet (2005) examined the power of music to convey social/behavioral concerns like racial stereotyping. They explored whether children who were exposed to Cape Verdean songs during regular class time showed less evidence of pro-white-skinned/anti-dark-skinned stereotyping than children who only learned Portuguese songs. The participants consisted of 193 randomly selected children (age seven to ten years) who were attending public schools near Porto, Portugal. Eighty-six children were from bluecollar families, 67 were from white-collar families and 40 were from more affluent families. In the Preschool Racial Attitude Measure II test, students had to check a boxed picture of a light skinned person or a dark skinned person in accordance with various descriptive adjectives. From this test, only twelve of the 36 ethnic attitude items (adjectives) were used. After the pre-test, half of the students then participated in a program consisting of eighteen, 60 minutes sessions of Cape Verde/Portuguese cross-cultural musical education. The other half of the group (control) participated in eighteen, 60 minutes sessions of  29  “ordinary session” (Portuguese music only). An ANOVA with a 2 (time) × 2 (age) × 2 (gender) × 3 (socioeconomic level) design was conducted on the raw data. The group × time interaction was significant, F (1, 185) = 12.28, p< .001. The main effect of age for both older and younger students was significant, F (1, 185) = 4.47, p<. 04. However, for younger participants only there was virtually no group effect with no apparent group × time interaction. For older participants, both the group effect and the group × time interaction were strong. The age × group × time interaction was significant, F (1, 185) = 4.35, p< .04. No differences were found between pre-test and posttest values in the control group. However, in the experimental group, post-test values were much lower than pre-test values. One limitation of this study is that the adjectives on which the effect was stronger referred mostly to external attributes of people (e.g., healthy, friendly, wonderful, ugly, wrong). The subjects responded with comparatively weaker effect to the more internal attributes of people because the words were difficult (e.g., nice, kind, sad, selfish, bad, stupid). It may be methodologically unsound to study stereotyping reduction using too broadly defined measurements, and perhaps some dimensions of stereotyping (e.g., physical, psychological, economical) were not taken into consideration. The biggest limitation of this study was the absence of follow-up to determine long-term stereotype reduction attitudes in the study sample. Do Rosario Sousa et al. concluded that the enjoyment and understanding of the music of a different ethnic group allowed students to recognize commonalities with the people of that group, therefore creating a sense of identification with “other” so as to lessen the racist attitudes toward that group. Despite the aforementioned limitations, the researchers indicate  30  that music can play a positive role in altering negative attitudes/behaviour around racial stereotyping. What I find interesting when I apply the findings of this study to my research, and highly contrasting, is that Do Rosario Sousa et al. used only pre-existing songs (with particular reference to the lyrics) for the children to study. The actual music was used merely as a device through which to convey a message rather than allowing for musical expression (e.g., playing instruments, composing melodies, writing original lyrics) to communicate the children’s understandings or ideas around racial stereotyping.  2.3.2 Music listening and memory = supportive research Music has also been linked to brain-based research in the area of memory and mood (Houston & Haddock, 2007). The purpose of this study was to explore people’s memory for music, specifically “whether a mood-congruency effect is attained using major and minor keys as memory stimuli” (Houston & Haddock, 2007, p. 201). Houston & Haddock attempted to “evaluate the specific properties of music that are responsible for the emotional responses observed in the listener”(p. 202). First, 60 test subjects were influenced into a particular mood by using an autobiographical recall task. They became happy, sad, or neutral after being asked to recall a memory with one of those moods. Minor or major scaled short songs were then played for the subjects. Those who were in the sad moods were able to recall more minor melodies (p< .03), while those who were “happy” could remember more major tunes (p< .001). This study points out that the mode of music carries emotional meaning for the listener and that mood might be able to influence music memory. Perhaps it also indicates that the tune that someone can be humming might be an indicator of what kind of mood they  31  may possess. While interesting in its own right, this study does nothing to support my research interests, yet it serves as another example of how music is being studied in relation to another aspect of learning, in this case, memory.  2.3.3 Music listening (and instruction) and spatial-temporal reasoning = supportive research Music has also been researched with arithmetic ability in Attention Deficit Hyperactivity Disorder (ADHD) children when used as an auditory distracter (Abikoff, Courtney, Szeibel & Koplewicz, 1996). The researchers found that auditory stimulation did not adversely affect the mathematics test performance of either the 20 children studied who possessed ADHD or the 20 non-disabled youngsters. In fact, music enhanced the arithmetic performance of the children who possessed ADHD, whereas the non-disabled children performed similarly under the three auditory conditions (music, silence, speech). The researchers reported that the music listened to was the children’s favorite music (presumably surveyed before the study), and no details were given regarding the quality of the silence or speech conditions. Newman-Keuls post hoc tests indicated that, “under the music condition, the children with ADHD had more correct answers than during the speech (p < .01) or silence (p < .05) conditions” (Abikoff et al., 1996, p. 242). No difference was found in the performance of the children with ADHD between the speech and silence conditions. Non-significant results from the simple main effects analysis for the non-disabled group indicated that the non-disabled youngsters performed similarly under the three “distracter” conditions. Finally, while the simple main effects analyses comparing the ADHD and non-disabled youngsters under each of the three distraction conditions yielded no significant differences, the order effects were significant in that the ADHD subjects who had music presented as the first condition had  32  more than twice as many correct answers as the youngsters with ADHD who received music as the second or third condition. “The results indicated that the children with ADHD who received music as the first condition had significantly more correct answers than did children with ADHD who received music in the second or third order, and significantly more correct answers than the non-disabled children regardless of the order in which they received music (p< .05). All other multiple comparisons were non-significant” (p. 242). This study leads us toward the next major body of research in this literature review. While this study found more significant results for ADHD children, it also indicated that mathematical skills were influenced positively by music listening. The second way that research is carried out in the field of trans-disciplinary studies between music and other disciplines is with the discipline of mathematics. Specifically, all research studies in this inter-disciplinary area have been related to links between spatial-temporal reasoning ability and music exposure (i.e., The Mozart Effect; Ivnov & Geake, 2003; Cockerton et al., 1997; Rauscher et al., 1993; Rauscher et al., 1995; Rauscher et al., 1997; Catterall et al, 1999; Doxey & Wright, 1990). The Mozart Effect is a highly debated phenomenon that gained public recognition after Rauscher et al. (1993) published the first of many research articles describing how listening to Mozart music (and later any “classical” music) before taking a spatial-temporal reasoning test enables people to perform better on these tests. Since this study was released in 1993, there has been a wave of replication studies that both support and discount Rauscher et al’s theories. In the original article entitled, Music and Spatial Task Performance, Rauscher, Shaw, & Ny (1993) presented their data research. Their findings were that 36 college students who  33  listened to Mozart music for ten minutes before taking a Stanford-Binet intelligence scale test had higher test scores (p < 0.002) than they did after listening to a “relaxation tape” or silence. A Scheffe’s post-hoc analysis indicated that the music condition differed significantly from the “relaxation tape” (p < 0.002) and from listening to nothing (p < 0.0008). Rauscher, Shaw & Ny reported another test between listening condition and the subjects’ pulse was carried out. Although no data were printed in the article, the researchers indicate that there were no significant differences. They assure the reader that they can rule out the causal relationship of arousal or musical preferences of the subjects on the increased test scores. Herein lies the main dispute many other researchers have had with the 1993 Mozart Effect study. This study fails to include the data that indicates that arousal is not a factor in the differences between auditory conditions. Another problematic element of this study is the ambiguous use of the “relaxation tape.” Many different types of sounds/music can be defined as “relaxation” and there was no mention of what this condition entailed in the research article. Also, the researchers did not indicate the order in which the subjects listened to the three auditory conditions. All of these problematic methodological factors could have affected the results reported. Many replication studies have been carried out. While some studies found similar results to the Mozart Effect, none have replicated the exact results and some have failed to demonstrate any similar results to the Mozart Effect at all (Steele, 2003; Hetland, 2000a). In these replication studies, it appears that many of the conditions in the experiment varied from the original experiment. Sometimes the same music was used, sometimes other classical music was used. Other studies of a similar nature used yet other styles of music, such as rock,  34  female “angelic singing voices,” and even disco. According to Hetland and Winner (2001), who synthesized 36 studies replicating the Mozart Effect through meta-analysis, a relationship does exist between musical and spatial reasoning because of the moderately sized effects of the replication studies. “It appears that spatial and musical processing areas of the human mind/brain are not entirely independent, but it is uncertain whether they influence each other because they are nearby, such that activation of one “primes” the other; or because they overlap, such that development of certain musical processing areas would simultaneously develop the particular type of spatial reasoning defined as spatial-temporal” (Hetland & Winner, 2001, p. 146). They conclude that more research is needed in this area, especially in terms of how the Mozart Effect impacts teaching and learning. In one such replication study, Ivnov & Geake (2003) contributed evidence to the positive existence of links between spatial-temporal reasoning ability and music exposure. Although not directly or clearly outlined in the article, the purpose of this study was to employ a wider use of the term “Mozart Effect” by investigating the effects of concurrent rather than prior listening by upper-primary school-aged children in a school setting. The study explored how listening to the music of Mozart temporarily contributed to the enhancement of temporal-spatial reasoning in school-aged children. Other issues pertinent to the study were if music specificity mattered, and if the effects of subjects’ musical experience were a possible explanatory variable for the effect. The study used 76 (34 males and 42 females) grade five and six students (age 10 to 12 years) from one primary school in Melbourne, Australia. The students were randomly distributed across three mixed ability classes. A brief Musical Background Questionnaire was given to establish the students’ prior music experience. A treatment(s) versus control design  35  was used for this study rather than pre-test/post-test design. Temporal–spatial reasoning ability was measured by an age-normed paper-folding test (PFT) with no time limit for subjects to complete the test. Twenty-eight students listened to Mozart’s Sonata in D major K.448 for two pianos prior to taking the PFT test and while taking the PFT test. Twenty-five students listened to Bach’s Toccata in G major, BWV 916 performed on piano prior to taking the test and while taking the test. Twenty-three students listened to background school noise (control group) while taking the test. A one-way ANOVA with LSD post-hoc tests addressed the group differences. The effect for Group was significant (p > .015). The mean PFT score of the Mozart Group (6.29) was higher than that of the Control Group (5.09). A Mozart Effect for upper-primary schoolaged children is demonstrated by this result. The mean PFT score of the Bach Group (6.08) was significantly higher than the Control Group score (5.09). This seems to demonstrate a Bach Effect similar to the Mozart Effect. A one-way ANOVA with musical background as the independent variable and PFT score as the dependent variable showed that the effect was not significant (p > .179). Results indicate that musical background did not contribute a significant amount to the variance in PFT performance in either the Mozart or Bach treatment conditions. Finally, age was not a relevant variable in this study as indicated by a Pearson’s correlation (r = .104, p > .372) with PFT scores. The researchers summarized, “The superior performance on an appropriately agenormed PFT by upper-primary school children listening to the music of Mozart compared with the performance of peers who listened to background school noise is consistent with the findings of previous studies which found evidence for the Mozart Effect” (Ivnov & Geake,  36  2003, p. 410). This study differed from other Mozart Effect research in that the subjects were ten to twelve year old children in a school setting rather than university students in a laboratory setting. Also, the control group in this study listened to familiar background noise of the school around them rather than an unnatural silence. Finally, the subjects listened to music before and during the administration of the PFT. The researchers suggest that, “these differences contribute to the generalizability of the Mozart Effect by extending both subject and setting” (p. 410). Arousal/motivation was not considered a factor in the enhancement of performance on the PFT because the researchers assumed that young students could not have particularly “enjoyed” the music. However, novelty or mood alteration may well have contributed to the variance and it is a generalization to assume that children could not enjoy listening to classical music. This assumption dismisses the many children’s capacity for musical appreciation, and this was a gross oversight by the researchers. The researchers claim that this was the first study that looked at the effects of listening to Bach’s music within a Mozart Effect paradigm. The result is consistent with the findings from a study on the Schubert Effect, suggesting that the effect may not be exclusive to the music of Mozart (Nantais & Schellenberg, 1999). Enhanced arousal effect is still under investigation when accounting for the Mozart/Bach/Schubert Effects, as it should be in light of the aforementioned oversight. Hetland & Winner (2001) also point out 19 studies with school-aged children where music instruction was shown to have a significant impact in spatial-temporal reasoning. These replication studies of music’s influence on spatial-temporal reasoning all demonstrate how listening to or sometimes studying music (instrumental or vocal) stimulates the same  37  areas of the brain used during mathematical acquisition (Schmithorst & Holland, 2004) (Catterall, 1999). From studies such as these, it has become commonly accepted in the general population that music and math “go together.” While this brain-based research is fascinating and clearly supports the claim that music listening and instruction are important to human growth and development, it does not indicate how, when, or why music might work together with mathematics in areas other than spatial-temporal reasoning. Nor does it offer evidence for enhanced understandings in both music and mathematics, or for mathematical cognition to be expressed through musical modes. That said, this research is appropriate to support my study in that I chose to combine music with math in the co-emergent project with grade three students since there is significant evidence to suggest that math and music stimulate similar areas of the brain while people engage in activities related to both music and math.  2.3.4 Music and general math ability = some support In Hetland and Winner’s (2001) chapter called, Cognitive transfer from arts education, in the book, Handbook of research and policy in art education (Eisner & Day, 2004), they mentioned a meta-analysis of six research studies that examined how music might stimulate mathematical thinking. This meta-analysis did not conclusively support music’s instrumental effect on mathematical ability, although more research on the questions may help stabilize results. “The meta-analysis … found and average effect size of r = .13 … the confidence interval did not span zero (95% confidence interval r = .03 to r = .23), and the t test of the mean Zr = 2.49, which was nearly significant … at p = .06. These findings suggest that there may indeed be a causal link between some forms of music instruction and some forms of mathematics outcomes” (Hetland & Winner, 2001, p. 155).  38  2.3.5 Music and high school biology and math = some support A number of studies have been conducted that demonstrate strong predictive relationships between musical experience and academic achievement (Schmithorst & Holland, 2004; Johnson & Memmott, 2006; Gouzouasis, Guhn & Kishor, 2007). Gouzouasis, Guhn & Kishor (2007) studied how achievement in music courses might predict academic achievement in English, mathematics, and biology—three common subjects for British Columbia’s provincial exams. They also examined whether or not there were group mean differences in academic achievement between students who participated in music and students who did not participate in music. The average correlation analysis from the years 2001, 2002, and 2003 in this study indicated a positive correlation between achievement in music courses and achievement in biology (r = .26), and mathematics (r = .22). This indicated that these two correlations were equal to medium effect size, while English (r = .16) was equal to small effect size. Considering that the degrees of freedom (df) for these analyses was over 50,000 for all three years, the critical values of r chart in the text Research Decisions (Palys & Atchison, 2008) indicates that the researchers assertions are correct. Upon completion of a t-test that assessed the statistical difference between group means (Palys & Atchison, 2008), it was found that in all three groups, differences were evident in academic achievement between the group of students that participated in Grade 11 music courses and the group that did not participate in any Grade 11 music courses. Students who had had greater achievement in Band 11 had greater achievement in all three examined subjects. The difference in math was more than ten percentage points with effect sizes (d = .38 to .61).  39  In biology, group differences were eleven to thirteen percentage points, and effect sizes were larger (d = .53 to .65) due to the smaller variation of scores. For English, differences were less significant with a range of two to nine percentage points, and effect sizes from d = .10 to .75. Students who participated in Band 11, String 11, Choir 11, or Music Composition 11 showed consistently greater achievement in mathematics and biology. However, the effect sizes were lower for the Band 11 participants (d = .06 to .53). Effect sizes for English were near zero, therefore there does not seem to be a systematic relationship between achievement in music and achievement in English. The researchers associated Band 11 with higher achievement in general, while other music course participation was associated with higher achievement in math and biology, but not in English. Surprisingly, Gouzouasis et al. did not find significant predictive relationships in academic achievement in regard to visual art achievement; moreover, with due respect to visual art, dance and drama, they did not emphasize this result in the conclusions because it was merely used to prove a distinctively music, and not arts learning, effect. The overall results of this study imply that some learning skills and characteristics related to music achievement are also specifically related to achievement in math and biology, and not English.  2.3.6 Music and reading ability = no support It has been suggested that reading ability has also been shown to be enhanced if students possess musical experience, as in the case of Butzlaff (2000). A meta-analysis of six experimental studies reviewed the empirical literature testing the claim that “there is an association between instruction in music (usually school-based) and performance in reading (as measured by reading test scores or by general tests of verbal aptitude)” (Butzlaff, 2000, p.  40  167). The researchers found that students studying music appear to have significantly higher scores on standardized reading tests, in the meta-analysis of the correlational studies, even though these studies do not account for causality or directionality. Alternately, the experimental studies yielded no reliable effect that music study enhanced reading improvement. The chi-square test of the heterogeneity of the effect sizes was significant (p = .003), demonstrating that the effect sizes in this sample are not normally distributed. This significant variation in the effect sizes indicates that the overall finding is not stable. The researcher suggests that the considerable variation in effect size is due to the intentions and motives of the researcher of the various studies in this meta-analysis and the political situation during the year in that the studies were carried out. “This suggestion of expectancy effects calls for a more stringent research methodology by the experimental researchers before it can be adequately addressed” (Butzlaff, 2000, p. 176). Hetland and Winner (2001) propose that because music notation and the written word have similarities in written code qualities, it could be possible that (1): “practice in reading musical notation paves the way for learning to read linguistic notation”(p. 154). It is also possible that (2): “listening to music trains the kind of auditory discrimination skills needed to make phonological distinctions”(p.154) and that (3): “music enhances reading skills only when students learn to read the lyrics of songs”(p. 154). Presently, however, there is no research that has provided evidence for these phenomena.  2.4  Qualitative research on music related integrated curriculum Dr. Hyun-Kyung Youm (2007), researcher of integrated music education and  Executive Director of Koomzaal Educational Institute in Korea, nicely summarized some resent research in the field of arts integrated curriculum in her research study entitled,  41  Processes used by music, visual arts, media, and first-grade classroom teachers for developing and implementing an integrated curriculum: A case study. Corn (1993) found that arts integration, implemented by an art teacher, a music teacher, and five classroom teachers, affected teachers and students positively in that they experienced personal and educational growth. Miller (1996) observed in her action research with first-grade children that integrating instruction facilitated topical connections and higher-level thinking skills for her students. McCullar (1998) observed that an integrated model for pre-service teacher training using fine arts (music, art, and physical education/ movement) and the core subjects (language arts, mathematics, science, and social studies) was successful at the elementary level, with the fine arts enhancing understanding of the core subject disciplines. Carpenter (2004) examined integrated music curriculum for grades K-6 and concluded "music standards can be taught and learned well with an integrated, interdisciplinary curriculum without losing sight of the goals of the state standards for music" (p. 35). Each of the studies cited provided evidence that integrated music instruction has a positive impact on both students and teachers. (Youm, 2007, p. 42) In Youm’s study, she describes the examination of a grade one unit where the Music, Fine Art, Media, and Grade One class teachers collaborated together throughout the unit. Her project sought to explore the development and implementation processes of this integrated first-grade curriculum. Through extensive interviews with the teachers, she found that they felt that student creativity and higher-level thinking were enhanced and that students made connections for learning concepts through the integrated unit. The teachers believed that the students also enjoyed every class. “Because integrated music classes make students' music learning vivid, interesting, and effective, it is strongly recommended that music and classroom teachers give their students the opportunity to learn within such a program” (Youm, p. 46). It is Youm’s literature review (partially quoted above) that has been more relevant in support of my study, than her study, itself. Qualitative studies like Youm’s and those referenced in her literature review illustrate the significance of integrated curriculum. The  42  studies involving music in integrated curriculum emphasize how music can enhance learning of the core subjects, or be used to teach the core subjects, or how music standards can be achieved through integrated units. The research did not point out how music can be used to communicate learning in the core subjects as my study seeks to explore. Another significant body of qualitative research about integrated curriculum is illuminated in Michael Parsons’ chapter, Art and Integrated Curriculum, in the book Handbook of Research and Policy in Art Education (Eisner & Day, 2004). He begins with a description of educators’ renewed interest in integrated curriculum since its suppression in the late 19th century with the scientific revolution (Ungerleider, 2003). He describes the complex needs required in our modern changing world and the subsequent needs for students to cope with these changes. He says that because, “students need a more integrated personality, greater awareness of self, and more understanding and tolerance of others” (p. 775), educators would do well to teach using, “a curriculum that encourages students to think about important ideas, to interpret them and relate to themselves, their own time and context” (p. 777). Parsons suggests that integrated curriculum offers this to students. This chapter also mentions the literature in the field of arts education and integrated curriculum. He thinks that the research in arts education “is scattered and practices are undertheorized” (p. 776). As I discovered during my research into the literature on transdisciplinary education/integrated curriculum, Parsons found that the many books and articles written on the topic of arts education in integrated curricula are most often descriptive accounts rather than reflective ones. Research practices in Reggio Emilia and Project Zero (2001), with their focus on documentation and reflection, take descriptions of projects and evolve them into legitimate research practices when description of the project methodology,  43  combined with theory produce significant contributions to teaching and learning. Parsons’ complaint is that there should be more studies with sound methodology like these to legitimize arts-based research in education. He ends his chapter with his own qualitative study example undertaken at Ohio State University where faculty, teachers, and students worked together in multi-disciplinary contexts to eventually produce large murals that explored the complex theme of Community Heroes. Overall, Parsons’ chapter on art and integrated curriculum is a succinct description of debate over attitudes and topics relevant to arts integration in discipline specific curricula. He discusses discipline-based learning versus concurrent holistic learning. He challenges assumptions that although some disciplines are difficult, their content need not be studied and mastered sequentially or in isolation. He suggests that educators should encourage students to seek personal meaning and understanding through inter-disciplinary activities and have the tools of those disciplines discovered and mastered in a practical problem-based context. “This means that students should grapple with the problems first and learn to use the tools as they find them helpful” (p. 778). In that regard, disciplines serve teachers and students as guides for meaning and “retain a significant educational role in an integrated curriculum, even though they do not constitute the primary educational goal” (p. 778). Parsons goes on to list a number of case studies on arts integrated projects that have done much to benefit children and teachers in the school context. Some common themes that arts and integrated curriculum have addressed in Parsons’ research are: social issues (e.g., the local community, contemporary issues in art, social activism, environmental activism, complex visual imagery, and media in art), psychological issues (e.g., gender issues, middle  44  school and identity, self stereo-typing, personal wholeness), and epistemological issues (e.g., the nature of the disciplines). According to Parsons, constructivism is the primary teaching methodology in integrated curriculum, and is accompanied by various inquiry strategies that guide the project’s investigation. Unlike this type of project methodology, my study uses a variation on the IBPYP Inquiry Cycle (Bruce, 2009; Short & Burke, 1996) – see chapter three – which is enactivist rather than constructivist. “Enactivist Inquiry” takes constructivist inquiry one (or more) step(s) further to where learning differs/changes/evolves depending on individual and group response to, interpretation of, interaction with the environment, the materials, the people, the questions, their own history, and their relationships. Parsons also elaborates some concerns that art educators and researchers have when dealing with integration. He talks about the potential of losing important aspects about art-the discipline, during integrated units with other subjects. Particularly in the case of visual culture, he muses, “If we integrate the study of art with other subject matter, will we lose our focus on the nature of the images themselves? Will art become only the ‘handmaiden’ of other concerns” (p. 781)? He also discusses concerns about integrated curriculum perpetuating “low-level learning” where art integrated projects do not have the depth and breadth required to cover areas of real concern to students, and to promote reflection, understanding, and meaning. However, by the end of the chapter, he assures us that integrated curriculum is a worthwhile and holistic approach to teaching and learning, but it conveys a demanding vision. It encompasses many academic disciplines, in an often messy/“ill-structured” (p. 785) inquiry-based framework. Parsons explains that this type of education calls for highly trained teachers who need to be versed in more than one discipline.  45  They must be prepared to step outside of “egg-box” curriculum, to collaborate extensively with their colleagues, students, and local experts, and to enter into a space sometimes unknown. These borderlands can be difficult places for some teachers to visit, but this holistic, artistic, risk taking is required by teachers to cultivate rich personal meaning and understanding for students through integrated curriculum. Finally, Parsons mentions the theory and research of Howard Gardner’s Multiple Intelligences (1983). He says that they are disciplined based, and in some contexts superficial examples of embodiment (Springgay, 2008), therefore, in principle Multiple Intelligences do not support curriculum integration. The legacy of Multiple Intelligences is that they have politically and ideologically influenced society that led to an increase in art education in schools. Ironically, Multiple Intelligence theories have paved the way for art integration in other subject matter even though it isolates the disciplines. As stated in section 2.1 of this chapter, Gardner’s more recent research with Project Zero and schools in Reggio Emilia is very much in support of integrated curriculum.  2.5  Summary of the related research and how it relates to the present inquiry The quantitative studies in this literature review demonstrate a one-way effect of  transfer, where listening to music or the isolated study of music is used as a tool to enhance the learning of another subject or stimulate an area of the brain so as to enhance test scores in a certain area. In that manner, they are instrumental (i.e., using music as an instrument to enhance learning in another area) in nature. None of those studies examined musical performance or composition (musical expression/communication) as the main indicator or effect of the studies significant result. Some researchers have examined how music may affect other areas of human behaviour/attitudes when experienced before or together with  46  those phenomena. However, no studies have been undertaken to indicate how music can express conceptual understanding of another subject’s content other than by stimulating an area of the brain used by both music and the other subject. This is the gap in the research where the present thesis will contribute in the field of trans-disciplinary project work. The qualitative studies in this chapter give detailed support of integrated curriculum. They describe the many ways in which learning of all subject matter can be enhanced when combining curriculum. Similar to the quantitative research revealed in this chapter however, no studies in educational research have explored how music can be used as a language of expression for cognition. There are other ways that music has been used to communicate ideas in other disciplines or areas of life. For example, from a religious perspective, some theologians have written about music communicating meaning via rituals (Reimer, 1963). Also, research has been undertaken in the field of musicology to show how music has been composed to communicate non-musical concepts. A review of the literature in musicology is beyond the scope of this thesis, but a few examples that come to mind where music is used to communicate story are in Sergei Prokofiev’s 1936 Peter and the Wolf, Camille Saint-Saens’ 1886 Le carnaval des animaux (The Carnival of the Animals), and Tchaikovsky's 1877 Swan Lake – among numerous other musical masterpieces. In these examples, characters are clearly represented by the use of instrumental tone colour and melodic motif, storyline is outlined through phrasing, musical sections, repetition, and song structure, and mood is implied through timbre of sound, instrumentation, and dynamics. Music has been used to communicate emotion or social messaging during times of political uprising as in the case of The Soviet National Anthem rewritten by Mikhalkov, El-  47  Registan, and Alexandrov in 1944. Songs of all genres and time periods communicate many culturally significant concerns through their lyrics but this is really a verbal language representation of meaning rather than communication through musical means. In the educational context, song lyrics have been used to communicate curricular content, influence learning, or demonstrate understanding, but in my search of the literature I found no studies where elements of music were used for any type of communication, especially cognition of non-music subject matter.  48  3  Methodological considerations, design, and teaching and learning processes What we see taking place in educational research is a gradual expansion of the methods that are considered legitimate for understanding schools, teaching, and learning. We are pushing towards pluralism. Eisner, 2006, p. 16 A/r/tographers re-represent their questions, practices, emergent understandings, and creative analytic texts and compositions as they integrate knowing, doing, and making through texts and/or performances that convey meaning rather than facts. Springgay, S., Irwin, R.L., & Wilson Kind, S., 2005, p. 903  3.1 Our context During this a/r/tographic exploration, I was the Junior Kindergarten to Grade Five music teacher at a small private IBO World School in the lower mainland in British Columbia. The fifteen children in the grade three class, along with their teacher, engaged with me in a co-emergent inquiry project where the children’s understanding of early multiplication was expressed using musical language. The class was made up of nine girls and six boys. All of the children were eight years old. Their academic abilities were diverse. Three students were in the process of being assessed for learning difficulties during the study and one student was already coded as gifted. The other students were considered average learners with strengths and difficulties in different areas. All of the students in this class (and the school) were from middle class, supportive families. Four of the children had been studying piano privately before the study and three or four students had dance class experience before the study. However, most students had no formal experiences with music learning outside of the music classes offered at school. All but three of the students in this class had been working with me for the last two years in the twice 49  weekly music classes and in the after school choir program. Subsequently, they had an ageappropriate understanding of music theory and could read and compose simple rhythms and melodies in treble clef, some better than others depending on their experience and general musical aptitude. Jenny, the grade three classroom teacher, moved to Canada from the United States six years ago. She had been teaching grades three to five for several years in private schools in the Vancouver area and in the California Public School System. Jenny earned her Master of Education degree in Differentiated Education in 2002. She is a dedicated teacher who has embraced the Primary Years Program and is familiar with the Reggio Approach. Jenny does not consider herself to be musically skilled, yet she was willing to collaborate with me on this thesis project without hesitation. Jenny is an outstanding educator who engages with her students, her colleagues, and her students’ parents with impressive integrity, enthusiasm, and professionalism. I learned a great deal from my interactions with Jenny. At our school, music was woven through the fabric of every day life. We had welcoming music playing over the sound system as we entered the building in the morning. This was followed by our “soft-start” whole-school, sing-along and gathering in the gym called “Morning Song.” There were regularly scheduled music classes for all students, a grade seven to ten concert band and contemporary music ensemble, and a Junior and Senior Choir offered in our after school activities. The aim of our school’s music program was to give children an opportunity to explore, perform, and create music both individually and in groups. Our school fully endorsed the central philosophy of the Primary Years Program of the International Baccalaureate that all subjects, including the arts, be included in transdisciplinary integration as part of the curricular Units of Inquiry.  50  As one of two music teachers in this school, I was encouraged to fully utilize my creative pedagogical skills while collaborating with my partner teacher (the grade 6 through 10 music teacher) in concert/performance planning and curriculum continuity across grade levels. In keeping with IBPYP philosophy, all teachers actively collaborated to varying degrees depending on the unit to create a rich, meaningful curriculum for students. Teachers had meeting time built into their schedules to engage in the conversations and planning necessary for this type of collaboration. Ample prep-time in teachers’ timetables allowed us to visit each other’s classrooms, have spontaneous meetings, and do individual planning. These time allowances were crucial for implementation of an integrated curriculum. It is worth pointing out here that time is likely the biggest obstacle that teachers in public schools face when trying to integrate curriculum. At my school we were blessed with fixed meeting times, a more relaxed timetable, and very open-minded administrators and colleagues. This flexibility helped my research a great deal since Jenny and I were able to meet during our prep-time to work through our planning throughout the project. The research study included five 45-minute sessions per week for six weeks. Four of the lessons were in double blocks (90 minutes) and the fifth class was on its own lasting 45 minutes. Jenny and I used the class’s two regularly scheduled music class time slots and three of my free prep time blocks for our research sessions together. Jenny graciously altered the normal classroom schedule to accommodate these changes. She borrowed the three extra blocks required for my thesis research study from their scheduled Unit of Inquiry time, so there was no pedagogical disruption to the children’s program. During the six weeks of my study, we were also working on a major whole-school musical theatre production called “The Return of the Glass Slipper.” This grade three class  51  had the role of the Royal Fanfare in the play; therefore, the fifth 45-minute block was used for the preparation of the Fanfare ensemble piece rather than the research project. It was a good use of time because as we entered the third week of the research study the children required the longer sessions for focused project development and enjoyed the break during our play preparation from the intense inquiry work.  3.2  It’s messy! a/r/tography = living inquiry + performative inquiry Art is the visual reorganization of experience that renders complex the apparently simple or simplifies the apparently complex. Research is the enhancement of meaning revealed through ongoing interpretation of complex relationships that are continually created, recreated, and transformed. Teaching is performative knowing in meaning relationships with learners. Irwin, 2004, p. 31  Figure 3.1 My “living system” of this research project A/r/tographic inquiry, a complex research methodology, encompasses my research design while embracing very similar methodologies—living inquiry and performative inquiry—that speak to me so clearly. This project was more than a post-modern romp through fragmentation (Fels, 1998), it was an enactivist, complex messiness that shook our 52  assumptions, challenged our understandings, and hurled us gently to the edge of chaos. This combination of methodologies is a collision in the borderlands and “mètissage” (sec 3.2.2) is the language spoken there (Irwin & de Cosson, 2004). I interpret a/r/tography as the writing of our experiences as artists/researchers/teachers. A/r/tographers “are living their work, representing their understandings, and performing their pedagogical positions as they integrate knowing, doing, and making through aesthetic experiences that convey meaning rather than facts” (Irwin, 2004, p. 34). In our context, the a/r/tographic experience is shared among participants in the following ways. Gimme an A! The children, their teacher, and I were all artist-musicians composing our understanding of concepts within music and multiplication (Gouzouasis, 2007). Gimme an R! To some degree, we are also all researchers in that we truly were investigating the answers to our questions as a community of inquirers. Through constant reflexive conversation the children, their teacher, and I “re-create[d], re-search[ed], and re-learn[ed] ways of understanding, appreciating, and representing the world” (Irwin, 2004, p. 28), while we unpacked our learning and expressed it musically. Gimme a T! We were also all teachers with adults and children sharing the lead depending on the need. Since leadership was shared amongst the group, from the larger context where Jenny, myself, and a few more confident children led, to the more intimate levels of community  53  where teaching occurred through subtle acts of communication like glances, examples, imitation, friendships, bossiness etc I, alone, was the “ographer” – the one collecting the data, and writing it in the form of this thesis. Although, depending on how you break down the a/r/tographic process, it could also be argued that through their graphic scores, compositional charts, and verbal reflections the children also participated in the “ography” of our inquiry. Of course, the fact that I am a musician (and that I undertook this research with an eye to artistic processes), that I am an arts teacher, and I am writing about our research, places our work firmly in a/r/tography. The theory behind a/r/tographic research methodology is messy when you look at all the complex methodological elements that make a/r/tography what it is: the knowing/researching (theoria), doing/teaching (praxis), and making/art-making (poesis) of the lived artistic experience. Paradoxically, the Inquiry Cycle (see section 3.3) is simple as a framework for project work, but it becomes complex when you begin to unpack the concepts and engage in the act of a/r/tographic inquiry. It’s delightfully complicated. A/r/tographers live in the arts—they are trained in an art form, they made livelihoods in an art form, they teach others their form of artistic expertise, and they perform research in, on, around, and about their art. In that sense, their research is informed by their lived artistic experiences in creating and teaching art. In other words, their art informs their research and their research informs their art. Gouzouasis, 2008, p. 4 Toes: hanging over the margin. Back: arched. Arms: waving for balance. Belly: clenched. Eyes: peering down over nose to see how far you’ve gone. A quiet moment of assessment, discovery: everything is okay, we haven’t gone too far, we are balancing here. Baby step back, look around, reassuring deep breath, eye contact, smile…  54  Engage! (I think I pulled a hamstring.)  3.2.1 Living inquiry In a/r/tography, visual, performative and written processes are enacted as a living practice of artmaking, researching, teaching, and learning in processes similar to our understanding of action research as living inquiry. Sites of living inquiry may interface, intersect and interrogate assumptions in order to inspire thoughtful action. A/r/tography is a fluid form of inquiry creating its rigour through continuous reflexivity and analysis. It is a form of inquiry interweaving theory, practice and poesies in contiguous ways allowing for deeper understandings to emerge over time. A/r/tographers position their work amidst the practices of artists, theorists, educators and researchers and engage with these communities through citational practices and a commitment to engagement with their artforms over time. A/r/tography Website, 2010 Working as an artist, researcher, and teacher or artistresearcherteacher (Gouzouasis, 2008) means negotiating one’s way through the renderings. Living inquiry is one of six methodological elements of a/r/tography that, like action research, requires us to consider knowledge as a state of being and becoming with one’s self and one’s community (Irwin & de Cosson, 2001; Carson & Sumara, 1997; Gouzouasis, 2006). Living inquiry is an ever-moving research/teaching/art-making practice that is difficult to capture in words. Like action research, living inquiry “is an endeavor to better understand the complexity of the human condition” (Carson & Sumara, 1997, p. xxi). Living inquiry is one of the renderings of a/r/tography, and although it can stand alone as a research methodology as delineated by Karen Meyer (2006; 2010), I have chosen to limit its function in my research only as per the a/r/tographic description. Living inquiry is an action, it is an inquiry in the process of doing, it is on-going, it is cyclical, it is the process  55  of living, like Gouzouasis (2007) said, it is “in the what I am doing” as a music teacher and musician. Maturana (1987) said, “whatever we do in every domain, whether concrete (walking) or abstract (philosophical reflection), involves us totally in the body… Everything we do is a structural dance in the choreography of existence” (in Davis et al. 1996). Inquiry is life, and a/r/tographic living inquiry is life writing. In the case of the children, Jenny, and myself, we were attempting to “write” (through both music and verbal language) our lived experiences of this co-emergent exploration of music as a “language” of expression for our understandings of multiplication. As I entered into the project with these children and their teacher my intention was simply to facilitate a music making experience with honesty and integrity, where all participants could contribute, challenge, and be challenged as a/r/tographic inquirers. Many times throughout the project leadership roles drifted between children and adults, like lead geese drafting for the flock in chevron flight. Our early discussions established reciprocity (Rankin, 1998) and as co-inquirers we conspired to answer the research question with open minds, using our past cognitive experience, our environment, our critical/creative thinking skills, our emotions, our aesthetic sense, and our relationships with each other.  3.2.2 Performative inquiry Performative inquiry provides a theoretical underpinning that supports the use of the arts as a viable vehicle for learning across the curriculum. Performative inquiry in the classroom calls for cross-curricular explorations that are embodied, relational, and intimate. Bringing performative inquiry into science, language arts, social sciences, or other disciplines opens new ways of working with students that encourage student agency and empowerment. Integrating the arts through performative inquiry engages students in meaningful curricular explorations, thus ‘enlarging the space of the possible’ (Sumara & Davis, 1997, p. 299). Fels, 2008, p. 9  56  Performative inquiry may be interpreted as inquiring while performing. It is an unfolding of ideas in the liberatory space of the borderlands (Smith, 1993), also referred to as the edge of chaos (Waldrop as quoted in Fels & Stothers, 1996). It can be seen as an understanding, an appreciation of the world expressed in some aesthetic form (Irwin, 2001). Performative inquiry as an action-site of learning is the re/creation of a world through which moments of recognition (i.e., interstanding) emerge (Fels, 1998, 1999). Drawing from enactivism, performative inquiry explores knowledge as “not separate from the learner but embodied within creative action and interaction” (Fels & McGivern, 2002, p. 8). It is a space where “interstanding and intercultural recognitions are possible” (Fels & McGivern, 2002, p. 6). The writings of Lynn Fels and her colleagues in performative inquiry, were also significant in informing my use of a/r/tography as research methodology (Fels, 1998; 1999). While performative inquiry may also stand alone as a research methodology, I choose to frame it within a/r/tography because philosophically I found them to be highly compatible. And where a/r/tography failed to describe the performative nature of co-emergent inquiry in the detail I would have liked, Fels’ conceptualization and articulation of performative inquiry detailed most accurately the edge of chaos where our inquiry placed us, and through this, explained in greater detail the nature of a/r/tography’s third space borderlands. Through music inquiry of a performative nature, my fusion of a/r/tography and performative inquiry as a research methodology allows for spaces of knowing, of being and becoming, while music-ing (Gouzouasis, 2007, p. 42-48). In educational performative research the data is enacted and collected by artists, researchers, and teachers. Its theoria (researching/knowing), praxis (teaching/doing), and  57  poesis (music-ing/making) is interpreted subjectively because the researcher(s) is/are (a) participant(s). The founders of a/r/tography, Rita Irwin and Alex de Cosson (2004), describe this expression during the complex messiness of performative lived inquiry as Mètissage. “Mètissage is the language of the borderlands … Metaphorically, these borderlands are acts of Métissage that strategically erase the borders and barriers once sustained between the colonizer and the colonized” (Irwin, 2001, p. 29). Music was Mètissage in our expression of multiplication in this inquiry, while “teaching [was] performative knowing in meaningful relationships with learners” (Irwin, 2003, p. 31). If we stand on the edge are we supposed to jump off? Do we desire the freefall? What is there when we land? How long must we ‘fall’? Is performative inquiry then, standing on the edge of chaos and backing away from the edge once we arrive there? How long do we stand there? How long can we stand to stand there? Do you jump into the chaos or just remain standing on the edge of it? Is standing on the edge similar to sitting on the fence? Is performative inquiry (like enactivist teaching/learning) really an edge, like a cliff or the seashore? It is more like a bridge between two lands, or the middle section of a Venn diagram, or the space between ones’ palms when hands are folded together as if in prayer. The spaces between the notes. So, we don’t really want to be IN chaos, because that is too chaotic. And we don’t want to be only in the order, either. We want to be able to be in both areas at the same time in some dialectic s/p/lace so as to be ‘pushing it’, but not quite crossing either line. Whose line/edge is it? Do we have varying edges depending on who is involved? And therefore, the diversity of action leads to diversity of edge definition. Where is your edge? Where is my edge? How do we then stand on the edge together? The adults can’t throw the basketball from the kids line ‘cause that’s no fair! The ladies tee offers a more equal opportunity for everyone to reach the green at the same time.  58  Living inquiry, as described as an a/r/tographic rendering, with performative inquiry, best explained the co-emergent nature of our lived and performed experiences during the research project and during the writing of the thesis. They enriched my application of a/r/tography as both a research methodology and over-arching guiding theme throughout the inquiry.  3.3  Procedure Using a/r/tography (Irwin & de Cosson, 2004, Springgay et al, 2008) as research  methodology, I explored how music could be used as a medium of expression (i.e., a language) for grade three students. As a musicianresearcherteachers, the students, their teacher, and I engaged in living/performative inquiry through a co-emergent multiplication/music project that offered students an opportunity to explore and express their understandings of the mathematical concept of grouping and early multiplication through musical means. My study explored the ways in which children attempted to communicate their understanding of multiplication using musical devices (such as rhythm, notation, melody and harmony, composition with consideration toward tone, timbre, mood, and other expressive elements of music). For assessment purposes the children’s oral and written interpretive statements accompanied their musical summative pieces of musical communication. The Inquiry Group (Bruce, Bishop, & Robins, 2002; Bruce & Bishop, 2008) from the University of Illinois published the “Inquiry Cycle” and other inquiry-based resources for teachers and students who are engaged in inquiry work in schools. The IBO references this inquiry cycle when running workshops for teacher training. In preparation for my inquiry project, I consulted with the PYP curriculum coordinator at my school, and she  59  recommended the work of Kathy Short (1996) and the Inquiry Group for the general frame for my research procedure. The IBPYP planning document is a more detailed version of the “Inquiry Cycle.” I chose not to use it as a procedural frame for my research because it includes many IB specific planning requirements that are not directly relevant to my research. Similarly, the Reggio Approach has some recommendations for procedure in project work. These recommendations are not prescriptive, but rather offer teachers pedagogical considerations when engaging with children in the inquiry process.  Figure 3.2 The Inquiry Cycle (Bruce, 2009) © Bertram C. Bruce, Building an airplane in the air: The life of the inquiry group. In J. Falk & B. Drayton (Eds.), Creating and sustaining online professional learning communities, 2009, adapted by permission. Using the Inquiry Cycle as our inquiry design we: asked, investigated, created, discussed, and reflected together in and around the research question “Can music be used as a language of expression for understanding multiplication? If so, how? If not, why not?” This cyclical process brought us through the knowing, doing, and making of a/r/tographic research in a number of ways. Our emergent understandings and questions were re-represented through our musical compositions, through our numerous discussions that became  60  documented here in this thesis in narrative form, and through our reflective artist-statement texts (the children’s texts were their documentation of their compositions). The inquiry cycle, as delineated by Chip Bruce (2009) and the Inquiry Group (Bruce, Bishop, & Robins, 2002), was a solid model to use because it did not determine the lived experience of the inquiry, but rather provided a map with which we could navigate our journey through the borderlands of inquiry. Reggio educator, Baji Rankin (1998), in her chapter “Curriculum development in Reggio Emilia: a long-term curriculum project about dinosaurs,” from the book The hundred languages of children, describes five main steps for teachers when facilitating project work. She says that the first task is to establish reciprocity with all project participants for maximum involvement, interest, and ownership. An extensive graphic and verbal exploration is required next, to determine where the project is heading. Then when the actual project is being developed it should be based on the children’s questions, comments, and interests. Rankin’s next step for project work is to allow the children to have plenty of time to continue to ask their own questions and find their own solutions as changes and further development occur. The last stage of a Reggio project is for the students to share their findings with others in the school community.  Inquiry Cycle Ask Investigate Create Discuss Reflect  Reggio Emilia Project Approach Reciprocity/ Verbal & Graphic exploration Verbal & Graphic exploration Project development Time for children’s questions and solutions Sharing with community  Table 3.1 Comparative model of the Inquiry Cycle and the Reggio Project Approach  61  3.4  Front loading/pre-assessment “What do we know?” Before beginning the project with the students in this Grade Three class, Jenny and I  had many conversations about the inquiry process and how co-emergent inquiry differs from constructivist guided inquiry used in IBPYP units. While still following the Inquiry Cycle (Inquiry Group, 2010), I wanted this project’s result to come from the children’s discoveries completely. This is enactivist co-emergent curriculum, rather than guided inquiry (constructivism) that produces pre-determined outcomes of knowledge and understanding. Although we had opinions and conjectures about the students’ findings during our inquiry, Jenny and I deliberately participated as facilitators and co-inquirers, hoping to support the children in their quests for discovery (Rankin, 1998). The children in this class had been working on the tools required for such an inquiry since joining our PYP school. Most of the class had been at our school since kindergarten or grade one. In any PYP classroom, all six of the Units of Inquiry trans-disciplinary themes are explored each year. For this reason, most of the children in this class were well acquainted with guided inquiry work and were prepared for the open-ended critical thinking required for such work. Jenny had already completed five full Units of Inquiry so far that year with her students. They had also recently finished an early multiplication unit that introduced the concepts of grouping and fast adding in multiplication. When the children and I started the research project, everyone in the class had some level of understanding of multiplication, some better than others. As I previously mentioned, for two years before the study began I taught this class music lessons twice weekly. My music classes are Orff, Gordon, Manhattanville Music  62  Curriculum Project, and Reggio inspired and include a great deal of exploration through individual and group work in music “centers,” composition/improvisation, traditional and graphic notation reading and writing, and consistent integration with PYP Units of Inquiry including integral unit integration and more supplementary integration. Generally speaking, these learners had extensive experience in thinking “outside the box.” It is worthy to note here that this class’s experiences were very different from average North American public school grade three classes. The research in integrated curriculum and project work indicated that, overall, children who have come up through the grades in our present educational institutions are not at all prepared for this kind of critical thinking, independent learning, and student driven inquiry (Fraser, 2000; Parsons, 2004). Project work/inquiry is a new concept for many students who have not been exposed to it throughout their careers as students. The traditional model of education has not consistently asked children to “negotiate shared meanings and to co-construct knowledge” (Rankin, 1998, p. 236) through inquiry work. In preparation for engagement in this inquiry during March and April 2009, while I put ethical review proposals and conference applications together, not to mention beginning intensive preparation for the whole school musical theatre production “Return of the Glass Slipper” (Strid & Donnelly, 2009), the grade three class and I began a review of their music vocabulary. At this early stage of an inquiry project, pre-assessment activities get us ready for the ask phase of the Inquiry Cycle. What do we know? To answer these preliminary questions we looked at six main elements of music that these students had been exploring in my music classes for the last couple of years, and in their own private world of music or piano lessons/practice. Music Vocabulary  63  1. 2. 3. 4. 5. 6.  Pitch - notes on the treble staff Simple rhythms, rhythmic patterns, and note values Meter signatures, measures, bar lines, and beat Song form and lyric content Chords and accompaniment. Basic harmony We played music games, wrote short music compositions, analyzed scores, and put  together xylophone ensemble pieces in order to exercise and extend our understandings of the concepts associated with the six elements listed above. After some weeks of focusing on these topics, we did a “butts-up” group activity to assess the students understanding of the six musical elements. A “butts-up” activity is when students kneel or sit around paper on the floor to record their ideas on a subject. Here we began to answer the question “What do we know?” which is an essential preassessment/front loading element of the inquiry process. In this activity, the children moved around the room writing on large sheets of paper their current understanding of these six musical elements. Here is what they wrote (NB: the Rhythm page was omitted from this exercise because the sheet became hidden beneath some materials on a table and the children didn’t see it to fill it out). Melody - “the leading top part of music” - “is the main part of the song” - “note value” - “main beat” - “melody is a song” - “Melody is like a song through feelings”  Meter Signatures - “what beat the music is in” - “time signature is how long each bar is” - “note value: how long it lasts” 64  - “how many notes are there?” - “is notes” Song Form/Lyric Content - “a form of a song through lyrics. Lyrics are like notes” - “The form of a song is the form like anything – the notes and content and stuff” - “Lyric is the words of a song” - “Notes of a song” - “Words to a song, expression of feeling” - “Is like a sound and a note” - “Words that match a song” Chords/Bass Lines - “AFBGC” - “Pattern of notes” - “Combined notes played in a block” Harmony - “Harmony is the bass line” (added bass and treble clef picture example) - “Harmony is a note of sounds and it is the bass line” - “Is basically the echo for the song” - “Is like music and sound” - “Harmony is background high pitched sounds or words” - “What is harmony?” - “Harmony is a high note” This data was collected in order to discover what kind of collective conceptual understandings and musical vocabulary the children possessed. From this information I was able to see that the children had a diverse and vague understanding of these concepts when asked to articulate them in writing. Having applied these concepts during music activities with the children, I knew that they had a better practical understanding than theoretical understanding. It looked like most students had a solid grasp of melody and lyrics. There was some evidence to suggest that they understood meter signature and harmony. The concepts of song form and bass line were not clearly explained by the students, and only one child attempted to explain chords (e.g., “combined notes played in a block”).  65  To clarify the children’s understanding of these music concepts, I asked them for examples and any more details they could offer by doing a “Think, Pair, Share” activity. In this activity the kids thought about examples or additional explanation of the concepts, shared their thinking with a partner, wrote them down, and then shared them with the whole group. All the children were easily able to give examples of melody and rhythm.  66  Many were also able to give examples of meter signature:  And song lyrics:  67  The concept of harmony remained somewhat tricky to define. A few children made attempts to give an example, or define their understanding of it:  Only two children attempted to give an example of chords (but since they are exactly the same example I suspect that somebody copied!):  The children understood bass lines in terms of contra bass bars to be played under a melody, but I have no evidence to believe that they know how a bass line is related to chords or a chord progression.  68  Finally, a few children were able to accurately show the concept of song form during this exercise.  These six music elements described above include considerably more depth of understanding than the provincial grade level standards in music education for Elements of Melody and Elements of Rhythm. As was evident in the pre-assessment activities (and during my regular music classes that year), all of these students met the BC Ministry of Education Prescribed Learning Outcome (PLO’s) expectations for grade three music. The only PLO that all of the children were not able to articulate in detail was how form should be described “in terms of repetition and unity of rhythmic phrases” (BC Grade 2/3 Music PLO, 2009/2010).  69  The “butts-up” group reflections by the students on these six musical areas was not as clear as I would have liked to see before asking them to use their musical vocabulary in an open-ended project with music as their language of expression. As a final pre-assessment on these six examples of music vocabulary, I was curious how confident each child was as an individual when understanding and applying these six concepts. I asked the children to chart their self-assessed understanding of the six music vocabulary areas using a ten-point continuous rating scale.  On average for the six vocabulary items 8/15 children in the class “got it.”  Not too sure  Makes some sense  1  5 On average 3/15 children in the class weren’t “too sure” about the six vocabulary items.  Got it On average for the six vocabulary items 4/15 children in the class thought they “made some sense.”  10  Figure 3.3 Music Vocabulary Self-assessment  There was a label title at the one-mark reading “Not too sure,” and another at the tenmark reading “Got it”! At the scale point marked five it read “Makes some sense.” The students charted their self-assessed understanding on each of the six music concepts using this 1 to 10 scale in order to give us some insight into their own perceptions of their understanding of these concepts. This way, we knew at the beginning of the inquiry project  70  whether or not we were using similar vocabulary. In other words, were we all speaking the same language with similar fluency, or not? While the “butts-up” activity implied that the students had only a moderate understanding of the concepts, the ten-point scale measured the children’s belief in their own ability to use and understand the concepts, even if they couldn’t express them in words. Eight of the fifteen children indicated that they felt they possessed a very good understanding of the music vocabulary items—their numbers ranging from seven to ten on the ten-point scale. Four children rated themselves in the middle near “makes some sense.” Only three students said that they were nearer to a three or four indicating that they only had a basic understanding of the musical elements at the beginning of our inquiry. A few students gave themselves tens on every music vocabulary item. As their teacher I felt that while the children’s high self-assessments were accurate for the most part, one student exaggerated his understanding in an effort to fit in and possibly protect his lack of confidence with the material. By the time our review study of these six musical vocabulary items was complete, the ethics review board had approved my research proposal and we were ready to begin the project. And so it began …  71  4  The Inquiry Cycle—The Narrative Unfolds The design of this a/r/tographic inquiry involved the components of the Inquiry Cycle—Ask,  Investigate, Create, Discuss, Reflect (see section 3.3) and the Reggio Approach's Progetazzione (emergent, child centered project work) (see Figure 3.1). The inquiry cycle was implemented with the following general outline.  Figure 4.1 Session Summary in Inquiry Cycle Design © Bertram C. Bruce, Building an airplane in the air: The life of the inquiry group. In J. Falk & B. Drayton (Eds.), Creating and sustaining online professional learning communities, 2009, adapted by permission.  72  4.1  Ask  We entered the inquiry. Or was it a continuation of the inquiry? We gathered in the music room during the grade three class’s regularly scheduled music time for our first 90minute session dedicated to this project. We began with a conversation about the Inquiry Cycle. Together we discussed what we know about the ask, investigate, create, discuss, and reflect stages of the Inquiry Cycle. Due to the children’s extensive experience in PYP Units of Inquiry, they were easily able to describe each stage of the inquiry cycle and, when prompted, offered examples of past projects where they used these stages of inquiry. Then I posed the original research question to the students and their teacher, “How can music be used as a language of expression for student understanding of multiplication?” As I wrote this question on chart paper, Christian interrupted me. He informed me that my use of the word “how” was an assumption that music could be used as a language of expression, and was, therefore, not an authentic inquiry question. As a group, we then modified the question to read, “CAN music be used as a language of expression for student understanding of multiplication? If so, how? If not, why not?” Thus, the co-emergent inquiry process influenced the research question itself. What fun!  73  Paige then asked me, “Do YOU know the answer?” This question was poignant for me because it illustrated the difference between co-emergent inquiry and guided inquiry. “To be honest, kids,” I replied, “I don’t know the answer! This is a bit of an experiment where we’re going to enter into a co-emergent open-ended inquiry project. Where we will all just figure it out together and I don’t know how it’s going to go.” “So, if we can’t make music for understanding multiplication then will you fail your project?” asked Katy. “Absolutely not.” I reassured her (and myself!). “The whole point of asking the question is to find out what the answer is. There is no judgment either way, if we can do it, great, if we can’t, it’s fine either way.” I paused for reflection, and then added, “What are your predictions? What do you think is going to happen?” “I have no idea,” mumbled Katy. “Chances are good it’s gonna work,” said Lara. “We’re probably going to be confused at first, but then we’re gonna get it,” articulated Ethan. “It’s gonna be cool (laughter),” said Lara again. “We’re gonna have a hard time, but hopefully it’ll work out,” intoned Katy. “Maybe some people will understand it and some people won’t.” “Yes, that’s possible. It might not be everyone’s language. Maybe if we work in groups that will help a bit,” I added. “I agree with all of your predictions. This is what happens in every inquiry unit that we do. We get the confusion at the beginning, and then by the time I ask you the summative questions at the end of the unit, you’ve got it,” Jenny confirmed.  74  “Yeah! That’s inquiry.” I chirped. The kids were keen to start investigating how we might answer the question. “Can we use instruments?” “Can we work in small groups?” “No, I wanna work as a whole class.” Again I explained that the project could unfold in a number of ways. The difference between IB guided inquiry and enactivist co-emergent inquiry is in the create stage. Both inquiry styles have a summative assessment that usually consists of asking the children to describe/answer the central idea or, in this case, the research question. While I hoped that the children would find some ways to answer the research question in the affirmative, it would not be a failure to discover that they could not. Both inquiry styles ask children to create a product by the end of their inquiry (in our case, a composition). However, there were no prescribed learning outcomes for this project. That is, how the students answered the research question was entirely up to them. Therefore, the learning outcomes would vary depending on the strategies utilized by the students. Guided inquiry imposed prescribed strategies that the children might use to answer the inquiry problem, as opposed to discovering their own strategies and direction in co-emergent inquiry. During the create phase, I pushed some compositional skills that allowed the kids’ ideas to coalesce in a musical way, but I was not assessing the children on their compositional skills as much as I was interested in the communication strategies for expressing multiplication that were discovered. While the students had many questions about our inquiry, the group readily accepted this open-ended approach. What ever happened, would happen. The question, “What do we know?,” extends from the pre-assessment, front-loading  75  phase of inquiry into the ask phase when seeking to fully understand the task. We required a detailed examination of the important words (“key concepts”) within the research question/central idea before answering the question itself. “Key concepts” in IBPYP Units of Inquiry are the big ideas within the question/central idea (Primary Years Programme, 2007). The term concept is not to be confused with the word element. The elements of music (e.g., rhythm, melody, form, etc.) have concepts within them. In the case of music “we teach music concepts that use the elements to explain how the elements work in a variety of music contexts”(Gouzouasis, 2003, p. 226). This evaluation of the important words in our research question prepared students with a robust comprehension of the question before moving on in the investigation. The children decided that most of the words in the question where complex and worthy of closer scrutiny. In a round robin “butts up” activity (see section 3.4) the children moved around the room deliberating and writing their ideas about each of the “key concepts” (e.g., expression, music, multiplication, language, understanding). This is what they wrote: (Can) Music: - is a language - is a form of expression - can be the way you feel - can help you understand things - is communication - is having fun - is a song - is something you play with an instrument - is always a tune - is sound - is a way of saying something to someone - is making sound - is a note, notes and other things  76  -  peaceful, angry, happy or sad sounds something to hear (fraap!) is a language through melody and beat  (be used as a) Language: - speaking something different - one language that uses no words is music - a way to communicate - writing, thinking, talking, chit chat, speaking, words - people from different places speak different languages - is different ways to communicate - language is culture - you could speak a lot of languages - a way of expressing yourself - some languages don’t use words (of) Expression: - is a facial expression - expression can be a way of communication - doing what you feel - showing/telling somebody how you feel - emotions shown - pain shown (for) Understanding: - knowing what they are speaking about - knowing how people feel - knowing different languages - like knowing what they’re doing - knowing - a form of knowing - taking a walk in someone else’s shoes - understanding when a person is feeling down - understanding why, doing something stupid - you could understand that people say something - understand what you see - knowing what something means - knowing what to do - seeing what other people see - understand what other people think - to know something - understanding is important - not getting mad at somebody (of) Multiplication: - skip counting backwards  77  -  fast addition games of math 2x2=4, 4x4=8 having fun a way of learning a way of speaking in a language of numbers is fast adding is math a way of figuring out a problem putting things in groups Next we looked up the “key concept” words in the dictionary to compare our  conceptions to the official definitions. After sharing our answers and the Webster’s definitions with the whole class, the children felt confident that they understood the research question.  4.2  Investigate  We began this inquiry by asking, “What do we know (about music and multiplication)?” Next, we posed and contemplated the research question itself. Then we moved into the investigation stage of the inquiry process and were ready to seriously consider, "What do we want to know (about how music can serve as a language of expression for understanding multiplication)?" The investigation phase began with a quick review of some elements of multiplication. Similar to the music front-loading preparation, Jenny (the classroom teacher)  78  and the class had completed their multiplication unit a few weeks previous. Since we had been focusing on music vocabulary during the pre-assessment phase, I had not had an opportunity to discuss the details of multiplication with the class until this point. In a large group discussion we considered the following properties of multiplication. Commutative Property: When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. (e.g., 4 x 2 = 2 x 4) Associative Property: When three or more numbers are multiplied, the product is the same regardless of the order of multiplication. (e.g., [2 x 3] x 4 = 2 x [3 x 4]) Multiplicative Identity Property: The product of any number and one is that number. (e.g., 5 x 1 = 5) Distributive Property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. (e.g., 4 x [6 + 3] = [4 x 6] + [4 x 3]) Banfill, 2009 The children were able to describe and give examples of these properties, but they were not familiar with their proper titles. During this group discussion we also talked about the multiplication tables, which evolved into an animated conversation about cell division and exponential growth. The ideas generated during this discussion later became the seeds for inquiry development for some of the students in the class. After talking theoretically about multiplication with the class, we introduced concrete manipulatives (e.g., blocks, buttons, and other counters) for the children to use to demonstrate a simple multiplication equation. As guides in this co-emergent curriculum, Jenny and I wanted the children to start taking their thinking of multiplication off of the page of written numbers and start to apply/express it in different ways. Teacher intervention in this way, like in Reggio Project work, is how teachers make supportive suggestions to further the process of discovery, especially if the children are “stuck” (Rankin, 1998).  79  We asked the children to choose a simple multiplication question and show it somehow using the manipulatives. Using objects to begin to express multiplication away from the written page served as an introduction into the use of alternative languages in a mathematic context. As described in chapter one, an enactivist approach encompasses constructivist scaffolding by allowing the children to move sequentially through stages of understanding. While enactivism and co-emergence sound very loose in structure, they are not. This kind of inquiry does not just throw people in with a “sink or swim” attitude. Co-emergence, as I interpret it, is an umbrella that embraces more traditional methods of teaching aw well as constructivist methods, yet goes beyond both by allowing for more personal discovery and student generated learning. In the Reggio approach, children have a hundred languages to express themselves (e.g., sketch, clay, paint, wire, blocks, etc). Jenny and I introduced concrete manipulatives (like buttons and marbles) to serve as stepping stones for beginning to express their understanding of multiplication in using a familiar language other that writing. This exercise also demonstrated the children’s current understanding of multiplication. Most children already understood grouping and fast adding. But some could not unpack the theory behind the concept.  For example, some children took unifix cubes and used them to represent the number in the equation 12 x 12 = 144. After some discussion, they discovered that in order for the 80  cubes to truly represent the equation, they were going to have to move them into groups rather than representing the number concepts only.  In essence, the question and the answer were one and the same. When representing multiplication using manipulatives, if the groups are obviously represented then one does not need additional material to illustrate the question as well. The question is in the answer.  Other groups were able to show their simple multiplication equations using marbles and connector blocks. For example, 5 x 5 = 25. A group of girls also pointed out that 5 + 5 = 10, 5 + 10 = 15, and 15 + 10 = 25, or as they concluded 5 + 5 + 5 + 5 + 5 = 25.  81  9 x 2 = 18. Nice and simple: two groups of nine are 18.  6 x 6 = 36 Red Block, 6 x 5 = 30 Blue Blocks. After the children described and discussed their processing and reasoning for using the manipulatives in these ways, it was clear that we were ready to move further into the inquiry. At the next session we had a large group brain-storming session about how we could further transfer our expression of multiplication—from abstract symbols, to concrete manipulatives, to music elements and concepts. Some kids suggested piling up guitars and other categorized instruments into groups and using them just like the blocks and marbles. After more discussion they considered that this would not really be music making even though it was instrument use. Katy suggested hitting one drum and then hitting another drum harder and showing 1 x 2 through volume differentiation. Karen suggested another idea. “We could have one song times another song.” “But one times one is one, so that wouldn’t make two songs. One PLUS one is two,” replied Ethan. “How about two songs times two songs is four songs?” offered Chan. “No, that wouldn’t work. How can we do that? It would be like two beats, but then somehow become four …” Katy replied. I paused the conversation to reiterate Katy’s question to the class. “How can we do that?”  82  Paige suggested making a song using words or notes in groups to add together. For example, she thought we could build three groups of 10 notes in a melody or 10 words in a lyric and perform them together (3 x 10 = 30). The discussion evolved further to relate the multiplication principle of grouping to what we know about rhythms and other musical vocabulary concepts. “So, this is where my knowledge in music ends,” Jenny, the teacher, mused, “but if we did illustrate multiplication showing 3 times 10 ¼ notes, which is equal to how ever many other kind of notes . . .” She paused, confused, but on the verge of understanding. “Should we do it slower?” This question spurred the conversation into a detailed delineation of subdivision. The children were not able to give a definition of subdivision of traditional music notation, but they knew how to make notes bigger or smaller. Katy was also able to describe subdivision of property. She told of how her parents were splitting their big lot into smaller lots to sell to people to build houses on. “That’s right!,” I said, “Subdivision is breaking something big into smaller parts.” I drew a rectangle on the board. “What’s half of a whole?” I asked. “A half,” many replied. “What’s half of a whole note?” I asked. “A half note,” said Ethan. “Fractions!” cheers Kalen, which evoked laughter from the class. It turned out that they had just begun a unit on fractions during their math periods. Serendipitous timing! Kalen’s exclamation reminded the children that they were in familiar territory again. This was, in terms of performative inquiry (Fels, 1998), a wonderful moment of recognition (i.e., learning). We ended the session by breaking down a whole note into two halves, four quarters, eight eights, and 16 sixteenths. Joshua subdivided all the way to 8192nds! The children thought this was hilarious and rolled their tongues to show how fast  83  those notes would be going. “Pretty soon it would be like bzzzzzzzz,” Ethan vibrates. “Then it would basically be like we were playing the notes altogether,” concluded Joshua, showing his understanding of sound vibration and frequency (Hertz). Our chart looked something like the following graphic.  Figure 4.2 Subdivision of a whole note  Our next session was back in the music room. Using the same simple multiplication questions used during their manipulative illustration, the students were invited to explore the non-pitched percussion instruments, xylophones and metalophones, recorders, the piano, drums, paper and pencils, boomwhackers, and their own voices (and anything else in the room that seemed appropriate) to begin representing their question. This freedom definitely placed all participants in the borderland space between order and chaos. With enthusiasm, most children moved around the room searching for the best materials to suit their needs. There were only a few children who required extra  84  encouragement to experiment with either the instruments or how they might express their question. While I was open to offering suggestions, this first session with instruments was an opportunity for the children to test early ideas and work together to explore the concept of multiplication through music. Some children tried a variety of instruments, but, oddly enough, by the end of this experimentation session almost all of the children ended up with small hand drums. The children had naturally formed partners or small groups for this first session. Using their hand drums, as well as some other small non-pitched percussion instruments, each of the groups came forward to share what they had discovered thus far. Four sets of partners showed their questions with beats representing the numerals, some with hand movements illustrating the “times” and “equal” mathematical symbols. For example, Katy and Paige played boom boom (two drum beats) times (they crossed mallets to make an X) boom boom (two drum beats) equals (they held two mallets parallel to each other) boom boom boom boom (four drum beats). The girls explained that they wanted to do 12 x 12 = 144, but it was too long to be counting out that many beats. That is why they chose 2 x 2 = 4. Sally and Colby showed 5 (drum beats) x (hands crossed) 5 (drums beats) equals (they held arms parallel) 25 (they played 25 drum beats). Ethan, Kalen, and Santiago used a similar strategy in their question of 4 (woodblock beats) x 6 (drum beats) = 24 (small cymbal crashes). However, these boys did not include the hand movements to show the mathematical symbols times and equals, nor did they say the words out loud. Tricia and Karen demonstrated 2 (drum beats) x 4 (drum beats) = 8 (drum beats). They also did not indicate any mathematical symbols using words or actions. Ginny, Kira, and Christian performed 4 (Ginny played 4 bass drum beats) “times!”  85  (they shouted) 3 (Ginny played 3 bass drum beats) “equals!” (they shouted again), then Ginny played a drum roll as Christian and Kira moved quickly to where Ginny was standing until together they exclaimed “12!” These five groups used what Jenny and I later identified as “communication strategy #1.” In this strategy, the multiplication numerical equation is represented in abstraction, but the concept of multiplication is not accurately represented. That is, five beats followed by five more beats do not portray five groups of five equaling 25. This strategy seemed to actually imply 5 + 5, which would equal 10. Some groups added creative elements that began to move the expression of these simple multiplication questions into the realm of the arts. For example: the elimination of technical words, the use of choreography to imply symbols, and the inclusion of frivolity for artistic flare. Joshua and Chan were the first group to demonstrate the multiplication principle of grouping when they shared their question 6 x 5 = 30. They broke the equation down into two smaller equations of (3 x 5) + (3 x 5). Without speaking, first Joshua played three groups of five beats on his drum, and then Chan played three groups of five on his drum. That was it. Although this example was very simple it clearly represented the multiplication question. Jenny and I called this “communication strategy #2.” This is when beats, either steady or in a rhythmic pattern, represent the numerical groupings of a multiplication equation. Without knowing it these boys also demonstrated common multiples when they broke the equation into two equations. Joshua and Chan shared a second example that illustrated 3 x 4 = 12. Joshua counted 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, while Chan counted 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3. They were  86  demonstrating how the 4/4 and 3/4 meter patterns, when played together on a steady beat, “meet up” (i.e., are equivalent) every 12 beats. This demonstration was modeled after a warm-up activity I had done with the class earlier in the day where we clapped 3/4 patterns over 4/4 patterns to see where they intercepted each other. I intuited that this might become a very good strategy to help us during the inquiry development, but I was not sure how. “Let’s talk about Joshua and Chan’s meter play.” I paused for a long time, thinking about what direction to go next. I was having a similar experience that Jenny had when our musical discussion went outside of her understanding. “Jenny, I think I need your math skills here. This is a part where I am really feeling the co-emergence, because I can’t explain this to you guys in a way you might understand it. Plus, how can we show this musically?” With the children’s participation, Jenny came over and gave us a mini-lesson that described common multiples. Together we wrote out 12 beats grouped into fours, sixes, and threes. 1 1 1  2 2 2  3 3 3  4 4 1  1 5 2  2 6 3  3 1 1  4 2 2  1 3 3  2 4 1  3 5 2  4 6 3  1 1 1  Table 4.1 Common Multiples of 12 comparison in number groups  Then she pointed out how the groups came back to their starting place every 12 beats, which makes 12 a common multiple of 3, 4, and 6. The children deduced that the next available common multiple of these three numbers would be 24, then 36 and so on. We concluded that musical expression using common multiples would be “communication strategy #3.” After these early attempts to express multiplication using musical vocabulary, we had  87  a group discussion to critically analyze the approaches explored. Joshua, a boy who is gifted in mathematical thinking, very quickly summarized our concern with “communication strategy #1.” Considering the example 2 x 4 = 8, he said, “It’s like we played two drum beats times four drum beats. But you don’t really, like, you can’t really multiply two drum beats by four drum beats because that’s really like two groups of four drum beats, or four groups of two drum beats.” The children were beginning to understand how the representation of grouping using patterns was an effective way to show their understanding of multiplication. In fact, in this musical context, away from the written page or math manipulatives, a number of the children were just beginning to make the connection between multiplication equations and grouping. This is not to say that “communication strategy #1” was incorrect. As a first attempt to answer the thesis question it is the most obvious and superficial way to do so. Overall the children did not feel that it expressed their understanding of the principles behind multiplication, but instead demonstrated the equation in its abstraction. After this somewhat disappointing realization, we flushed out the same question (2 x 4 = 8) together as a group until we felt we had found some musical means to demonstrate the concept of grouping behind this equation. Since an enactivist educational approach encompasses constructivist philosophy (Carson & Sumara, 1997), this kind of teacherleading felt appropriate here when it seemed to Jenny and I that the children needed some prompting and encouragement to take their thinking about grouping to another level. “Tell me more about what this question means,” I asked the class. “Times!” “Two groups of four is eight.”  88  I drew 2 x 4 on chart paper, followed by two groups of four circles.  “Or four groups of two,” Santiago added. I asked the children how they might use rhythm to play this. Joshua demonstrated, “You could do: clap clap, clap clap, clap clap, clap clap.” He clapped his hands in the rhythm of four groups of two with a rest in between each group. I wrote his rhythmic pattern on the board:  “How we represent this musically is up to us! How can we show four groups of two with more than clapping?” “Piano! Snare drum! Xylophone! Shaker!” Children yelled from the group. The next thing we knew Kira was off to the piano, Chan was on the snare drum, Kalen was on the xylophone, and Lara was on the shaker. This ensemble tried to play the pattern on the board by each taking two of the eight notes. After some trial and error this little ensemble had an impressive rhythm established with a good sense of flow and musicality. Shake, shake, (rest), tone, tone (rest), plunk, plunk (rest), thump, thump (rest). Shake, shake, (rest), tone, tone (rest), plunk, plunk (rest), thump, thump (rest), and on and on – an excellent example of “communication strategy #2.” Using musical means in this way is the same as using the concrete manipulatives to express multiplication. That is, music (i.e., drums, shakers, etc.) and manipulatives (i.e., 89  marbles, blocks, etc.) are some examples of the hundred languages of children. Therefore, in beginning to answer the thesis question we were slowing discovering how music can be used as a language of expression for the understanding of multiplication. By the end of week two and the beginning of week three, we were ready to start moving into the create phase of the Inquiry Cycle where the student’s ideas could develop into wider composition that would express multiplication using musical means. There is an invisible boundary between investigate and create since in a sense there is extensive creation during investigation and further investigation required during creating. To move the children toward thinking about composing and our musical vocabulary, we revisited the traditional Hanukkah song called “My Candle” that we had performed at the Winter Concert a few months earlier in the year. The children were able to recall that the performance piece had eight verses, a group of singers, accompaniment parts on Orff instruments and non-pitched metallic instruments, a bass line, choreography with props, dynamics, and a rondo (ABACA) form. In preparation for composing their own musical expressions of multiplication, I asked the children to keep these elements in mind while developing their compositions. By week three, the children’s groups began to form. The factors that influenced their groups were (1) their approaches to answering the questions, (2) instruments preference, (3) the concept they were investigating, and (4) friendship groups.  90  4.3  Create  This is where things got messy. Again. The difficulty of setting the a/r/tographic experience to words is especially evident in this deeper stage of the inquiry process. Doing the “ography” of artmaking/researching/teaching is challenging because there are so many layers of detail and complication involved. At this stage groups of children (e)merged, personalities collided, the camera rolled, initial ideas were fleshed out and expanded, questions were asked, explanations were given, performances were critiqued, minds changed, arguments occurred, and the children’s compositions to express their understanding of multiplication using music were in process. Inquiry arose within inquiry. Within each creative decision was another cycle of inquiry. We discovered micro-cycles inside of macro-cycles. While the children grappled with the larger inquiry question, they had to ask, investigate, create, discuss, and reflect again and again while decision-making during the process of project development.  91  Figure 4.3 Inquiry cycles in inquiry cycles © Bertram C. Bruce, Building an airplane in the air: The life of the inquiry group. In J. Falk & B. Drayton (Eds.), Creating and sustaining online professional learning communities, 2009, adapted by permission. Since this action research project was co-emergent, during the create phase of the procedure the participants were given a lot of time for exploration and development. The students spoke/wrote using the musical vocabulary available to them, employing the language of music in their attempted expressions of the mathematical content. The stage was theirs for performative knowing--to perform their understandings, for being deeply emerged in the compositional processes, and for becoming composers/creators of new, uncharted knowledge in music and mathematics. “Create is the stage where children need to begin their independent explorations. Some children are able do this, some need lots and lots of coaching . . . and for some, it may  92  never truly work well for them simply because of their learning style and ability, ” reflected Jenny, the classroom teacher, while the children were off working in their groups. “Because you keep bringing them together and touching base and discussing ideas, the kids are getting a solid foundation for the project and frequent spring-board stops, just in case. This is great for most kids and you can see the transition for some at these levels and their progress into understanding. Some remain lost. Some are holding fast to their own ideas.” At this stage, Jenny and I tried not to introduce the children to new ideas that might have imposed our views on project direction. We led the children in a review of the six big ideas/themes that came to light during the three previous stages of pre-assessment front loading, ask, and investigate. The Big Ideas 1. 2. 3. 4. 5. 6.  The four properties of multiplication Cell division/exponents Subdivision Grouping/common multiples Fast adding The times tables  Jenny and I were there to ask or answer questions that helped the children reflect on their choices, but while we were co-inquirers in the project, the thesis question asked for the expressions themselves to come from a grade three level. We facilitated discussions about group strategies for expression and made anecdotal records. We offered “permission” for material use, and provided instructions for logistics or safety concerns, but overall we took a giant step back. Except when we didn’t, or couldn’t. Our intervention served to support the children “on their quest for a solution but did not impose upon them an adult one” (Rankin, 1998, p. 227). As co-investigators, Jenny and I  93  often pondered the children’s discoveries with honest questions seeking our own understandings of the subtle complexities encountered while digging into multiplication expressed through music. Because I had more music expertise than Jenny, I was more involved in the create stage of the inquiry. After the inquiry Jenny reflected, “I tried to remain more of an observer than a guide in that stage [create] of the project. I took notes and observations - and tried to get report cards done sometimes, but I did not intercede as much as I may have with my own projects. I was comfortable with this. When I felt it truly necessary to stick my nose in, I did! I circulated with the purpose of gathering observations rather than assisting and guiding, unless the kids asked me directly for ideas or opinions or help. The only thing I was uncomfortable with was what to expect from them in the end, as I didn't have a vision of how I would have conquered that project. I think this feeling came from my lack of musical knowledge.” During the investigate stage of the Inquiry Cycle, Jenny and I had labeled three of the communication strategies used in the children’s first attempts at musically expressing multiplication. As the create phase progressed, we observed that the children found a total of seven approaches to communicate their understanding of multiplication using music (see table 4.2 below.)  94  1. 2.  Abstract representation Patterned grouping  5 x 2 = 10 versus 5 + 2 = 7 5 x 2 = 10 5 x 2 = 10  3.  Layering of patterned groups Common Multiples  4.  Elementary algebra  5.  Subdivision  3 x 4 = 12 4 x 3 = 12 2 x 6 = 12 2 x 5 = 10 1/4 x 4 = 1 1/4 ÷ 1/4 = 1 1 = 1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4  6.  Lyrics  7.  Reciting times tables  Do 5 beats and 2 beats equal 7 or 10? 2 beats (in a pattern) + 2 beats + 2 beats + 2 beats + 2 beats = 10 beats 5 patterned beats on piano + 5 patterned beats on the xylophone = 10 beats 12 is a common multiple of 3, 4 and 6 and could be grouped and layered accordingly. Substitution: C = 2, 5 beats of C = 10 - Showing the relationship between the whole and parts - dividing and multiplying fractions, - finding and adding common multiples for fractions - comparing equivalent fractions Using Lyric content to explain the concept of multiplication. Reciting times tables through lyrics with beat.  Table 4.2 Music communication strategies for multiplication The following narratives are transcripts of the children’s explorations of the music communication strategies and their creative processes during the create stage of the inquiry: Tricia and Karen “We don’t know what to do.” Tricia and Karen wandered up to me after the groups have been working independently for a few minutes. “Which big idea are you focusing on?” I asked. “That’s what we can’t decide on!” they moaned. I reminded them about the times table recitation they were experimenting with earlier in the week. “Oh yeah. … we could do 3x4,” said Karen. “Sure, you could do the entire four times table,” I suggested. They looked at each other and returned to the storage room where they had been working. Later, when I popped  95  my head in the storage room, the girls informed me that they were going to do the whole three times table (communication strategy #7). Demonstrating what they had been working on, the girls kept a wobbly beat by hitting two very long boomwhackers (hollow plastic tubes cut to lengths of pitches in the C major scale) together as if in a sword fight while laboriously reciting the three times table up to 3 x 5. They kept working on it and by the end of the session Tricia and Karen each had two shorter boomwhackers to hit together independently, presumably for increased stability and to keep a steadier beat. Although the composition had not advanced beyond this initial stage, it was rhythmic and had a nice flow. In the next session, the girls indicated to me that they simply had no ideas on how to develop their composition. Remembering their song from our last session I used two of their boomwhackers to keep a steady beat while reciting the three times tables to five. I complimented them on their good start. “This would make a great A section in your composition,” I said, and then I asked them to join me on the carpet near the chart paper to talk about song form and expansion. As we reviewed the different sections of a song in ternary (ABA) and rondo form (ABACA) the girls started having ideas about how they could include other instruments into a contrasting B section. Without wanting to further dictate the direction of their composition I encouraged them to go back to their space and work on it. The girls returned to the storage room for a half hour or so, then came back to ask me to come and see their progress. They had written out the A and B sections on a piece of paper. The A section was still the three times table to five, but rather than simply reciting it using words, they had eliminated the words and the accompanying steady beat. Instead they  96  were using beats on the boomwhackers to whack out the numerals in the question and the products (communication strategy #1).  It was a confusing shift. The B section was a continuation of the three times table from six to nine. This time they included a tambourine and a woodblock to tap out the numbers in the equations and the answers (all the way to 27 taps!). That is, tap tap tap (3), bonk (times), tap tap tap (3), bonk (equals), tap tap tap tap tap tap tap tap tap (9).  97  When they were done playing, I asked, “What do you know about one, two, three, times one, two equals one, two, three, four, five, six?,” clapping the numbers. “That it’s just the equation and not really multiplication,” Tricia answered pointedly. “This,” she pointed to the chart paper, “is just showing that you know the equation. That you know what the numbers are in the equation, but you just don’t show how you DO multiplication and stuff like that.” “Right. I knew that you knew that! But, musically . . . I think you’re stuck.” She looked at me with a wry smile, and then nodded brightly as if to say, “Duh!” “I remember how you recited the times table earlier today. Could you bring that back in?” “Yeah . . . especially for the higher numbers because then it’s not so du-du-du-du-dudu-du-du-du-du,” Tricia answered. Karen remained quiet, listening while running her pencil up and down the groove of a ruler. “Great. Okay, and for this next part, the B section, how could you show a pattern that might be two groups of three or three groups of two?” I asked. Karen piped up now. “Maybe we could have different instruments. Like one playing one two, then another one playing one two.” “Yeah, cause then the instruments are showing the groups.” Tricia interjected. The girls and I talked some more about the different sections of their song and brainstorm some ideas for how to make the song more musically interesting. We agreed that they should keep all of the elements they have come up with so far. That is, the recitation (Strategy #7), the equation representation (Strategy #1), and the grouping (Strategy #2). As I  98  left the storage room, Tricia and Karen picked up their boomwhackers for some more practice and planning. Santiago and Christian I released the children to the task of beginning to use their musical vocabulary and communication strategies to create their compositions about multiplication. Christian roamed around the busy music room with hands in pockets watching the other children getting organized. “As I said, you can work in groups, you can work alone. Basically, you can do what you want,” I called over the growing noise. Ethan, a skilled pianist, begins playing the “Mission Impossible” theme song on the piano, which drew Christian over. After asking Ethan to focus on the task at hand, I tried to engage Christian by suggesting that maybe he and Joshua could work together. Joshua, however, chose to work alone. Overhearing this, Santiago, who until this point has been searching through the percussion instruments approached Christian and said, “Hey Christian, you could work with me.” And so, their alliance was formed. It took some time before the boys saw that the room was truly open for their use. “Boy, oh boy!” exclaimed Santiago, when he saw the plethora of hand drums available to him. He selected the big blue Egyptian sadaf drum and carted it around the room for a few minutes while visiting with other groups. Christian got some paper and a pencil from the table and settled down on the floor. He wrote “equation brainstorm” at the top of the page just as Santiago joined him. They discussed what possible questions they could choose and Christian wrote a few of them down while Santiago played his drum.  99  They became distracted because Santiago flipped the drum over and reached his arm deep into it. Christian was lying on the floor. When I moved toward them they both refocused, and Santiago suggested 7 x 3. Not a minute later, Christian was lying on the floor again and Santiago implored him to get to work. “Your friend is hoping to get moving on the project,” I intervened, “so, maybe you could help him out and respond to his request.” They got up and went in search of more drums. This pattern of focus and distraction happened over and over again as the session wore on. Much later, while I was working with other groups, Christian asked me if I could make the room quieter because he was having trouble hearing himself “testing his own music.” I suggested that they go outside where there were less people since it’s tricky in the music room with so much experimentation. The boys then “re-discovered” the drums (what were they doing for the last 20 minutes?!) and started moving them outside onto the lawn.  100  This move proved to be motivating. I overheard Santiago declare, “This is so cool. We made a drum set!” Then the boys set to work creating a simple composition based on their equation 7 x 3 = 21. At the end of this session Christian was playing seven groups of three beats on seven different drums (communication strategy #2). Santiago’s approach was to hit a combination of drums randomly, 21 times. At the time I thought this indicated that he didn’t fully understand the concept of grouping, but I later came to realize that he “got it”—he just much preferred to hit those drums! The fun and freedom of exploring with drumming was a huge factor for those boys. I tried to offer some suggestions and ask a few questions about how they could advance this concept and use it to expand their piece. Lost in the world of drums, they didn’t respond to my questions, so I left them to work hoping that they were thinking about it. This partnership worked consistently with their question over the next two create sessions. Their attention inevitably wandered on and off task every few minutes with light saber drum stick battles, flying Frisbee drum pads, long fanciful story-telling, and wandering around to visit other groups. But, in the end their composition evolved into an attentiongrabbing performance piece that clearly, albeit simply and concretely, communicated their  101  understanding of multiplication by demonstrating a simple pattern of seven groups of three along with dynamic drumming. Sarah and Carly Two small hand drums just aren’t the right tools for the job, Sarah decided. The two girls returned to the instrument shelves and Sarah muscled an alto xylophone down to the floor. Carly kept her drum. Quickly going about the business of gathering everything they needed, they sat down to make some attempts at playing 6 x 5. First, Sarah played five beats on both low and high C of the xylophone, and then Carly echoed it on her drum. They discussed whether or not they should show the times sign by crossing their arms. “Naaaa.” Over and over they played the five groups of beats, then they tried playing the product of 6 x 5; all 30 beats in a row. In a few minutes, this process began to sound musical. Sarah played:  Carly answered:  They did this three times through, and then played 30 beats together. Good! They smiled at each other, satisfied. Just then, I walked into the room and bellowed over the noise, “Remember, there’s paper and pencils on the table if you need it! Write down your work.” The girls headed over to the table to get started.  102  Working now with five groups of six rather than six groups of five, Sarah blew a B note six times on the recorder and Carly responded with her own six drum beats. It became obvious by watching these girls that Sarah was the leader in the partnership. Carly seemed quite happy with this arrangement and the girls worked hard for the whole session counting, tapping their toes, and playing their instruments. By the time the girls were ready to share their discoveries from the first create session with the class, Sarah moved on to her fourth instrument – a djembe. Carly remained on her initially chosen, small hand drum. “We’re doing 5 x 6 = 30, but we’re adding 6 +6 +6 +6 +6, like that, five groups of six.” Sarah described. Carly softly tried to interject to demonstrate that she also knows what is going on, but didn’t manage to actually say anything. When they began to play, instead of simply playing the nice rhythmic groups that they had rehearsed, the girls didn’t use a steady beat and said “plus” between each group. Thus, it ended up sounding more like math than music. During the next create session Sarah began combining the instruments that she had experimented with during the first session and wrote excitedly on their sheet. Carly watched Sarah for a long time, sitting quietly with her small hand drum in her lap. Eventually, Carly pulled a contra bass bar over and was now able to play both instruments. 103  These girls worked very hard at putting their composition together. They added more instruments, and became experts at tapping their toes to the beat. They added extensive detail to their sheet on who does what, and even added some words (lyrics or directions?). Later, Carly told me that although it wasn’t assigned, the girls had written down two ideas about multiplication so that they could “make sure that we understood multiplication.” At one point in this session Sarah took Carly’s hand drum away from her because it looked like it was too difficult for Carly to hold simultaneously with the two mallets required for the other instruments. But, by the end of this create session the composition became more detailed and musically challenging.  104  Since Carly was missing her hand drum, the pair started finding more percussion instruments to add into their song. They revisited the five times table with a guiro, woodblock, djembe, and finger cymbals. Using “communication strategy #1,” the girls illustrated the first three equations in the five times table. After some discussion, they changed it to use “communication strategy #2,” so that they were adding the groups in the questions.  105  They wrote out the following new section.  During a private session with me, the girls showed me all of the parts of their song and their notes. We watched some video clips of the girls’ explorations from previous sessions. I encouraged them to combine everything that they have composed so far into one smooth composition – including the definition ideas that they have written about multiplication for their own reference. They looked at me blankly when I suggested this, so I showed them some video data of Tricia and Karen reciting their A section with a steady beat. “What do you think?” I ask hopefully, visualizing a great lyric section. They played a steady beat and read what they wrote. They nailed it! Not much more work was needed for the completion of this piece. Carly retrieved her hand drum, and they spent the next class practicing and perfecting the flow of their composition. What a team!  106  Paige and Chan After Joshua turned down his request for collaboration, Chan sat quietly experimenting with the contra bass bars when Paige found him. She had also failed in her attempts to include herself in some previously formed alliances. Ugh, the dreaded moments of childhood when you don’t have a partner and it seems like everyone else does. Those who were your friends yesterday might not be today. And so, in an effort of confidence or desperation, you ask to join, you work alone, or by default you join forces with other outsiders. Chan and Paige auspiciously came together in this way, and in short order they worked effectively together. This interesting partnership enacted a brilliantly intricate unearthing of mathematical concepts and obscure musical expression. “Can we do more than one question?” Paige asked as I walked by. I assured them that they could. “We’re gonna do 20 x 2 = 40. And 40 x 2 = 80. And Chan had an awesome idea. What we’re gonna do is the C is gonna be 10, and the F is gonna be, uh . . .” she paused. Chan finished her sentence, “20.” After some experimentation, Chan discovered a solution to having to tap out many beats for larger equations. Although he was only eight years old, in an effort to solve a immediate problem, he discovered the grade 6/7 algebraic operation of substitution. When they had a chance to share their findings from this session with the class, they showed 20 x 2 = 40 using “communication strategy #1.” They illustrated the equation with 20 F notes plus 2 drumbeats. Then Chan hit the drum 40 times to show the answer. Having already seen a number of examples of this, the class quickly pointed out the shortcomings of  107  this approach. We discussed how this is just one of the many ways that multiplication can be represented using music.  When they described to the class their approach for 40 x 2 = 80, we realized they hit upon something profound. Chan explained, “Just pretend that C is 10, pretend that F is 20, and that D is just normal, just one.” Paige played the pitches, F F C C C D, then said, “So that would be 80, cause . . .” “No, that would be 81 with the D,” Joshua, our attentive mathematician, interrupted, “cause D is 1.” I asked them to do it again so that we could count. “We might not need the D,” suggested Chan. Paige played the pitches and I counted—F “20,” F “40,” C “50,” C “60,” C “70,” D “71” . . . a pregnant pause while Paige looked up at me. Chan grabbed Paige’s mallet, his hand on top of hers, and hit D nine more times! “72, 73, 74, 75, 76, 77, 78, 79, 80!” The class began to count out loud, and then we all laughed!  108  “Oh yeah, we needed 10 of those,” reflected Paige sheepishly. During the next create session Chan told me, “We want to make it harder.” Paige started writing a very complicated script of notes and numbers on the page. When I came around to take a look, I couldn’t make heads or tails of it, but Chan and Paige played it for me with great understanding.  Using the contra bass bars, two boomwhackers, and a djembe, they definitely succeeded in their efforts to make it “harder,” but because the chart was not clear and the composition itself had no rhythmic or melodic flow, it sounded clinical rather than musical. I suggested that they write out another chart to explain their intensions to the audience more clearly. However, by the time they were ready to share their findings with the class again at the end of the session, this harder section had been put aside. This time they told us that B was 40 and the boomwhacker was two. For the question 40 x 2 = 80 they played the B flat once and the boomwhacker once - the end! This could be interpreted as 40 + 2 =42, or 40 x 2 with no answer represented - again “communication strategy #1.” Paige and Chan were clearly struggling with both their musical communication skills AND the concept of  109  multiplication. They were on the right track, but there was a gap in their expression from theory to practice.  They had a second example to share that day with detailed number substitution for contra bass notes, but were unable to express it in a way that the class could understand. Fortunately, I had time for a lengthy private session with Chan and Paige later in the week and we were able to work through their composition together. We sat down with their pages of written work and tried to make some sense out of their ideas. They told me that F = 20, and they played it once. I translated/wrote: 20 x 1. After some discussion, both of the children were able to transcribe this equation into algebra: F x 1 = 20. We went through the rest of their substitutions together. C = 5, and they played it twice. We translated/wrote: 5 x 2 = C x 2 = 10 D = 2, and they played it five times. We translated/wrote: 2 x 5 = D x 5 = 10 A = 1, and they played it 20 times. We translated/wrote: 1 x 20 = A x 20 = 20 In their attempt to musically and mathematically delineate 40 x 2 = 80, they actually created the four multiplication equations above, and one additional question: 20 + 10 + 10 + 20 + 20 = 80. It turned out that they weren’t representing 40 x 2 at all, rather they were  110  exploring numbers for which 80 is a common multiple (communication strategy #3). Not to mention their exciting discovery of algebraic substitution (communication strategy #4). I took their work one step further by pointing out that in the addition question: 20 + 10 + 10 + 20 + 20 = 80, they were using three groups of 20 (20 x 3 = 60), and two groups of 10 (10 x 2 = 20), therefore 60 + 20 = 80. The two children understood this. It was “communication strategy #2.” Then I elaborated that if F = 20 and B = 10, then F x 3 =60, and B x 2 = 20. In an wonderful display of comprehension, Paige then took it a step further during this conversation to point out that since F = 20, B = 10, and D = 2, then B x D = F! The kids giggled with delight when I mentioned that without any numbers in the equation there would be no way to figure out what the letters represent. What a riddle! These two children spent the next two sessions figuring out how to make all of the parts of their composition flow together with musicality. They added a steady beat, an introduction, and an ending, and were soon ready for their performance. Joshua, Lara, and Katy Joshua worked alone, pacing the room with his recorder in his mouth. Lara and Katy headed outside with their recorders. The girls had been working with the big idea of subdivision since Katy suggested it during the investigation phase of the inquiry. On the B note on their recorders, together they played a whole note, then two half notes, then four quarter notes, then eight eighth notes (i.e., communication strategy #5). “Do you know any more notes on the recorder?” I asked. “Yeah, we know all the notes!” replied Katy. Ten minutes later the girls played the same subdivision pattern on a descending B, A, G, F line. “You’re on to something really good there, but it’s only the beginning of an idea.  111  How can you extend it?” I probed. A few minutes later the girls descended from B to F while subdividing a whole note, and then ascended back up again in the opposite pattern.  Meanwhile, Joshua was helping Ethan and Kalen consider chord progression options for their composition. When I had the chance to ask Joshua what he had come up with so far, he played the identical pattern that Katy and Lara played earlier, except he subdivided on B all the way to sixteenth notes. When I asked him what math concept he was representing he thought for a very long time before replying. “Well, it would be like one equals two halves, equals four quarters, equals eight eighths, equals sixteen sixteenths.” “Could you make it into a multiplication equation? Or if it’s division, could you make it into a division question?,” I asked. Again, he sat for a very, very long time in silence looking away, thinking. No answer. Eventually, I left him to his thoughts. When we shared our findings that day, Katy and Lara played their ascending and descending subdivision line, which they were now calling “cell division,” drawing on another of the big ideas from our earlier investigations. I asked Joshua if he would like to share directly after them because I suspected that they might find each other’s discoveries interesting.  112  “Sure,” he said to answer my question, then continued, “but was that actually cell division? Because cell division is doubling, and subdivision is just breaking it into smaller pieces but still having the same amount.” After reviewing our notes from previous discussions about cell division and subdivision, the girls agreed and stopped referring to their song as “cell division.” The next time the children came to work on our project, Joshua was absorbed into Katy and Lara’s group. The three children sat outside in the shade figuring out how to turn their subdivision ideas into an actual musical composition. Their practice required a big djembe on which they kept a steady beat. This helped them keep the subdivision more accurate because they had previously tried to keep track of the beat by counting on their fingers.  Four sections of a challenging song about subdivision came together during this session. In each section a different person kept the beat on the drum while the other two played up and down the subdivision pattern. One person would start on F playing sixteenth notes and move upward, and the other person would start on B playing a whole note and move downward. I asked the children to think about how to get the sections to flow together musically.  113  In our next large group discussion we summarized the strategies that the children were using for their communication of multiplication through music. We began to unpack the theory behind Lara, Katy, and Joshua’s subdivision composition after their teacher, Jenny, asked, “Now, is that even showing multiplication?” I attempted to reason on the group’s behalf, “It’s fractions, and division, which if you just take it the opposite direction is multiplication, right? But, let’s write down what we’re thinking. Katy?” As Katy described their song, I drew the subdivision of a whole note on the board, and the corresponding fractions next to the notes: 1/2, 1/4, and 1/8.  Figure 4.4 Subdivision of a whole note Jenny scrutinized the picture. Her eyes widened and she whispered to me, “When you multiply fractions they get smaller.”  114  I laughed and shared the secret, “Your teacher is worried. Multiplying fractions is grade 6 work. We’re afraid! We don’t want to go there!” “Yes, we do!” the children exclaimed. “Well, let’s talk about it. When you multiply 1/2 x 1/2, what do you have to do?” I asked. “You get 1/4,” said Joshua. “Why?” I asked genuinely. “That’s confusing,” Ethan replied. “That’s ‘take away’,” said Lara. “I don’t understand it myself,” Jenny replied. “Times means ‘groups of,’ so half times half is one half of a group of half.” Joshua enlightened us. Our wheels were turning as we considered this explanation. He continued, “Half times a half is like a half group of a half . . . if you take a half group of a half you get a quarter.” Jenny summarized, “Multiplication of fractions is division…” “Oh my god, that’s confusing,” interrupted Katy. “. . . and division of fractions is multiplication, right?” Jenny finished. “This is supposed to be music not math class!” complained Ethan in jest. The class burst into excited conversation since this new direction took us all into uncharted waters. “Hang in there guys, we’re on to something here!” I called over the commotion. Joshua was the only one in the room who understood what is going on! After pacing around the edges of the carpet where the class was sitting he approached the white board and took the marker out of my hand. He drew the equation as he spoke:  115  1/4 ÷ 1/4 = 4/4 = 1 1/4 x 4 = 4/4 = 1 “You have to do either 1/4 divided by 1/4, or 1/4 times 4 to equal a whole. So, 4 ‘groups’ with one 1/4 note per group equals a whole.” Everyone was a little dazed by this intellectual leap. We teachers barely understood what we just learned—more like “remembered” from our own elementary school education-and the children only slightly saw where Joshua was going. We discussed how to multiply and divide fractions a little more during the session, but mostly we were lost – except Joshua. I summed up this discussion by asking Joshua, Lara, and Katy to try to find the multiplication problems that fit with their composition. This group did not, however, revisit these mathematical concepts. Perhaps it was because only Joshua fully understood them, or perhaps it was because there was only one class left before we were to share our compositions. Instead of finding multiplication equations for their song, this group used their remaining time to find alternative instrumentation, to add an introduction and transitions between the sections, and to write out their chart coherently. Ginny and Kira They lingered with the chimes and congas. They looked around dubiously at the lively classroom; children were moving and playing and talking and busy. They squinted at the noise. “Can we go outside?” They lugged their heavy instruments out the door.  116  Ten minutes later, Ginny and Kira were no longer partners; instead they were working alone but parallel to each other. Kira sat in the doorway with a hand drum watching the other children. Ginny came over and said, “I can’t think of anything.” I asked her what question she was working with before. We looked on the board for the big ideas the class had already come up with. We talked about the two times tables. She did not appear impressed with any of my suggestions or questions. Again, I invited her to just take some time to explore. She picked up the chimes and headed back out the door. Soon Ginny returned to ask me to come and see what she has worked on so far. We passed Kira on our way out the door. She looked morose. Should I intervene? This initial create session was “supposed” to be about self-directed learning and cooperative group work – if they chose to work in a group. These girls did not thrive on either of those concepts. I didn’t want to interrupt Kira’s process, but also wanted to support her learning. As we passed by, I suggested to Kira that she get some paper and begin recording her ideas. Ginny showed me her rendition of 36 x 2. It consisted of 36 toe taps, followed by two swipes of the chimes. Then she made a little “equals” sign with her fingers and then clapped 72 times. Using “communication strategy #1,” she had a start.  After Ginny finished sharing her work with me, I stopped in to see what Kira was doing. She had a paper, pencil, and hand drum and was busy writing out what I thought were  117  some ideas for the project. It turned out that what she was writing were song lyrics about friendship and her difficulties in working with Ginny. In an effort to redirect her focus out of this lament, I praised her lyrical skills and asked if she could communicate an idea about multiplication using song lyrics. I drew Ginny into this conversation. The girls tossed around a few ideas--Ginny creatively, Kira reluctantly. Somehow, during this process of brainstorming ideas, Ginny and Kira had become partners again. However, by the end of the session, no song lyrics had been written. Instead, Ginny was playing with the same equation she had showed me earlier, and Kira was keeping a steady beat. I can only presume that any efforts to write song lyrics were abandoned at some point in favor of Ginny’s original question, but at least they were finding ways to continue working together. During the next create session the girls gathered the materials necessary to continue developing their song: a kick drum, wood blocks, jingle bells, djembe, xylophone, and chimes. They decided to change their question to 25 x 2 after discovering that there were 25 bars on the chimes. If they played one swipe down the chimes then that would comprise one group of 25 (communication strategy #2).  118  Eventually, with much arguing on their part and refereeing and facilitating on my part, Ginny and Kira found a few other ways to play 25 x 2 = 50 using “communication strategy #2.” For example, 5 x 5 = 25, therefore they created 5 groups of 5 to show 25. Rather than playing this pattern twice, they decided to put two beats in each group (5 x 2 = 10). This expression became 5 x 10 = 50 when they played the two beat pattern five times on five different instruments. Their equation became: (5x2) + (5x2) +(5x2) +(5x2) +(5x2) = 50.  119  The girls were ready to take this idea to a higher music level of understanding by adding an introduction and an ending. Kira wanted to do a stand up comedy sketch for their introduction. She practiced it enthusiastically. The A section would become (5x2) + (5x2) +(5x2) +(5x2) +(5x2) = 50 played on five different instruments and the B section was 25 x 2 = 50 played on two swipes of the chimes. Ginny spun her drumstick three times, and then struck a pose for the ending.  120  Unfortunately, the girls didn’t have a chance to perfect their piece because both of them were absent for the remainder of the project, although Kira was available for a reflection session with me at the very end of the project. Ethan and Kalen Ethan bolted to the piano when we broke from the discussion. Kalen was close on his heals. We were just beginning our explorations in the create phase of the inquiry and these boys were keen to get started. Or were they keen to play the piano? The “Star Wars” theme song became the “Mission Impossible” theme song became . . .“Ethan, that is not focused,” I said, not for the last time. Kalen, not a pianist like Ethan, sat at the high end of the piano plunking a few notes with his two index fingers while Ethan read over the notes we made on the board. He then played five quick notes on middle C. Not satisfied, he played a C major triad five times instead. The boys didn’t speak. Kalen watched as Ethan then plays all the triads in the C major scale ascending in a rhythmic five beat pattern. He has unintentionally used “communication strategy #2” to express 5 x 8 = 40. Joshua and Christian wandered over. For their entertainment Ethan bust into the “Indiana Jones” theme song. Kalen spoke to the other two boys excitedly as Ethan played. Santiago arrived to collect Christian back to their work and Joshua resumed his wandering recorder subdivision exploration. Next Ethan tried to play a whole note on C major triad, two half notes on D minor triad, four quarter notes on E minor triad, eight eighth notes on F major triad, then four sixteenth notes each on G major, A minor, B diminished, and C major triads. Subdivision, “communication strategy #5,” was Ethan’s tactic of choice, so he headed over to the table to  121  get paper and a pencil and wrote down his idea. Kalen and Ethan had not yet discussed any musical or mathematical ideas as partners. When the boys came over to tell me about their ideas, Ethan described how he would do the subdivision for the first part (i.e., communication strategy #5), then he would play ten chords on the piano, and Kalen would bang eight on the conga drum, and then they would play 80 beats together showing 10 x 8 = 80 (i.e., communication strategy #1).  “You’re going to hit it eighty times?” I asked politely, but dubiously. I wanted these boys to move beyond strategy #1 since Ethan was such an able musician and Kalen was an open-minded boy. “Maybe there is another way. Try to explore,” I prompted, as the boys headed back to the piano. Joshua overheard these boys practicing strategy #1 and similarly critiqued their obvious expression of multiplication. Ethan looked a little confused. Kalen seemed to be entertained by it all and happy to contribute where and when he could. After this interaction, I observed that the boys were just sitting around noodling on the piano. They seemed stuck. I asked them if they had considered writing song lyrics. “I don’t really like singing. I just like playing,” replied Ethan. “I’m bad at singing. How about YOU sing?” Kalen laughed. “Sure,” I said, “You could accompany a singer. Give me a chord, any chord.” 122  Ethan played C major and I improvised lyrics about the Associative Property on do, re, and mi of the C major scale. The boys laughed again. “You can make up your own lyrics,” I told the boys. Kalen protested a little saying; “my brother is better at this sort of thing than me.” Ethan, with expressions of incredulity on his face, headed off for more paper.  I gave the boys a mini lesson on chord progressions and we walked through how to put chords together in a lucid manner. Soon Ethan was playing one bar of C major, followed by one bar of F major, followed by one bar of G major. Kalen kept the pulse with his finger in the air and acknowledged the chord changes with a sweep of his hand. Eventually, this evolved into a nice chord progression of their own composing, which I transcribed for their reference.  123  When I came back to this partnership, the boys had written song lyrics that described some uses of multiplication in real life. The lyrics were meant to go along with their chord progression (i.e., communication strategy #6). Multiplication is the math that always rings a bell. How would we live without multiplication, we’d never be able to sell. Things need money; things need cash. Multiplication is a bash! Multiplication, Multiplication, Multiplication. Oh yes!  Both boys insisted that they couldn’t sing it. When it came time for them to share their song with the class that day, the boys handed me the lyric sheet. Much to the class’s delight, we realized that they intend for me to be the singer for their song! I managed to make up a  124  little melody that followed a guide tone line from their chords, but I suggested to the group that we all learn it next time. During our next session, the boys needed some assistance on how to expand their song from just this one verse. After some thought and discussion, they considered adding some multiplication question examples, an introduction, and an ending. Using “communication strategy #2,” Kalen stepped up in this session and helped to create two good questions for a B and C section: 5 x 5 = 25 played on the woodblock, and 2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16, 16 x 2 = 32, 32 x 2 = 64 using boomwhackers. The piece de résistance in this session was Kalen starting to compose a melody on the xylophone that fit with Ethan’s chord progression. The boys wrote out a chart and practiced their composition so that the parts flowed nicely together. Soon they were ready to share the finished masterwork with the group.  4.4  Discuss  Each small group shared their musical expressions, first informally with the class and then again in front of a larger audience of parents and other classes from the school. After their performances the students explained how their compositions expressed concepts in  125  multiplication. Questions were asked and feedback was gathered after the two performance opportunities. Tricia and Karen After their initial explorations during the first half of the create stage of the inquiry these girls did not have much additional time to prepare their composition for the performance because Karen missed the next two sessions. Tricia practiced what she could during those sessions and the girls put it all together the day before the performance. For their final performance in front of invited classes and some parents, Tricia and Karen presented a polished version of their three-section song entitled “M and M.” The A section was the recitation of the three times table up to nine (communication strategy #7). This was clearly chanted with a lively steady beat played with two boomwhackers, again in sword fight style.  The music stopped while the girls picked up their woodblock and tambourine and proceeded to play the contrasting B section. It appeared to be the three times table again, but this time represented using “communication strategy #1.”  126  The instruments were put down again and the C section began - an interesting twist. They played nine groups of three beats on the boomwackers – “communication strategy #2.” The three beats were two whacks on the whackers with one beat of silence between the beats – whack whack rest, whack whack rest – nine times. Some might interpret this as 2 x 9, but their composition transcription indicated 3 x 9.  127  “We did the three times tables: 3x1 till 3x9. We broke it up into 3x1 to 3x3 and stuff like that,” explained Karen simply. “First we told the equations, then we showed the equation without saying it, then we showed the grouped beats,” added Tricia. While their verbal description was brief, and their creative musical compositional skills challenged, Tricia and Karen understood the concepts of multiplication and were able to express them using their somewhat limited musical vocabulary. Christian and Santiago The boys wrote a detailed chart to accompany their composition that described their project’s development. The style of their song was declared “heavy metal” as was reflected in the title, “Kryptonite.” Santiago started the song with a drum roll followed by a cymbal crash. The next section changed from the first writing session in that Christian no longer  128  played the seven groups of three on the drums, but instead used only the drumsticks while Santiago played along on the drums (communication strategy #2). This created an interesting timbre, and was a pleasant artistic choice.  Then the boys improvised on the drums, with some aggression, for a few seconds (perhaps this was the “heavy metal” part) while Christian made his way around the “drum set” to the cymbal. The boys abruptly stopped the random drumming and pretended like they were about to hit the cymbal really hard, but instead Christian calmly repeated the seven groups of three on his drumsticks. This was a dramatic dynamic change – another good example of careful consideration of performance choices. Finally, the boys hit the cymbal together with flare, and their song was complete.  129  When Christian and Santiago had the chance to perform their composition before an audience it did not go as smoothly as the session before with only the class watching. Perhaps their performing nerves got the better of them, but they were out of sync on the first section, Christian “exaggerated” (as Santiago later described it) the cymbal at the end of the song so as to almost cause damage to it, and they were unable to explain their process to the audience in a clear, articulate manner. Sarah and Carly “The title of our song is Music and Multiplication. It’s also called M plus M,” Sarah told the crowd shyly. Their performance was polished and confident with each section flowing perfectly into the next.  130  The introduction started off with 4 beats to set up the tempo. Then they chanted the lyrics for this section (communication strategy # 6). When it was finished they moved straight into the A section by reciting the equation (6 x 5 = 30) and then demonstrated it using six groups of beats separated into three groups of two on the drums and contra bass bars (communication strategy #2). This flowed into the contrasting B section which first chanted the five times table to three (communication strategy #7) and then demonstrated it using rhythmic grouping on the guiro and djembe (another example of communication strategy #2). They ended the piece with a well-timed “bing” with finger cymbals.  When they finished Carly held the microphone and they explained their song.  131  “First we tell: What is multiplication?” described Sarah, “And then we tell you that 6 x 5 is 30.” “Which ‘is’ the question!” burst in Carly with enthusiasm. “Then we tell you 6 +6+6+6+6 equals 30. Then we did the B section, which was 5 x 1 = 5, 5 x 2 = 10, and 5 x 3 = 15. We showed that, like, 5 = 5, and 5 + 5 = 10, and 5 + 5 + 5 = 15. Then at the end we showed . . .” Sarah paused, thinking of how to describe it. Carly took over and said, “What she’s trying to say to us is that we got these at the end,” she picked up the finger cymbals and continued, “then we shook our heads three times, like this,” she shook her head three times, “and then we did this” she said enthusiastically as she played the finger cymbals one time. “That’s what she was trying to say!” Everyone laughed. “Right on, girls!” I cheered. Paige and Chan “Can you keep the steady beat for us?” Paige asked me during their last practice session. I agreed, knowing that their composition needed some glue since their musical algebraic representations where not flowing together naturally. After some discussion they decided that I should play the pattern “du, du, du-de, du” on the djembe. Later we asked Ethan to join us on the drum part as well. When it came time for the sharing, Chan and Paige were able to play their composition easily and explain it very clearly in both the class performance and the large audience performance. Chan introduced the composition, “Okay, our question is 40 x 2 = 80. And the title of our song is ‘The 80’s’.” This pronouncement got a laugh from the adults in the crowd.  132  For the entire composition, Paige played the contra bass bars and Chan played the boomwhackers. Their song began with an introduction where they both played C on their respective instruments. They started with a slow, steady beat that sped up faster and faster until together they stopped with one final “whack.” I then began the steady rhythm pattern over which the two children added the various parts that - to them - demonstrated the many multiplication equations in their composition. After their five equations that used “communication strategy #4,” Paige improvised on all of the contra bass bars for a few seconds. Then, using eye contact for direction, they hit one last note together for their ending.  Paige explained, “So, Chan played F and that was 20 times 1, then I did C twice so that was 5 times 2, um, then Chan did D 5 times, so that was 2 x 5, then I did A 20 times, and I did it in a pattern just like the djembe was doing, like one, two, three-four, five, six, seven, eight-nine, ten, like that up to twenty. And that was 1 x 20. Then Chan plays B 2 more times.” Chan interjected, “So, the F is 20, the C is 5, the D is 2, the A is one, and the B is 10.”  133  Paige took over the explanation again. “So what he means is that . . . Chan plays F with the boomwhackers, right? So, we made them into things. We did algebra. So, his F would be 20 and if you play it once that would be F x 1 = 20, and we did that for all of them. Like, my C is 5, so that’s C x 2 = 10. His D is 2, so that’s D x 5 = 10, then I play A which is just normal, like just one, so that’s A x 20 = 20. And then Chan plays B twice and that’s B x 2 = 20 ‘cause B is 10.” Chan added, “And then we did the ending.” “Yeah. That’s basically it!” chimed Paige. Joshua, Lara, and Katy “The name of our song is ‘Math plus music equals fun!’ It’s a subdivision example.” Katy said into the microphone, then ran back to her place with Joshua and Lara and prepared to perform. Their introduction (A section) was a seven-note run from low C to B on the recorders. Then Lara started a steady beat on the boomwhacker and djembe – whack, boom, boom, boom, whack, boom, boom, boom. Katy and Joshua began the B section on the recorders, which had Joshua playing from a whole note to eight eighth notes going down  134  from B to F, while Katy played eight eighth notes to a whole note from F to B. After leaving one bar open they then played the opposite pattern. This was “communication strategy #5.”  The music continued to flow while the drum part was passed from Lara to Katy. Joshua played a small recorder transition that they called the C section. The D section consisted of Lara playing a whole note on high D followed by four half notes on B and another whole note on D, while Joshua played eight eighth notes on low D then eight quarter notes on G followed by eight more eighth notes on low D.  135  Another contrasting (C) section played by Lara transitioned the players to the next contrasting (E) section. Joshua now kept the steady beat as Katy and Lara played together from high B down to F while subdividing from a whole note to eighth notes and back up again. Katy took a turn at playing the C section transition, then the three children played the F section, which was a simple scale from high B to low C on their recorders. They added a special coda (i.e., ending) by each picking up a boomwhacker and taking a turn hitting one beat each until they then played all together on the fourth beat. These children played their composition without any major problems, although the beat tended to drag some times during their performance. They explained the song and the theory behind subdivision clearly and concisely to the audience. Overall, it was a very impressive music composition and expression of multiplication through music. Ginny and Kira Unfortunately, Ginny and Kira dropped beneath both Jenny and my radar sometime during the middle of the project. Not only did they miss the majority of the create and discuss sessions due to absences from school and participation in other important student services programs at the school, but we did not seem to be able to find time to monitor their progress during the development of the inquiry as well as we were able to with other groups. Therefore, there was no discuss phase for this group during the research project. Ethan and Kalen “So, the title of our song is Ye old Billabong!” laughed Ethan into the microphone, stirring the crowd. “Just joking. It’s actually ‘MegaMath!’ ARE YOU READY TO ROCK?!” he exclaimed in a rocker voice. The audience laughed, the children with amusement, some  136  adults with discomfort. “Where did this come from? And is it something I need to be concerned about?,” I wondered to myself. The boys moved excitedly to their places and, focusing well despite their potentially distracting announcement, began their composition. Ethan played eight shakes on a shaker egg and then we burst into the chorus (A section), with Ethan performing chords on piano, Kalen on the xylophone, and Katy and me singing the lyrics. The chorus used “communication strategy #6” to express multiplication.  Ethan called “Woodblocks” and Kalen, using “communication strategy #2,” played five groups of five on the woodblocks for the B section. We repeated the A section again before moving onto the C section. The boys used long C boomwhackers to play 2 x 2, 2 x 4, and 2 x 8 using “communication strategy #2.” First Ethan played a group of two, and Kalen repeated it. Then he played a group of four, and Kalen repeated it. Finally, he played a group of eight, and Kalen repeated it again.  137  The chorus returned once more. This third time through Katy and I sang the melody very clearly, in fact it solidified into an “official” melody by this third pass. Kalen also nailed his part and was able to play the exact same melody as we were singing on the xylophone. This was a moment of performativity as we “sang/played” ourselves into knowing.  For the ending, the boys demonstrated 4 x 3 = 12 using a sword fight with boomwhackers. With dramatics the boys fought with the whackers for a few more seconds until Ethan cried “Last man standing!” and the song was complete.  138  Ethan described the song to the crowd section by section, but did not go into details on how it portrayed multiplication. He was very excited and shouted into the microphone a few more times for entertainment value. Kalen tried to add his opinion a few times, but was not able to get a word in edgewise with such a spirited partner. Whole Class “Discuss” During a number of large group discussions near the end of the create phase and into the discuss stage of our project various forms of feedback were offered from the children in class. We talked about the importance of flow in our songs and how to make them more “musical.” We added in extra players to a few groups to help support the ensemble. For example, Katy and I became the singers for Ethan and Kalen’s piece, and Ethan and I became the drummers for Chan and Paige’s piece. Also, Lara, Katy, and Joshua joined forces along the way. We discussed the pros and cons of the seven communication strategies and the challenges involved with composing music with a mathematical framework. Some audience members wanted to know if the children made up their own multiplication questions. The class assured them that they did. Someone asked if they had help from the teachers. I briefly explained the concept of co-emergence and told them we all helped each other, but that the groups did have guidance if they needed it. One junior kindergarten child asked why we wanted to make songs in the first place. Christian answered, “That was the whole purpose in order to find out how this works. To see if our question will work, cause maybe it won’t. We wanted to see if the [thesis] question was true or false. Basically it was like grade six math.”  139  Finally one boy asked how they knew which numbers to use. Christian again answered for the group by telling him that we basically picked the numbers randomly and then researched ways to show them in a musical manner. Overall, all of the children were able to explain and discuss their own progress in grappling with the thesis question. They were also able to critically analyze each other’s work, offer suggestions, and consider the input of others when making decisions about their project.  4.5  Reflect  Reflecting on reflections, I think that the children, Jenny, and I reflected on our process throughout the project as well as at the end of it. That made us reflexive researchers, which is a strong component, (i.e., learning outcome) of the a/r/tographic process. As I described during the create stage, we engaged in Inquiry Cycles within Inquiry Cycles. This allowed us to modify our thinking throughout the process. We attempted to answer the thesis question to the best of our ability by altering our course if we saw ourselves diverging in a way that proved negative. We had large group discussions nearly every session throughout the development of our projects where we questioned our strategies from many angles. It  140  became a constant self/peer/teacher-assessed procedure, which did not allow for any group to be unsuccessful in answering the thesis question. In one large group conversation during week four near the end of our inquiry, I asked the children if they had, indeed, been able to use music as a language of expression for their understanding of multiplication. “Yes!,” came a chorus of responses. Every child, to varying degrees of complexity, had been able to show that music can communicate multiplication. (See the Discuss section 4.4 of this thesis for the answer to: “If so, how?”) No one had to explain “If not, why not?” Even the children who used simplistic forms of “communication strategy #1, #2, and #7” were still able to answer that question in the affirmative. Had there been some children who were unable to use music to express their understanding of multiplication, I think it would have been because (1) either they did not understand multiplication, or (2) they had little to no musical vocabulary from which to draw. This was not the case for any of our students. The discuss stage appeared to overlap with the reflect stage in that some main experiences/remembering from our inquiry were considered in both stages. Throughout the six weeks of the project, together the children, Jenny, and I continually talked about what was transpiring and had transpired, about who had said and done what, and about what funny, challenging, exciting or otherwise memorable learning had occurred. In preparation for the children’s individual reflection interviews, I described to the children the reflection questions that I might ask them to think about. During the description of the questions, the children gave responses as a group that did not come up during their private reflection sessions. For example, when asked if they learned more about music through this project, Lara said she learned what subdivision was about and Katy said she  141  discovered that “most music is actually math in the ways that it is patterned.” They told me that they solidified their understanding of multiplication, (e.g., grouping and fast adding), and they learned the difference between multiplication and division. In the last week of the project I was able to engage most of the students individually in an oral interview that gave them an opportunity to reflect on their project and the learning that occurred. It was here that we considered how effective our communication of multiplication was through the language of music. Unfortunately, I did not have enough time to complete all of the oral reflection interviews due to the time of the school year. Students and teachers were very busy with year-end tasks, not to mention the school musical that was going up the next week, as well as general pre-holiday fatigue. Some children were absent for early summer holidays since our school used a year round calendar and it was already July by the time we were ready to reflect. With these restrictions, the children and I reflected on our musical journey as best we could. Tricia and Karen Neither Tricia nor Karen were available on the day that we did the reflections, so there is no data to support this section of the inquiry cycle for these two girls. This is too bad, because I would have liked to know what the experience was like for them. However, during our reflection on the learning journey of these two girls, Jenny and I felt that they struggled with independence and creativity. They relied on an idea given to them (i.e., by me) early in the create stage after seeking my help and were at a loss as to how to expand their expressions from the first attempts. They used the three (3) times tables and “set it to music” with a steady beat and recitation.  142  Karen and Tricia applied “communication strategy #1” as their primary method. They also used “communication strategy #2” to better understand how grouping can be expressed through rhythmic patterns and “communication strategy #7” as a memorization tool for the times tables. They understood exactly what they were doing and why, but were unable to explore further and develop new understandings or musical representation. This could have been due to some anxiety that prevented free exploration in self-directed learning or it could have simply been their learning styles. They accomplished the task, and they learned many things as they went along their journey, but their exploration was not expansive. Christian and Santiago When our inquiry was over, Christian was absent for a few days for personal reasons, so he was not able to reflect on our experience. Santiago sat down with me and told me what the project had been like for him. He said that he was the one who provided most of the music and Christian was the one who showed the multiplication. He felt that everyone learned that music can express multiplication, “’Cause music literally is multiplication. You have to multiply beats and create a piece of music by multiplying times 4 times 3, and at the same time making music, because you plan it out and then you play it.” What Santiago may have been trying to point out is that music and math, while not exactly the same, intercept on a conceptual level because they both deal with patterns and patterning. Santiago was pleased that he got to use the snare drum, but he felt that performing was hard because he had stage fright. “I was a little bit nervous.” He told me, “It was the first time I ever played drums in front of a big crowd.” His reflection was brief, but he concluded, “One thing’s for sure, it was fun!”  143  Jenny and I felt that Christian and Santiago learned valuable lessons through play and exploration. In addition to the practice and learning of music and mathematics concepts, their time on the drums with the freedom that the create sessions offered was important for creative thinking, social development, and personal expression. Christian and Santiago were able to express their understanding of multiplication using music through a simple version of “communication strategy #2” that showed grouping through a rhythmic pattern. Although both of these boys came into the inquiry with a strong understanding of multiplication they did not challenge themselves mathematically, as was evident in their choice of equation: 3 x 7 = 21. Their use of applicable music vocabulary was not as expansive as I would have expected from them considering their musical abilities and skills in teacher-led music lessons. From my observations of this group, like those of Karen and Tricia, I think Christian and Santiago’s learning styles were challenged by the open structure of co-emergent curriculum. Also, their academic development in music may have been challenged by this assignment in that they did not yet have an extensive creative skill set developed at their age and experience level to compose music in a more detailed way. Finally, I think that their social/emotional developmental stage combined with their personalities to dictate an atmosphere of fun and entertainment, also perhaps one of desultory rebellion or mischief, rather than an attitude of focused work and concentration. Sarah and Carly When Sarah and I sat down to reflect on her composition, she struggled to explain her process. This could have been because of her ESL status or possibly due to some fear at  144  sitting together one-on-one with me for the reflection. Either way, she was not the lively creative girl that I was used to observing in our inquiry sessions. While she did not go into much detail she was able to tell me that she learned how music could be used to explain multiplication and that multiplication is grouping. She said that when we started the project she only “kind of understood about grouping and fast adding” and that the project increased her understanding of multiplication. She said that she found some of the initial explorations challenging, but overall she enjoyed the inquiry. In this manner, even though she did not state it explicitly, she seemed to understand the notion that music and mathematics share many concepts at the level of patterns and patterning. Carly was quite articulate during our reflection. She laughed and said, “I think so,” when I asked if she was able to express multiplication through music. She talked at length about her, and Sarah’s, problem solving techniques and creative decision-making when composing their song. She described how they critiqued their work and made changes when they wanted to use more than “communication strategy #1,” when they graphically notated their score, and when they sought out complexity and better musicality/flow in their composition. Carly said that the most exciting part of the project was the performance and the hardest part was the practicing. This was a duo ensemble that showed a solid understanding of all stages of our inquiry and was fully engaged throughout the process. According to Jenny, the approaches used and the girls’ attentiveness for the work were right on standard grade level expectations. Their final composition was balanced between concrete and abstract understanding, which is ideal for their developmental level.  145  Carly and Sarah took their mathematical equation and practiced expressing it in several different ways. They played in rounds (canons), in unison as partners, and then individually. They rotated through instruments in a clear and conscientious way and incorporated “communication strategies #2, #6, and #7.” Sarah and Carly’s composition was well thought out and designed specifically to fit the task of expressing their equation. Any confusion experienced during their first attempts at working with the concept of multiplication through music was overcome as the girls moved successfully through the inquiry process. It was interesting how Sarah was strong during the investigate and create stages of the inquiry, but then became quiet and without significant contribution during discuss and reflect. Carly, on the other hand, thrived during the discuss and reflect stages, but followed Sarah’s lead during the first part of the inquiry. These two girls complimented each other well. Paige and Chan Paige told me readily that she was able to use music to express multiplication. She reflected that during our inquiry project not only did she and Chan find out that music could be used to show multiplication, “I never would have thought of it,” she said wistfully. However, they also discovered a thing called algebra (communication strategy #4). “It’s like a math question where you have to figure out what the letters mean,” she clarified. In a lengthy explanation she describes all of the algebraic substitutions they used in their composition. Paige also discovered that the contra bass notes could be used for more than just bass lines. They could be used to play the main melody. She said that the most challenging aspect  146  of the project was “going up and doing it in front of everyone,” although she countered this with “but it was fun!” Chan’s reflection paralleled Paige’s. He told me about their discovery of algebra and described all of the number substitutions they used. The algebra “invention” came about when they were trying to find a way to show large numbers without having to actually play a note that many times. That was how “just pretend F is 20” and other substitutions came about. With a very clear understanding, he explained to me how the grouping and substitutions worked. When I asked him if he learned anything new about music he answered, “Well, this music was kind of like normal music but only with multiplication in it. You can do music with multiplication, so you use multiplication to make music. Like using multiplication and music together, it’s like a multiplication song.” Chan also reflected that he thought music could be used as a tool to figure out answers to math questions. “So like, if you did 4 x 2 and you don’t know the answer you could just have a drum and hit 1234,1234 and that would equal 8,” he reasoned. For Chan, the most interesting part was doing multiplication and music at the same time. Paradoxically, the most challenging part was the same thing, how to express multiplication through music “because some questions were easy but some were hard for me. Sometimes you have to do it in different ways.” “What a duo!” laughed Jenny, when we reflected on this group. Chan and Paige were fantastic examples of abstract minds struggling to express themselves in a concrete manner. While the “invention” of algebra came from Chan, most of the music compositional creation came from Paige. She also did the majority of the explanations, although through his reflection Chan demonstrated a great deal of understanding about the compositional process  147  as well. This “power” imbalance may have been due to Paige’s vibrant personality and Chan’s gentle one. Jenny and I found this group’s approach fascinating. The children used an algebraic puzzle to express their understanding of many small multiplication questions within their larger equation of 40 x 2. While the duo had a deep theoretical foundation behind their expression, they struggled with the compositional skills and music vocabulary necessary to flesh it out with logistic and articulate style. There song did not demonstrate flow, nor did their performance exemplify a competent level of musicianship. Song writing ability aside, these two children very successfully expressed multiplication—and more—using music as a language of expression. Joshua, Lara, and Katy Lara was not available the day of our reflections, but Katy and Joshua sat with me to talk about their project. Katy had a great deal to say about her experience during this research project. She told me that she learned that music had fractions in it and that it could be used to show math. She learned that music could be used to show all kinds of things, in fact. For example, in our project she reflected that some people used notes to represent numbers (e.g., algebra), and some people wrote song lyrics, and some people “did” a problem, yet they all showed their understanding of multiplication. “It really worked!,” she concluded. Katy described her group’s processes from their formation, to working through conflicts, to power struggles and leadership issues, to creative decision-making, to grappling with Joshua’s complex multiplication of fractions. “That [diving and multiplying fractions] was really neat!,” she said.  148  Katy ruminated that having to share the classroom instruments was challenging because her group’s composition sounded slightly different every time they practiced it. She told me that while her group had some stage fright before they did their final performance it was still a lot of fun. When I asked her what big idea she would take away from this project she replied, “Well, the answer to the whole question! That music can be used to show that kids understand multiplication.” “Did you?” I ask. “Yes, I did.” She confirmed. Joshua also reflected on his group’s experience of showing subdivision through music. He said that since subdivision is a kind of division, and division is the opposite of multiplication, then they were indirectly demonstrating multiplication through their composition. This was true when you consider the multiplication of fractions by whole numbers as was discussed during the create stage of the inquiry. He did not feel that he learned any more about math or multiplication through this experience, but he did learn that music could be thought of in many different ways besides how it is usually used, specifically in its related use of patterns in both music and mathematics. It appeared that this group deviated from the task of expressing multiplication because they chose to work with subdivision. However, upon closer inspection their discoveries revealed a more complex mathematical concept that no one but Joshua understood. That is, 1/2 ÷ 1/2 = 2/2 = 1 versus 1/2 x 2 = 2/2 = 1, 1/4 ÷ 1/4 = 4/4 = 1 versus 1/4 x 4 = 4/4 = 1, 1/8 ÷ 1/8 = 8/8 versus 1/8 x 8 = 8/8 = 1. In a sense, this might imply that Lara and Katy did not succeed in answering the central question of the inquiry project.  149  Considering that the concept of subdivision was uncovered during one of our first conversations about multiplication in which Lara and Katy were active participants, I would not suggest that was the case. Through music, those girls were able to express a concept that they did not understand thoroughly. The subdivision example also showed another math concept—how music in notation may be considered as equivalent fractions. All three children understood this concept and were able to describe it within the composition as well as when talking about their composition. What I would have liked to have seen from this group was a deeper consideration of the multiplication involved or the use of another “communication strategy to express a similar concept or an additional multiplication example. This trio ensemble ended up choosing one big idea at the beginning of the project (subdivision) and without truly understanding how it qualified as multiplication used it as a foundation for a music composition without further addressing the mathematics within it. That was possibly because it was too difficult a concept for most grade three children to comprehend. They didn’t spend much time thinking about ways to communicate the mathematical concepts as much as they spend composing and practicing a strong music composition. Ginny and Kira Kira said she learned how to express multiplication through music. When I asked her how, exactly, she reflected back on her first attempts with Christian and Ginny doing 3 x 4 using “communication strategy #1.” She learned that multiplication meant, “fast adding,” and she said she hadn’t learned that before this project. The most challenging part for Kira was trying to find ideas to use because she and Ginny had many conflicts. Overall, Kira did not  150  have more detailed reflection to offer at this stage. Unfortunately, Ginny was absent on reflection day. Jenny felt it was important to note that although Kira and Ginny had two teachers and a whole classroom of peer support, they simply could not stay focused long enough to get their heads around a comprehensive finale. However, even these two children, who possessed a basic understanding of music and multiplication, were able to create a project that made sense to them and showed their understanding of the concepts in the best ways they could. Ethan and Kalen Kalen, a boy of few words, had the following comments. About getting started, Kalen reflected, “First we were going to just do a math question with different instruments, but I knew Ethan wanted to use the piano. We wrote our questions on paper.” On their choice of instruments he stated, “We found the right instruments that would fit. We used the wood blocks, the piano, shakers, and xylophone. We chose them because they went so good together.” About music, Kalen related, “I learned that you could just make up stuff, put it all together, and make a song.” On multiplication, he said, “At first, I wasn’t really good at it so I really learned a lot of it. Like, I was doing the woodblock, first I did one group of 5 on the wood blocks, then another and another until I got to five.” That said, Kalen was unable to describe anything that was written on their chart (perhaps it was really Ethan’s chart). However, he was able to verbally describe everything the boys played in their composition.  151  However, he did comment on what he felt were the most interesting/most challenging parts. “The most interesting was making stuff up. It was all pretty challenging at first, like figuring out the stuff.” With regard to the melody, Kalen said, “I knew that he [Ethan] wanted me to do a different thing, and at first I was just using C and going up and up until B. So he said I should do it the way he wanted it so I just did it.” Finally, when he elaborated a big idea, Kalen exclaimed, “It was fun!” When it was Ethan’s turn to reflect he said that he and Kalen were able to express multiplication using music, but that they took a different approach than the other groups. “We didn’t really do a question, we had song lyrics about how multiplication is used in life and then we did two grouping examples. And our finale!” (communication strategies #6 and #2). Ethan told me that he learned that it’s not that easy to make a song. “You have to make up the words, and the melody, and what you’re gonna do, and who you’re gonna do it with. We learned that music could be combined with multiplication to make a song,” he reflected. Overall, Ethan said that rather than learning more about multiplication himself through this process, he felt that their composition was educational for the audience because “we were teaching about multiplication through our song.” Writing the words and the melody was the most challenging part for Ethan. He admitted that the lyrics were written on their first try without further editing. The best part for Ethan was the boomwhacker sword fight finale. Jenny and I felt that this group showed they could compose for a purpose and that they demonstrated a solid understanding of multiplication through their work. They wrote a  152  chorus to express the importance of multiplication in daily life and included a few concrete demonstrations of multiplication. While Ethan was an exceptionally talented pianist, his creative ability to express concepts through music was very similar to other children his age. It is important to note that Ethan did most of the creative composing and directing in this partnership during the first three stages of the inquiry. Kalen was willing to do what Ethan or I asked, but his contribution to the learning was limited. Fortunately, by the end of the project he was able to verbalize many aspects of his participation and learning.  Questions that arose When I pointed out that on the Inquiry Cycle there is an arrow going from reflect to ask and around again, the children thought of many more ask questions in response to this final reflection on the thesis question. For example, they asked: 1. Why? 2. How did you figure it out? 3. How is it possible that 5 x 3 can be shown using rhythm? 4. How is subdivision multiplication? 5. If music can be used to express multiplication, can it also be used to express division? 6. Can music be used to express other math concepts? 7. Can music be used to express language arts? 8. How about geography? 9. What about the other three arts - can dance, visual art, or drama be used to express other subjects? 10. Can music be used to express music?  153  As reflect moved back into ask, Jenny and I, like the children, came up with many more questions that would be fascinating to explore. 1. If something as abstract as multiplication can be expressed through music by Grade Three students who developmentally often struggle to get beyond concrete thinking, what might children at various ages be capable of and how could educators begin to meet those needs or tap into that aspect of discovery? 2. If this group had been given a written assessment to show their learning of multiplication and music at the end of this task, what conclusions would their educators have come to discover? 3. If an independent party had observed their final performance and filled out a rubric on their learning, would that person have understood the starting and ending points of each child’s learning throughout this process? Is it significant to understanding the children’s learning? 4. What is the best way to assess and understand student learning in this context? 5. As teachers in a co-emergent inquiry project, it was sometimes difficult to “see the forest for the trees” until we had a chance to view the video data and anecdotal records with some time away from the children. This liberated our examination of their learning from immediate reactions to student’s behavior or personality conflicts and we could see it more objectively. This led us to wonder, were we assessing students fairly, accurately, thoroughly, inclusively, and holistically? Do most teachers have the time, expertise, and resources to document and assess in this manner?  154  For Jenny and me, watching the students perform their work for the school community was bittersweet. In some ways, it looked as if the children were simply tapping away at instruments and it was difficult to discern their learning. They gave explanations of their piece after their performance and some audience members could say, “Ahh, I get it.” But sometimes it was more confusing with the explanations provided by the children. Granted, Grade Three students are not the most experienced orators, nor were their explanations written, edited, or practiced – all factors which could have enhanced audience comprehension had we put the time into preparing it.  4.6  The Student’s Compositions To see and understand the learning that happened in this project, one must scrutinize  far more than the final expression—the music compositions. Understanding whether or not these students were successful in expressing multiplication through music requires a thorough examination of the entire project. That said, the following section comprises full transcriptions of the children’s compositions. First, a review of the communication strategies in Table 4.3 is provided for the reader’s ease of application. Next, Table 4.4 provides the reader with a list of the communication strategies that were employed by each duo and trio ensemble as they performed their mathematics in musical ways. Finally, I transcribed each group’s musical composition using a computer software program called PrintMusic 2010, and they are provided herein (all songs © Corry Moriarty 2009). A further detailed summary of the learning that occurred during this project takes place in Chapter 5.  155  1. 2.  Abstract representation Patterned grouping  5 x 2 = 10 versus 5 + 2 = 7 5 x 2 = 10 5 x 2 = 10  3.  Layering of patterned groups Common Multiples  4.  Elementary algebra  5.  Subdivision  3 x 4 = 12 4 x 3 = 12 2 x 6 = 12 2 x 5 = 10 1/4 x 4 = 1 1/4 ÷ 1/4 = 1 1 = 1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4  6.  Lyrics  7.  Reciting times tables  Do 5 beats and 2 beats equal 7 or 10? 2 beats (in a pattern) + 2 beats + 2 beats + 2 beats + 2 beats = 10 beats 5 patterned beats on piano + 5 patterned beats on the xylophone = 10 beats 12 is a common multiple of 3, 4 and 6 and could be grouped and layered accordingly. Substitution: C = 2, 5 beats of C = 10 - Showing the relationship between the whole and parts - dividing and multiplying fractions, - finding and adding common multiples for fractions - comparing equivalent fractions Using Lyric content to explain the concept of multiplication. Reciting times tables through lyrics with beat.  Table 4.3 Music communication strategies for multiplication (a review for the reader)  #1  #2  Tricia & Karen  Tricia & Karen Christian & Sebastian Sarah & Carly Paige & Chan  #3  Ginny & Kira Ethan & Kalen  #5  #6  #7 Tricia & Karen  Sarah & Carly Paige & Chan Joshua, Lara, & Katy  Ginny & Kira  #4  Sarah & Carly  Paige & Chan Joshua, Lara, & Katy  Ethan & Kalen  Table 4.4 Summary of application: music communication strategies  156  157  158  159  160  161  162  163  164  165  166  167  168  169  170  171  172  173  174  5  Synthesis  Figure 5.1 Mind/body reunion – [a portion of “enactivism” from Figure 1.1]  5.1  The art form influences the research influences the art form Fels’ (1996) description of the edge of chaos clearly defined the lived experience of  this research study, the writing of this thesis, and the undertaking of my Master’s program as a whole. Throughout the a/r/tographic process, I have lived in both order and chaos, but mostly I have dwelt in the in-between; in the liberatory borderland spaces that emerged—that were created and expanded—through inquiry (Smith, 1993; Irwin, 2003). The Inquiry Cycle played an important role in my process of writing this thesis. The Inquiry Cycle itself is a formal structure (see Figure 4.3), which metaphorically parallels many forms of music (i.e., from rhythm, tonal, and melodic patterns to phrases to periods to entire segments of music compositions). Music is highly formal and influenced by a broad variety of forms across numerous styles and historical periods.  175  Since form and research exist in a reciprocal relationship, the form may influence the research, which in turn influences the form of the artistic process and outcomes (Eisner & Barone, 2006). A/r/tographic inquiry as a research methodology, embarked on through a coemergent research design, allowed for a reflexive process of being and becoming, of wonder and discovery, as I engaged with children during our inquiry, and as I wrote the thesis. I “wrote-myself” as I cycled through the inquiry process of asking, investigating, creating, discussing, and reflecting through every step of the writing process. I “wrote” myself, my history, my hope, and my experiences before and during the study, and my journey to understanding throughout the entire inquiry process. The elements of this thesis that describe the history of music education and my concern about the rigidity of the “institution” that education has become are a/r/tographic autobiographical statements, in the form of arts-based educational criticism (Eisner & Barone, 2006), which serve as a sub-plot in this narrative. Those musings provided the motivation for the study in the first place. During the writing of this thesis I often traveled between clarity and confusion, between order and chaos, crossing those malleable borders into a temporal place of understanding. Throughout that experience it was important to stay calm and be brave, and to trust the process (McNiff, 1998). A/r/tography as a research methodology, and co-emergent inquiry as a research design for this project, opened spaces for learning that I knew were possible in theory when working in this capacity. However, until I dove into the work— traveled to the borderlands—I had not had many glimpses of their depth before (1) engaging in a Master’s program that supported arts-based educational research in general, and (2) in undertaking this a/r/tographic research study specifically.  176  5.2  The puddin’ Like Katy said, “It really worked!” The children in this grade three class were able to  communicate their understanding of multiplication using music as a language of expression. Due to our enactivist approach to the inquiry cycle and the detailed discussions throughout the inquiry process, we saw that “through conversation – not through debate or dialogue – the new emerged. That is, in debate or dialogue contrary beliefs assert themselves, while in conversation contrary beliefs talk with one another, the differences talk” (Doll, 2010, p. ii). Our conversations granted the children the creative freedom, confidence, descriptive language, and cognitive skills necessary to communicate multiplication using their music skills. Some children’s expressions had more detail and expansive development than others, but nevertheless, all of the children succeeded in their task. Our findings from this research inquiry present a number of scholarly contributions. First, we offer credence to the use of interdisciplinary teaching at the early childhood level. Our research provides opportunities to reach students in a way that positively influences their learning of both music and multiplication. It also positively influences the student’s expression of their understanding of these two subjects. Our work helps to support enhanced learning opportunities for primary students by providing meaningful connections between subjects and by providing alternative “languages”—in the Reggio sense of the word—for the expression of learner cognition. Our research sought meaning pathways for synthesis between theory and practice in more than one subject and in an integrated context. So far, the International Baccalaureate Organization does not provide extensive research to support their assertion that trans-disciplinary education is sound education. In their documentation (e.g., Primary Years Programme, 2007; Primary Years Programme,  177  2009a; Primary Years Programme, 2009b) they have cited the educational theories of John Dewey, Elliot Eisner, Howard Gardner, and the work of Project Zero (from Harvard Graduate School of Education), but there are no research studies that support integrated curriculum. Our research on music as a form of expression offers the IBO more support for their claim that trans-disciplinary themes in education opens doors for understanding and expression. The IBO “recognizes that educating students in a set of isolated subject areas, while necessary, is not sufficient. Of equal importance is the need to acquire skills in context, and to explore content that is relevant to students and transcends the boundaries of the traditional subjects” (The primary years programme, 2009, p. 8). Secondly, this research contributes to the development of new notions and forms of music instruction—created by and for children—that advance both music and related learning. As is evident in the data, we demonstrated how music (1) can be taught in and of itself, (2) can be thought of as a medium (i.e., a “language,” as in the Reggio definition) for the expression of concepts in multiplication, (3) can be used as a vehicle through which to be instrumental in attaining knowledge of both musical and mathematical concepts (see table 5.1), and (4) when linked with mathematics, can show learning transfer and access related learning between the two disciplines (see section 5.4). This research was not looking to separate, compartmentalize, or appropriate disciplines in egg-carton style (Eisner, 2004). I chose mathematics and music to study together in this inter/cross/trans-disciplinary inquiry project because they have already been linked to access similar parts of the brain during cognitive processing (see Chapter Two). However, until now they have not been experienced together in a synthesized manner that allows not only for transfer between the disciplines, but also for them to interact  178  conceptually. Music in and of itself is not mathematics, and mathematics in and of itself is not music, but the two converge, at least conceptually, when it comes to understanding some common concepts in both music and mathematics (i.e., patterns and grouping of patterns, substitutions, common multiples, and equivalent fractions). How music is used versus how music is thought about mark the difference in music being instrumental to learning and music sharing conceptual connections with other subjects. That is, how music is used implies an instrumental relationship (i.e., music helping to teach/learn/express another subject), while thought of implies recognition of the conceptual connections between music and mathematics (i.e., patterns, substitutions, common multiples, equivalent fractions – concepts common in both music and multiplication). I believe our research succeeded in demonstrating both of these phenomena. The instrumental nature of music in most of the studies mentioned in chapter two used listening to music to influence learning and/or enhance cognition. Our research used music composition to not only influence learning/enhance cognition, but also to express the student’s understanding of multiplication. Additionally, our research uncovered many shared concepts between the two disciplines for Grade Three children in the present study (see table 5.1). Concept connections between music and mathematics were apparent in our study as children moved between the two subjects (e.g., in pattern grouping, fast adding, substitutions, equivalent fractions, common multiples). Examples of this were in Joshua’s subdivision example that uncovered fraction multiplication and division, and Chan’s use of substitution. Another curricular link was evident through Ethan’s song lyrics illustrating the practical uses  179  of multiplication (i.e., social studies/life skills), and Katy’s link from real estate to mathematics to music when describing subdivision. Finally, through this research we participate in ongoing academic conversations on the present structure and role of the music teacher (and other “specialists”) in our schools and the teaching/learning that occurs therein (Doll, 2010; Parsons, 2004; Eisner, 2004). See section 5.4 for more on alternative roles of specialist teachers in our schools.  5.2  Reflections on the Inquiry Cycle  What did we know? How did we learn what we learned?  What did we want to know?  What did we learn?  How did we express what we know?  Figure 5.2 The Inquiry Cycle deconstructed as a cartoon  5.2.1 Front Loading/Ask: What did we know (about the research question)? During these two early stages of our inquiry, I was impressed with the amazing clarity of the children’s thought processes in elaborating the concepts within the elements of music and their comprehension of the meanings of the “key concepts” in the research question. The children were thoughtful and deliberate in articulating their ideas about what 180  these words meant (e.g., the six music vocabulary/elements, and the important words in the research question). This detailed elucidation set the tone for investigation, and it gave the children confidence to explore and play with ideas. It is worth noting that the students who indicated a score of seven or below on the music vocabulary self-assessment exercise were the students, generally speaking, who had some difficulties with the creative development with their composition. This told us that fluency in the language is a contributing factor in children’s ease for language acquisition.  5.2.2 Investigate: What did we want to know (about the research question)? When the children, their teacher Jenny, and I began finding and talking about the six big ideas (i.e., concepts within multiplication that helped to narrow down the research question into more manageable content) during the investigate stage of the inquiry project I started to get excited. It was with some trepidation that I had entered into a project of this nature with children not knowing what might unfold. Although, through my experience with this class over the previous two years I knew that I could trust them to explore with discipline and creativity. As a music teacher who trusts children and the inquiry process, I knew that what I was asking of the children was developmentally appropriate for their age and experience. Still, there is an element of risk present in any open-ended co-emergent inquiry that the participants, for many reasons, would not want to venture into the borderlands. The Big Ideas 1. 2. 3. 4. 5. 6.  The four properties of multiplication Cell division/exponents Subdivision Grouping/common multiples Fast adding The times tables/equations  181  From these big ideas came the content of our compositions. The children moved through more than one big idea until they settled on the one or two that they liked best. The most common idea to express the children’s understanding of multiplication was grouping showed through patterns. It is unclear if the children were aware of the commonality between music and mathematics through their use of patterns, because it did not come up during our conversations. Nevertheless, most of the groups in some way or another used patterns, and patterning concepts, to express multiplication. Fast adding, the times tables/equations, and subdivision were also musically articulated. No one attempted cell division/exponents after Lara and Katy’s confusion between subdivision and cell division, and the wording of the properties of multiplication were so complicated that the children did not attempt to use them in their compositions. Some aspects of the four properties came up naturally during the children’s work. For example, Sarah and Carly discovered that their questions were the same when the numbers were reversed (e.g., commutative property), and Ethan pointed out in our very first conversation that anything times one is one (e.g., multiplicative identity property).  182  5.2.3 Create: How can we express what we know (about the research question)? Jenny and I were delighted when the seven communication strategies (Table 5.1) started to come to light as we worked with the Big Idea(s) of the students’ choice.  1. 2.  Abstract representation Patterned grouping  5 x 2 = 10 versus 5 + 2 = 7 5 x 2 = 10 5 x 2 = 10  3.  Layering of patterned groups Common Multiples  4.  Elementary algebra  5.  Subdivision  3 x 4 = 12 4 x 3 = 12 2 x 6 = 12 2 x 5 = 10 1/4 x 4 = 1 1/4 ÷ 1/4 = 1 1 = 1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4  6.  Lyrics  7.  Reciting times tables  Do 5 beats and 2 beats equal 7 or 10? 2 beats (in a pattern) + 2 beats + 2 beats + 2 beats + 2 beats = 10 beats 5 patterned beats on piano + 5 patterned beats on the xylophone = 10 beats 12 is a common multiple of 3, 4 and 6 and could be grouped and layered accordingly. Substitution: C = 2, 5 beats of C = 10 - Showing the relationship between the whole and parts - dividing and multiplying fractions, - finding and adding common multiples for fractions - comparing equivalent fractions Using Lyric content to explain the concept of multiplication. Reciting times tables through lyrics with beat.  Table 5.1 Music communication strategies for multiplication The create stage brought us to the edge of chaos, and some of the participants actually went over the edge into chaos. Like the border between investigate and create, our borders varied and changed depending on our personalities, learner styles, expectations, need for order, or acceptance level of what could be perceived as disorder. The children, their teacher, and I each responded to the edge differently; each seeing our own edge depending on our world view, our comfort, our experience, and our skills with this type of engagement. That said, through conversation and reflection (i.e., the a/r/tographic reflexive process) we were able to work through most challenges and discover that the border between order and chaos was permeable and malleable. Additionally, we experienced the borderland expanding 183  to become more than a thin boundary-line between two worlds. Instead it became a territory of its own with all of the institutional complexity and cultural richness afforded to a thriving community. Like in the new ground forged by co-emergent inquiry, “those living in the borderland of a/r/t recognize the vitality of living in an in-between space”(Irwin, 2003, p. 33). A/r/tography becomes that third space between theory [multiplication] and mètissage [music] while opening up the spaces between artist-researcher-teacher. There are spaces between spaces between the in-between. There are multiple borders diffused again and again. And yet, all the while we do not dismiss the lands that create the blurred perimeter of the borderlands. Irwin, 2003, p. 31-32 We have, through this process of inquiry, become engaged in a shared reciprocity that encourages students to re-imagine a curriculum that is responsive to each student’s presence. Fels, 2008, p. 8 Several benefits revealed themselves while we engaged in co-emergent inquiry. The children had the opportunity to (1) trust themselves and their ideas, (2) practice working cooperatively, (3) learn/exercise both music and math skills/vocabulary, (4) gain confidence in their learning through ownership of it, and (5) access personal meaning, relevance, and authenticity through this complex co-emergent pedagogical approach. The students in our co-emergent inquiry each responded differently to the approach for many reasons. Participants had differing levels of musical and mathematical knowledge and experience (i.e., music/math vocabulary and music/math skill levels), therefore finding common ground was sometimes challenging. The open-ended structure of the create stage was also challenging for the children who (1) found cooperation difficult, (2) thrive on structure, (3) who are prone to “fooling around,” and (4) who think concretely and/or are inexperienced with creative thought processing.  184  A few problematic aspects arose during co-emergence. Some power dynamics during group work (i.e., leaders/followers, dominant/submissive personalities) caused some children to contribution more than others, and meant that space for self-expression and/or personal learning time was not always equal between participants. Also, some children’s special learning needs/styles were not catered to in this type of learning experience. While this was not problematic in and of itself, it generated some confidence issues for those not accustomed to abstract conceptual expression and caused some children to doubt themselves, their ideas, and the creative process. Some children were able to travel to the borderlands, but not cross over into chaos. Some entered both worlds freely. Some children were reluctant to leave home at all. If the children struggled with focus, “got stuck,” or just didn’t want to travel, Jenny and I played the role of tour guides, scaffolding them through their zones of proximal development – that is, helping to bridge the gap between what they could do on their own and what they could do with support (Vygotsky, 1978; 1962). Sometimes it felt like we played Charon, the boatman in Greek mythology who ferried the souls of the dead over the river Styx to the underworld. When the children were unable to take themselves through the creative problem solving process they required a great deal of support. At times it was challenging for me to recognize when the children needed my support versus when they needed to be left alone to think and explore. It was also challenging to determine when the children needed me to work along side them as a cocollaborator in the inquiry, making my suggestions just like any group member would, without taking over. With attempts toward balanced facilitation, Jenny and I negotiated our way through the landscape of this new s/l/pace that we helped to create (Springgay et al.,  185  2005). When we in Reggio say children have 100 languages, we mean more than the 100 languages of children; we also mean the 100 languages of adults, of teachers. The teacher must have the capacity for many different roles. The teacher has to be the author of a play, someone who thinks ahead of time. Teachers also need to be the main actors in the play, the protagonists. The teacher must forget all the lines he knew before and invent the ones he doesn’t remember. Teachers also have to take the role of the prompter, the one who gives cues to the actors. Teachers need to be set designers who create the environment in which activities take place. At the same time, the teacher needs to be the audience who applauds. Malaguzzi, 1994, p. 5 While writing musical compositions to attempt to express multiplication was the goal of this project, the compositions themselves were not necessarily the intended learning outcome of the inquiry. Technically speaking, on the first day of the investigation stage all of the children were able to articulate their understanding of multiplication to varying degrees in their initial attempts at communicating through music (see section 4.2). Similarly, by the end of the project, although some students’ compositions were not as expansive and creative as others, it did not necessarily imply that they were unable to demonstrate their understanding of multiplication through music. Even the most basic expressions succeeded in communicating the children’s understanding of multiplication. The same children who struggled with the multiplication concept of grouping during our work with manipulatives were able to musically express 2 x 3 only by using “communication strategy #1” (i.e., tap, tap, “times,” tap, tap, tap, “equals,” tap, tap, tap, tap, tap, tap). Consequently, not only was “communication strategy #1” an accurate expression of their (limited) understanding of multiplication, but over the course of the inquiry they learned more about grouping. Thus, their expression evolved from “communication strategy #1” to “communication strategy #2” (i.e., pattern grouping, and  186  layering of patterned groups). The mathematical learning that occurred in the course of our research study took place throughout the process of the inquiry, in all stages of it, and was not limited to the final product. As was evident through our on-going conversations throughout the inquiry cycle, and from the children’s reflections, we discovered that a great deal of learning about multiplication/mathematics took place early on in the inquiry (e.g., pattern grouping, fast adding, substitutions, equivalent fractions, common multiples), while some children discovered more about mathematics (e.g., the multiplication and division of fractions) well after their composition was developed. Similarly, the lessons learned about music (e.g., form, patterns, lyrics, flow, melody, chords, instrumentation, graphic and traditional notation, and steady beat) also occurred throughout the inquiry process.iv During the create stage we also became more fully aware of the difference between (constructivist) guided inquiry and (enactivist) co-emergent inquiry. The two inquiry cycles are nearly the same except for the create stage, and it is on this stage that the rest of the inquiry cycle depends. In guided inquiry, the teacher has a good idea of what the children are going to come up with for their final product, as well as how they are going to get there, because the students directly explore prescribed learning outcomes and well-considered lines of inquiry. In co-emergent inquiry, no one knows exactly what will happen once the exploration begins because the students will determine the learning outcomes as they discover them. Co-emergence offers ownership, authenticity, meaning, and relevance to students in an inquiry. The summative assessment in both guided and co-emergent inquiry could be as simple as an explanation by the children stating their understanding of the central idea/research  187  question. This assessment could be the same using either inquiry model, but the final product would be different in the two styles of inquiry. In this study, the central idea/research question could have been answered either positively or negatively, as in “Yes/No, music can/cannot be used as a language of expression for multiplication cognition, because …” No matter if the answer to the question was negative or positive, the children still would have been able to answer/describe the central idea. Reggio projects (progettazione) are conceptual or theme-discovery based (e.g., amusement park for birds, or dinosaurs) without any predetermined learning outcomes or summative assignment/product. All lines of inquiry and project direction come from the children throughout the development of the project, with the help of their teachers. Coemergent inquiry merges guided inquiry and Reggio project approach. I wonder: if we hadn’t started with the predetermined research question, would our exploration of multiplication and music have turned out similarly? We might have just started with a theme such as music and multiplication. Allowing these words to get tossed around by the class in a discussion would have allowed it to emerge in any number of directions. Would the children have written compositions? I suppose the possibilities are endless. My co-emergent inquiry using the Inquiry Cycle was a combination of guided inquiry and progettazione in that with some structure our project emerged in a similar way, yet with individual expressions in the different approached taken by the groups.  5.2.4 Discuss: What did we learn (about the research question)? Our findings indicate that the expression of conceptual understanding of multiplication using music as a language is not only possible for children, but has enhanced the depth of their understanding of the concepts within both music and multiplication.  188  In essence, during this co-emergent inquiry the children were performatively “writing themselves” (e.g., consider M.C. Escher’s 1948 image of “Drawing Hands”) - or should I say “music-ing themselves” - into an understanding and expression of multiplication, while simultaneously employing the skills of composition and song-writing while gaining a better understanding of them. The children who did not understand grouping and fast-adding before the inquiry learned these concepts through the inquiry process (e.g., Chan, Kira, Ginny, and Kalen), while some improved their understanding of grouping and fast-adding (e.g., Paige, Sarah, and Colby). Other children demonstrated that although they had a solid conceptual understanding of multiplication, their lack of creative thinking skills and/or musical versatility limited their ability to translate their understanding using music as a “language” (e.g., Tricia, Karen, Santiago, and Christian). Finally, some children grew in their musical composition skills without enhancing their understanding of multiplication at a grade three level (e.g., Joshua, Lara, & Katy, and Ethan). Two advanced math anomalies that were well beyond grade-level expectations occurred during the create phase, and were delineated in the discuss stage of our inquiry. Joshua did not improve his mathematical ability through this inquiry, but he did uncover some unexpected and challenging concepts. Early on in the create stage, when I asked him about the math questions involved in his subdivision example he was silent and looked away for a long time. Judging from my past experience with Joshua’s learning style, this behavior led me to believe that he was simply thinking about my question and it required a great deal of time for him to come to a conclusion. It was during the following create session that Joshua took the marker out of my hand and explained the multiplication and division of  189  fractions. He might have learned how to multiply and divide fractions during our inquiry, but more likely he related his learning from previous experiences (i.e., the advanced math classes he took three times per week) and applied it to this context. Joshua was also able to articulate the concept of equivalent fractions. The other children understood this concept as well, but only after Joshua verbalized it. Chan’s use of substitution was another advanced mathematical concept that arose from our inquiry out of need. He and Paige needed to show 80 without having to hit the note 80 times. This natural development was a textbook example of the meaningful benefits of integrated learning compared to discipline-based learning. “Tools have meaning only when their usefulness is understood … this means that students should grapple with the problems first and learn to use the tools as they find them helpful” (Parsons, 2004, p. 778). Chan “invented” the idea of substitution out of a need for a tool to do a job. This mathematical concept was well beyond grade level expectations for grade three students, and, more importantly, it demonstrated how integrated curriculum experienced through coemergent inquiry opens pathways for advanced learning. It also demonstrated concurrent learning where the tools (i.e., elements and concepts) of a discipline are learned in personalized context, which fosters a deep understanding for the meaning of the tools of a discipline when learned through experience and application, rather than in isolation.  5.2.5 Reflect: How did we learn what we learned (about the research question)? When reflecting on our project the children described some reasons for engaging with the research question – in other words, why were we doing this? The children’s answers were, (1) to write songs expressing multiplication in order to teach an audience about multiplication, (2) to use multiplication for artistic conceptual inspiration for song writing,  190  and (3) to learn about music and mathematics, that is, to “compose-ourselves” into learning musical or mathematical concepts while composing. During the reflect stage the children, Jenny, and I came up with many more questions that could lead into further inquiry that could be possibilities for further inquiry. 1. If music can be used to communicate multiplication, can it also be used to communicate division? 2. Can music be used to express other math concepts? 3. Can music be used to express language arts? 4. How about geography? 5. What about the other three arts - can dance, visual art, or drama be used to express other subjects? In the final reflection stage, Jenny and I considered that as teachers in a co-emergent inquiry project, it was sometimes difficult to “see the forest for the trees” until we had a chance to view the video data and anecdotal records with some time away from the children. Having the sessions documented in this way allowed us to observe the learning going on during the sessions more clearly, by freeing us from the immediate distractions of student’s behavior or personality conflicts. Teachers as co-inquirers get caught-up in the action and sometimes are unable to observe important shifts in the project direction when we are on the “front lines” with the children. This led us to wonder whether or not teachers can assess students fairly, accurately, thoroughly, inclusively, and holistically while being so involved in the hands-on teaching/learning process? We wondered if we were responding first to our feelings about the student’s behaviors as opposed to their learning.  191  Furthermore, Jenny and I wondered about how most teachers in public schools with hectic schedules and large class sizes might be able to find the time, expertise, and resources to engage with children, not to mention document and assess their learning, in this coemergent manner? There are barriers to the type of teaching/learning explored in this thesis that would require considerable alternatives to the structure of current teaching/learning environments so as to accommodate trans/cross/inter-disciplinary inquiry projects in schools. It has been my experience that one such barrier to engaging in co-emergent inquiry in schools deals with the timetabling methods in place that have many classes filtering through the specialist teachers programs on a rotating basis. This prevents collaborative relationships with students, teachers, and parents. Once I had as many as 585 children go through my classroom for two 40 minutes classes in a seven-day rotation – that was 1170 children. The pressure that this kind of teaching load puts on a specialist teacher eliminates the possibility for building healthy relationships with people outside of the frenzied schedule, let alone find meeting time for planning, preparation, and reflection. The time required for intense collaboration with students, teachers, and parents in inquiry projects is often unrealistic in the school music-teaching context due to the heavy timetabling requirements, limited budgets, and lack of vision for alternative structures at the administrative level. The other barrier I found when re-thinking the roles of specialist teachers in elementary schools deals with program legitimacy. Music/art/drama/dance class in most public schools is considered to be separate from the real program. These classes exist primarily to provide classroom teachers with prep time. Interest in what actually occurs in these classes is negligible – as long as specialist teachers follow the provincial curriculum standards in our subject areas. We are welcome to “add-on” to the on-going programs in  192  children’s classrooms, but we don’t have time to authentically collaborate in integrated curriculum. More consideration about the barriers around co-emergent inquiry, along with the questions asked by the children during our reflection stage, are of interest for further research in the future. Art is the visual reorganization of experience that renders complex the apparently simple or simplifies the apparently complex. Irwin, 2001, p. 31 Malaguzzi was very aware of this unity and synergy between theoretical declarations and quotidian practice, and he often urged us to be mindful of these two moments and to be consistent… a school is a very complex and fluid organism … I believe that by recognizing meaning in all the small gestures of the quotidian we can see how visibility and depth are given to important concepts. Strozzi, 2001, p. 59-60 Reggio educators seek to make the ordinary extraordinary. The work that came from the children in this a/r/tographic research project was not unusual for eight-year-olds. But somehow their work was also profound. The packing and unpacking of the theoria (research) in our inquiry uncovered its complexity, the praxis (teaching) of vocabularies (elements of music and multiplication and the concepts therein) returned it to simplicity in its brokendown parts, and poesis (music-making) rendered it both complex and simple as the students synthesized the material. In short, an audience could easily consider our compositions to be “cute little ditties,” unaware of the meaning attached to the conceptual considerations in the compositional process. Most of the children’s expressions were standard songs with age-appropriate use of the elements of music like rhythm, melody, form, texture, and dynamics (Gouzouasis, 2003; British Columbia Ministry of Education, 2009/10). Interestingly, many of the children’s  193  songs (Chan and Paige, Christian and Sebastian, Joshua, Katy, and Lara, and Sarah and Carly) could be compared to the music of aleatoric and contemporary composers such as Phillip Glass, Arvo Part, and John Cage, because their compositions demonstrated similar minimalism concepts, unpredictable phrasing, and atonal melodic choices. The children’s ordinary application of their musical skills at an eight-year-old level produced extraordinary results that shared characteristics to the works of world-class musicians.  5.4  Vision for the future Having entered into this Master’s program, and undertaken this research, because of what  I perceive to be problematic rigidity within the “institution” of education, I am hoping to be able to instigate institutional change through my research contributions. As I mentioned in chapter one, my frustrations were a motivating factor behind my research study, and were an important aspect of this a/r/tographic exposé during my living practice throughout the undertaking of the research project, as well as during my continued practice of teaching music to children. I can see beyond the role of the traditional music teacher. When children come into my IBPYP/Reggio inspired music room they are free to experiment with materials. The questions I pose have any number of possible answers/expressions with a variety of directions to explore. Everyday is different. Although the music teacher (or art, PE, French, Spanish, or dance teacher) is not often the primary leader in classroom inquiry, s/he can engage students in inquiry in meaningful ways that link to the classroom curriculum or standalone. Music class can be about so much more than merely “music education,” in the clinical sense of the term. Listening, singing, composing, moving/dancing, playing instruments, visually representing sound, improvising, recording, performing – all of the  194  many elements of music can be seen as “languages of children.” By “languages” Reggio educators mean the many ways in which children communicate their ideas about the subject or concept on which they might be focusing. Some schools are committed to an engagement of shared inquiry, care, and enthusiasm in their teaching/learning environments. Many schools have a wide healthy, holistic ethos with an attitude toward learning that encourages inquiry, considers alternatives for the timetabling of longer sessions for project work, integration, and co-emergent exploration, and upholds equity and reciprocity among participants, adult and student alike. As a music educator, I task myself in implementing a balanced approach to music education whereby David Elliott’s praxial music education (see section 1.6) is combined, re/fined, and re/defined with elements of Aesthetic Education’s subjective view of the symbols of feeling, Reggio’s hundred languages of children, and co-emergent inquiry’s student generated approach. Such a program might enable children to develop more fully as individuals within a balanced program that fully expresses their musical (and other) understandings. Written and oral languages are the dominant communication modes in schools. Traditional education reinforces the communication styles of some people (i.e., those skilled in written and oral languages), but often ignore the languages of others. Very little consideration is given to how effective alternative languages of expression could be for cognition, and how other languages can communicate in ways that written language cannot. Moreover, it is not often recognized that many people thrive in areas besides written and oral modes of expression in school. Consider Howard Gardner’s Multiple Intelligences – sometimes referred to as the “seven talents”(Gardner, 2003) or the nine “smarts”(Carlson-  195  Pickering, 2006). If children were encouraged to speak in their languages of strength perhaps we would have more academic successes and celebrations of learning, rather than insisting on forcing round pegs into square holes, or triangle pegs into star holes. Fortunately, “Inquiry,” “the project approach,” “student generated curriculum,” and “developmentally appropriate practice” are becoming common terms in the field of education. Reggio inspired practice is also becoming accepted in elementary schools, and some public school boards in British Columbia’s Lower Mainland are starting to implement the Primary Years Program of the International Baccalaureate. My hope is that these pedagogical approaches will become the rule rather than the exception to it. I propose that music is more than a catalyst for brain stimulation or relaxation leading to learning readiness, or a vehicle to attain knowledge of other subject matter or musical skill in isolation (see chapter two). Musical expression (e.g., composing, improvising, playing instruments, and singing) is a “language” with which to communicate learning. 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NY: Teachers College Press.  205  Endnotes  i  Alexander Durig’s (1994) interpretation of Suzanne Langer’s (1951; 1957) theories  about the meaning of the word “language” support Reggio’s use of the word “language.” He suggests that Langer could be interpreted as saying that a re-consideration/re-definition of the type of logic used to infer meaning in symbols (since “formal logic” does not fit with nondiscursive symbols of meaning, rather informal logic and social inference does) would allow music to communicate a type of conceptual meaning, which could be called “language.” He suggests that meaning, logic (re-defined), and emotions form a holistic, triangular, relationship where one cannot exist without the others, and that through this relationship, meaning can be expressed through the “language” of music.  ii  Gouzouasis (2007) has posed the argument that some aspects of music may possess  some form of semantics. For example, he says that when singing the song “Oh Susanna,” one may recognize that the chords have a prescribed order (i.e., syntax) and they possess meaning—in how they function and support the melody, e.g., many pitches in the melody, with the exception of the stepwise passing tones, are directly taken from the chordal accompaniment; in a sense, the melody “outlines the harmony.” The fact that musicians can agree where the tonic, subdominant, and dominant chords need to be when we accompany the song implies that we may be able to agree on the implicit “meaning” of the chord functions, the tonal patterns of the melody, and how the chords may relate to the melody.  206  iii  That said, it should be noted that music is not a universal language, but is  contextually specific to culture, musical style, or an individual musician. While most every culture has sounds that are deliberately organized, it is not globally accepted in the field of musicology and ethnomusicology that “music” exists in every culture. This is because “organized sound” around the world has difference purposes and meanings (Ibsen al Faruqi, 1985). Some cultures consider their organized sound to be for prayer or hunting or ritual, and it is not “music” as we understand it in the majority of cultures around the world. Therefore, it is debatable whether all cultures could use music as a language of expression, as I am suggesting within the context of this study.  iv  It might be important to mention that I in no way imply that the children learned the  entirety of their mathematical understandings during this study. What is mentioned here only refers to the learning—aspects of music and mathematics—that students discovered during the inquiry process. Obviously, these children had numerous mathematical learning experiences in other areas of their lives.  207  Appendix A Ethics Review Board Permission  208  

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