KINESTHETIC MATHEMATICS:
MEANINGFUL APPLICATIONS IN THE CLASSROOM
MEGAN JOHNSTON
A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
MASTER OF ARTS
GRADUATE PROGRAMME IN INTERDISCIPLINARY STUDIES
YORK UNIVERSITY
TORONTO, ONTARIO
OCTOBER 2010 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition
395 Wellington Street 395, rue Wellington OttawaONK1A0N4 OttawaONK1A0N4 Canada Canada
Your file Votre rGterence ISBN: 978-0-494-80625-8 Our file Notre reference ISBN: 978-0-494-80625-8
NOTICE: AVIS:
The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats.
The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission.
In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these.
While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis.
1*1 Canada IV
Abstract
Kinesthetic mathematics: Meaningful applications in the elementary classroom examines kinesthetic ways to enhance the elementary Ontario Mathematics Curriculum through the use of the Ontario Drama and Dance Curriculum. I accomplish this within the context of a case study in a primary classroom in a public school in the Greater Toronto
Area. I designed creative movement/creative dance activities to create ways to enhance the Ontario Mathematics Curriculum and then collected evidence that supports the effectiveness of those ideas. It is my aim to make my research available for use by all teachers, regardless of their creative movement and/or creative dance training. V
Acknowledgments
"Be the change you wish to see in the world."
Mahatma Gandhi
With all my heart, I give thanks to the following:
• The Great Spirit, my center, strength and inspiration on this journey.
• Mary-Elizabeth Manley, Margaret Sinclair and Walter Whiteley, my
guides through this learning process. Your investment, guidance and
kindness will always be remembered.
• Ms. T. for offering your perspective and your class for this study.
• The students, for participating in this study.
• All my teachers, past and present, I have learned so much from you.
• Kathleen McCann Johnston and Edward Johnston, my parents and great
examples on this journey. Thank you for your love and support through
this process.
• Jake, Molly, Jesse, Rose, Claire and Elizabeth, my siblings. I am very
fortunate to have you in my life. Thank you for your love and support.
• Francis Douthwright, my husband. Your love, support and encouragement
has helped me more than words can express. Thank you.
• This work is dedicated to the memory of my mother in law, Anna
Douthwright. The torch still burns and is carried on. vi
Table of Contents
Abstract iv
Acknowledgments v
Table of Figures x
Chapter One: Introduction 1
Rationale 1
Study Questions 2
Chapter Two: Literature Review 4
Teaching Mathematics through a Kinesthetic Approach 4
Family Math for Young Children 4
Math in Motion 5
Methods of Teaching Dance 6
Creative Dance Classes 9 Dance in the Schools 10
Laban's Movement Theories 12
The Ontario Kindergarten Program 16
Kindergarten Mathematics 17
Kindergarten Dance 18
Teaching Movement in Alternative Schools 18
Montessori Education 19 vii
Waldorf Education 20
Arrowsmith School 22
Roger Williams Middle School 22
Linwood A+ Elementary School 24
Whittier Community School for the Arts 24
Learning Theories 25
Piaget's Cognitive-Developmental Theory 25
Multiple Intelligences 27
Brain Plasticity 28
Chapter Three: Methodology and Methods 31
Aboriginal Research Methodology 31
Case Study 32
Study Overview 33
Participant Observation 35
Participants 36
Setting 38
Scheduling 38
Materials 39
Curriculum 41
Data Collection 43
Photographs 44 viii
Student Journals 44
Data Analysis 44
Ethics 45
Chapter Four: The Study 46
Lesson One 46
Lesson Two 50
Lesson Three 54
Lesson Four 61
Lesson Five 66
Lesson Six 71
Lesson Seven 75
Chapter Five: Emergent Themes 81
A Special Case 81
Creativity 84
Diverse Representations 88
Emerging Competence 92
Chapter Six: Answers to Questions and Conclusions 102
First Question 102
Second Question 107
Third Question Ill
Fourth Question 113 ix
References 125
Appendix A: Lesson Plans 1-7 129
Appendix B: Checklists 148
Appendix C: Post Survey for Students 150 X
Table of Figures
Figure 1 14
Figure 2 48
Figure 3 49
Figure 4 52
Figure 5 53
Figure 6 54
Figure 7 57
Figure 8 58
Figure 9 60
Figure 10 60
Figure 11 61
Figure 12 62
Figure 13 65
Figure 14 65
Figure 15 66
Figure 16 68
Figure 17 70
Figure 18 71
Figure 19 76
Figure 20 77 xi
Figure 21 78
Figure 22 79
Figure 23 79
Figure 24 82
Figure 25 82
Figure 26 83
Figure 27 83
Figure 28 85
Figure 29 89
Figure 30 90
Figure 31 91
Figure 32 91
Figure 33 92
Figure 34 94
Figure 35 109
Figure 36 110
Figure 37 110 1
Chapter One: Introduction
My connections to dancing and teaching have always been important to me. My secondary education led me to study for a BFA in contemporary dance at Concordia
University in Montreal, Quebec, followed by a BEd at York University in Toronto,
Ontario. The seed of integrating the arts into traditional curriculum subjects was planted by my BEd Course Director, Kathy Gould Lundy. I explored dance integration in my practicum placements as well as in my first two years of teaching in the Toronto District
School Board. As a teacher in the public system it became impossible to ignore the persistent feeling within—a feeling that the way the curriculum is taught by teachers and learned by students was not reaching its fullest potential. I surmised that by integrating movement into the traditional curriculum subjects, student learning would improve.
At this time, I decided to write a proposal to York University's Interdisciplinary
Department in Graduate Studies. I wanted to further explore the linking of kinesthetic learning with the core curriculum. I hoped to do this by bringing together the disciplines of education, dance, and mathematics.
Rationale
It was my intention to create ways to enhance the Ontario Mathematics
Curriculum and collect evidence to see if my ideas would work. It was also my intention to make my research applicable for use by all teachers regardless of their creative movement/creative dance training. The purpose of my research was to investigate how to increase integration of kinesthetic activities in the mathematics curriculum. The case study was done in the context of a case study in a Junior Kindergarten/Kindergarten 2
classroom. This research is relevant and valuable to education because it focuses on movement, the first modality that young children use to communicate feelings about themselves, the environment, and other people. I was investigating the impact of activities that may help diverse learners within the classroom. This research was also an attempt to develop students' potential for creative communication.
It was my hope that the students in the study would benefit from learning mathematics with movement and that my research would help teachers understand what kinds of activities are interesting to students and what activities help students learn best.
Study Questions
The following are the four study questions that I posed for this study:
1. Are the students engaged in the learning of mathematics when it is learned
through creative movement/creative dance?
• I hypothesized that students would become more engaged in the learning of
mathematics, especially when learning with their bodies (part or whole).
2. How might student attitudes and feelings towards mathematics change when
learned through creative movement/creative dance activities?
• I hypothesized that student attitudes and feelings would be positively affected
when creative movement/creative dance activities were integrated into
learning mathematics. Attitudes and feelings towards mathematics would be
shown by increased enjoyment of learning mathematics.
3. Will students' self confidence in mathematics class be improved when it is
learned through creative movement/creative dance? 3
• I hypothesized that students would feel more confident about themselves
during math class. I also hypothesized that the self-confidence developed in
math class would filter into other curriculum subjects and interactions with
others.
4. What sequence of activities supports students'learning?
• I hypothesized that the way I sequenced the lessons would support the
students' learning of the material presented.
By answering these questions, it is anticipated that the research which focuses on creating and teaching meaningful learning experiences for students will be furthered. 4
Chapter Two: Literature Review
In this chapter, I will discuss the literature around the themes of teaching mathematics via a kinesthetic approach, methods of teaching dance, the Ontario
Kindergarten Program, alternative schools and their approaches to teaching mathematics and dance, and learning theories.
Teaching Mathematics through a Kinesthetic Approach
Mathematics can be taught and learned in a variety of ways. The next two subsections describe some non-traditional programs for teaching and learning mathematics to illustrate the importance and need for providing children with opportunities to engage in movement when learning mathematics.
Family Math for Young Children
Family Math for Young Children (FMYC) is one of the programs offered at the
Lawrence Hall of Science, University of California at Berkeley. One of the goals for this program is to bring together families with young children to both investigate and enjoy mathematics. The families are encouraged to work and learn together. One of several guiding questions for this program is: "Can a child's understanding of mathematical ideas grow through physical movement?" (Coates & Franco, 1999, p. 169).
FMYC is designed to meet the needs of children between the ages of four and eight. Various class components include two guided activities for each session, a movement component, interest centers and discussion time for families to share mathematical ideas and understandings. 5
Examining the guiding questions of this informal family learning program, the results that I extracted from reading about the program included that it was important for a family to learn to integrate all of the aspects of a child's development. In the article, researchers noted that "When parents have the opportunity to discuss and give input, they broaden their own repertoire of parenting and teaching strategies" (Coates & Franco,
1999, p. 173). In the FMYC model, which included movement, mathematics, and parental involvement, parents came to understand the importance and need for providing children with opportunities to engage in movement. Parents developed a repertoire of activities to help their children attain practical skills that serve as tools to broaden their children's creative expression and deepen their understanding of mathematics as well as other areas. It was also found that parents developed a deeper understanding of themselves.
Math in Motion
One approach to teaching mathematics concepts is through the use of physical activity. Math in Motion is a coined term that describes the integrated mathematics and physical activity approaches. As Goodway, Rudisill, Hamilton and Hart mention, "This approach was developed as a result of the concern expressed by early childhood educators who found it difficult to keep at-risk and delayed children engaged in more- traditional mathematics activities" (Goodway, Rudisill, Hamilton & Hart, 1999, p. 175).
In this article it also states: "Although empirical evidence for this curriculum is still emerging, the Math in Motion activities have been reported by the teacher as effective for children with a wide range of backgrounds and needs" (Goodway, Rudisill, 6
Hamilton & Hart, 1999, p. 175). Some of the teachers have reported that these activities have been useful for children in grades pre-K-2 who are developmentally delayed, from both urban and rural environments, or at risk of developmental delay or school failure; and for students with limited English-language skills. Teachers reported in the article
"Math in Motion" by Goodway, Rudisill, Hamilton & Hart, 1999, that when they have used the Math in Motion curriculum activities in the classroom, the students have expressed a joy of learning that was not present when they taught in more traditional ways. The teachers also reported that the children seemed to engage in the activities at a deeper conceptual level. Because of the characteristics that children who are "at risk" bring to the classroom including lack of impulse control, short attention spans and high activity level, the physical context of the Math in Motion activities are very suited to their needs. The article states:
It is often said that movement is the language of early childhood, yet we
seem to spend more time resisting children's desire to move rather than
taking advantage of this valuable resource. We believe that physical
activity is a wonderful medium in which to teach mathematics concepts
(Goodway, Rudisill, Hamilton & Hart, 1999, p. 181).
Methods of Teaching Dance
In a Western context, a variety of dance forms exist, including ballet, modern/contemporary dance, creative movement/dance, tango, salsa, tap, jazz, and ballroom, to name only a few. In the sections that follow, I will compare traditional methods of teaching dance with recent initiatives to introduce dance in the schools. Also, 7
theories about movement will be introduced.
Dance and dance education is defined by Gale Kassing and Danielle M. Jay in
Dance Teaching Methods and Curriculum Design as follows:
Dance education can be defined as educating the learner through the media
of dance, dance making, and dance appreciation . . . Dance is the human
body rhythmically moving through space and time with energy or effort.
Dance engages the dancer's physical, mental, and spiritual attributes to
perform a dance form as a work of art, a cultural ritual, a social recreation,
and an expression of the person. A dance form initiates from physical
movement, rhythm, content, style, aesthetics, traditions, and mental and
spiritual meanings that may be social, cultural, or religious. (Kassing &
Jay, 2003, p. 4)
Movement and movement education is far more broad than dance or dance education. Movement is considered to be the result of, or the process of moving. Kassing and Jay describe movement education as follows:
Movement education focuses on learning movement with and without
manipulatives (balls, sticks, and other props). This approach is the basis
for physical education in the elementary school. (Kassing & Jay, 2003, p.
202)
Kassing and Jay later go on to define creative movement:
Creative movement focuses on exploring the elements of time, space, and
energy (force) and creating shapes through everyday and dance-specific 8
locomotor and non locomotor movement to gain movement competency.
The aim of this dance form is to preserve the child's spontaneity of
movement while enhancing movement vocabulary and repertoire.
(Kassing and Jay, 2003 p. 215)
The definition for creative dance by Ann Green Gilbert from her book Creative
Dance for All Ages is described in this way:
Creative dance (which may also be referred to as creative movement) is a
form of dance used commonly with children, to explore concepts of body,
space, time, and energy (Gilbert, 1992).
Creative dance and creative movement are often synonymous, each taking on characteristics that make them suitable for a variety of instructional situations. Creative movement and creative dance are distinguished by the expectation of the class. In creative movement, the purpose of the class is an exploration of the understating of the elements of movement. In a creative dance class, children explore the elements of movement that are structured with a beginning, middle, and ending. This structural element of a creative dance class enables creative movement to unfold into creative dance, due to the dance expanding into choreographic form.
Creative dance is appropriate for preschool through elementary school. Rudolf von Laban was instrumental in the development of creative dance due to his movement analysis known Labanotation. The dance methodology upon which the Ontario
Kindergarten Dance and Drama Curriculum is based, as is the 2009 Ontario Grades 1-8
Dance Curriculum, comes from the Laban theory and is defined as creative dance. In 9
educational frameworks in the West, the dance form taught in the primary division is creative dance. For the remainder of the document I will be using the term creative dance when writing about the kind of dance that I taught for this study.
Creative Dance Classes
Some general features of a creative dance class may include the dance space being taped out or defined by markers. This device is used to safely control the area in which the children can move. Depending on the setting, children are expected to wear comfortable clothes or dance attire. Footwear to be worn can range from bare feet, running shoes, todance slippers. Creative dance classes can occur in a range of spaces including a dance studio, gymnasium, church hall, or classroom.
General space is the space through which all dancers move and personal space is the space directly around the dancer's body. For the young child, the teacher may support the understanding of personal space through the use of a personal carpet square, a laminated paper with the child's name on it, or tape markers on the floor, or by imaging a bubble surrounding the body.
In creative dance classes, exercises and activities are repeated to reinforce movement, can serve as ritual, and aid in the teacher's control of the class. These repetitive components offer foundational skills paired with reassuring activities that children both enjoy and come to expect. Some traditional rituals of a creative dance class include entering the class, forming a circle or standing in lines, a warm-up, taking turns executing movements, learning new activities or movements, and a warm-down and closing. 10
Warm-ups in a creative dance class can vary in content and can include a variety of exercises to increase body awareness, strength, flexibility, and balance. Warm-up activities may be performed while lying on the floor, sitting in tailor position, while kneeling or standing. Often during the warm-up, the elements and concepts to be used in the class are introduced.
During the body of the creative dance class the activities usually contain some exploration of the elements of dance which include space, time, and force. The teacher guides the students through activities by cueing, asking questions, and at times giving examples. Individual activities in the class usually result in a culminating activity. A common culminating activity would include students having the opportunity to create movement sequences/sentences either alone or in a group. The sequence generally would need to include a clear beginning, middle, and end. Students are usually given the opportunity to observe others dancing and time to share what they have observed.
The warm-down and closing of the creative dance class may include free dance time. During this time students will explore movement and review elements and concepts learned from the lesson. A more formal review of the elements and concepts taught may also be part of the warm-down. Many times, a teacher guided relaxation exercise will be included in the warm-down to allow students to prepare for leaving the dance space.
Dance in the Schools
Over the past thirty years, The Kennedy Center Professional Development
Opportunities for Teachers Program in Washington, D.C. has been assisting K-12 educators to integrate the arts into classrooms. Approximately 25% of the program's 11
focus is to provide both dance and classroom teachers, methods of approaching the teaching of core curriculum with dance as a tool. Emphasis is on creativity rather than technique. Due to the growing number of ESL students, teachers must face the reality of learning to teach through nonverbal ways. Elder (2007) notes what The Kennedy Centre
Vice President for Education, Darrel Ayers, says about nonverbal teaching: "Some of these students become disenfranchised because they don't understand what the teacher is saying . . . dance breaks down that barrier" (p. 20). Melanie Layne, an Arts Integration
Resource Teacher at Bailey's Elementary School for the Arts and Sciences in Falls
Church, VA, agrees with Darrel Ayers. She notes:
I am convinced that kinesthetic learning is a natural way for students to
engage in learning while building self-control, confidence and community
within the classroom. It is a way of teaching that is especially imperative
for students who are learning disabled or are ESL learners and may have
difficulty learning rational schooling methods. (Elder, 2007, p. 21)
By attending workshops at the Kennedy Center, Melanie Layne has not only gained contacts, lesson plans, and ideas of how to integrate movement into core subjects; she has also gained the confidence to experiment. She comments:
Due to the teaching artists' leadership, instruction and inspiration during
the workshops, I now am able to use dance to teach vocabulary, patterns
and cycles in science, imagery in poetry and geometry in math. (Elder,
2007, p. 21)
Similar to the Kennedy Center Program in Washington, the Manhattan City 12
Center in New York, NY, is helping dance become central to teaching the core subjects.
Sponsored by the New York City Board of Education, the goal of this institute is to train dance teachers to become staff developers in their districts. Dance teachers learn the tools to teach classroom teachers how to integrate dance into the core curriculum subjects.
Dance integration is old news to New York City. Flender (2000) quotes Jannas Zalesky,
City Center's Education Director who said: "Integrating dance into the curriculum is not a new idea for City Center. For more than 10 years, we have been working in this way"
(p. 68). The purpose of the workshops at the City Center is to create lessons whereby dance, kinesthetic activities, or creative dance can be used by the teacher when teaching other subject areas. The City Center also offered a service of five Saturday workshops designed for K-9 teachers entitled "Connecting Movement with Math, Science and
Literacy." The participating teachers were expected to work collaboratively in groups and develop lessons that address the New York State standards and New York City curriculum.
Laban's Movement Theories
Rudolf von Laban (1879-1958) was a movement theorist, choreographer, dancer, and one of the founders of modern dance. His work helped garner recognition for dance as an art form, and his discoveries within the theory and practice of dance and movement have significantly influenced scholarly dance pursuits. Laban established what is known today as Laban Movement Analysis (LMA). Part of LMA concerns Laban's use of the five crystals, which are also known as the Platonic Solids and the potential for movement that lies within these solids. A variety of movement scales were developed within the five 13
crystals that include Laban's Dance Elements including space, time, weight, and flow.
Effort actions which use space, time and weight that enhance the scales include the dab, flick, punch, slash, glide, float, press, and wring.
Laban's ideas caught my interest due to the visualization component of imaging a surface surrounding the body that offers the dancer a sense of boundary and safety. As a dancer, I have always thought of structure as liberating and a key element to my creativity. Structure for me offers room to explore, choices to make, a time frame, and literally—endless possibility. Creativity without structure can be lethargic and sloppy. It is because of Laban's ingenuity in using crystals as a stimulus for movement that I appreciate his approach so much. This part of his methodology provides the structure needed to allow for creative exploration to occur.
Laban created movement scales that are based within the Platonic Solids. Platonic
Solids are named after the philosopher Plato and belong to a special category of polyhedra. A polyhedron is a solid with faces that are polygons. Two faces meet at an edge and three or more edges will meet at a vertex. Polygons are figures that are bounded by line segments; for example, quadrilaterals and triangles are polygons. The faces of polyhedra are polygons—for instance, the five faces of a triangular prism (a common roof shape) include three rectangles and two triangles. Platonic solids have faces that are regular polygons—polygons that have equal edges and equal interior angles. The equilateral triangle and the square are two commonly known regular polygons. The following image of the Platonic Solids (see Figure 1), was retrieved from The Internet
Encyclopedia of Science. The Visual Geometry Project's video entitled The Platonic 14
Solids, characterizes the platonic solids as thus: 6
dw&6t&h&t**m jixsahzdinrm
Figure 1. The Platonic Solids
1. All faces are regular polygons.
2. All faces are the same or congruent.
3. All the corners are the same; the same number of faces meet at each vertex
in exactly the same way.
A cube is an example of a platonic solid because it fits all the characteristics as stated above. Every polyhedron has a dual polyhedron with vertices and faces interchanged; for example, the cube and the octahedron are duals—the cube has six faces and eight vertices while the octahedron has eight faces and six vertices. The dual of every
Platonic Solid is another Platonic Solid. Thus, the Platonic Solids can be arranged in dual pairs. Creatively and intellectually speaking, I am fascinated with the range of motion and mathematics that corresponds to these solids/crystals. Within my case study, I will use the 15
term solid as opposed to Platonic Solid because the 3-D solids to be explored are not all
Platonic Solids.
As an example, the first movement scale titled the Dimensional Scale is based within the Octahedron and has the dancer moving in one dimension and reaching to six different directions through this sequence. This scale is based on the dancer moving along the three-dimensional axes of the body: length, width, and depth and their related axes and planes within the dancer's kinesphere. During this sequence, a central pull, and a right and left as well as a forward and back pull of centrality and stability is experienced.
The second scale, titled the Diagonal Scale, is based within the cube; three-dimensional movement is created around the diagonal cross of axes. More twisted and unstable movement is created with this scale due to the fleeting movements within it. Because of the opposite movements created in these scales, these two scales and the solids associated with them (the cube and the octahedron) are duals of each other.
Laban also developed an Effort grid, which includes the body efforts of weight, space, time, and flow. Each of these efforts corresponds to a symbol. When all the symbols are put together, the grid emerges. Laban also indicates that movement can occur on planes including: the vertical plane combining up/down and right/left movements; the saggital plane, combining forward/backward and up/down movements; as well as the horizontal plane with the combination of right/left and forward/backward movements. All of the planes and parts of the planes (for example high left or forward low) correspond to an individual symbol. 16
The Ontario Kindergarten Program
Currently, the Ontario Kindergarten Program is designed for a half-day class, five days per week. The six strands in the Ontario Kindergarten Program are: Personal and
Social Development, Language, Mathematics, Science and Technology, Health and
Physical Activity, and The Arts. The philosophy behind the Kindergarten program is based on the premise that children's early learning experiences are paramount to their future well-being. Kindergarten is the foundation for future learning and behaviors. It is the goal of Kindergarten educators to understand that students are arriving in school with diverse backgrounds, and to create a secure and respectful environment whereby all students are nurtured.
A kindergarten classroom usually consists of a carpeted area where the students and the teacher gather and have circle time. During circle/carpet time, mini lessons in mathematics, language arts, science and technology, or the arts are given. The teacher generally sits on a chair while the students sit on the carpet. There is usually a white board or a low easel for the teacher to use during her/his lesson. A typical kindergarten classroom has a variety of centers, e.g., listening, nature/science, computers, arts and crafts, painting, water play, blocks, stories, games, math, puzzles, and writing. Students are generally allowed to move between centers throughout the course of a day. The centers change in content over time and they are designed to meet aspects of the
Kindergarten Curriculum. During center time, the teacher circulates to the centers and conferences with the students. Assessment occurs during this time, through anecdotal records of conversation and teacher observations. Depending on the time of year, center 17
time is also used for formal evaluations. The kindergarten program also allows time for library, physical education, and music classes as well as for outdoor playtime. Snack time occurs every day in a Kindergarten classroom.
Kindergarten Mathematics
Mathematics in Kindergarten is based on a student's desire to make sense of her/his world and to help children demonstrate and develop their mathematical understandings. The foundation of the Kindergarten mathematics program is based on problem solving and reasoning. Children explore and become involved in mathematics through being challenged by rich mathematical problems that are presented to them.
These problems arise from real life situations. Each problem has a variety of ways of being approached as well as answered. Because of this, solving these mathematical problems requires persistence. As stated in the Ontario Curriculum:
Through active participation in mathematics investigations, including
problem solving and discussions, children develop their ability to use
mathematics as a way of making sense out of their daily experiences. (The
Ontario Kindergarten Curriculum Revised, 2006, p. 40)
In order to provide balanced instruction, the mathematics program is divided into the following five strands or subheadings: Number Sense and Numeration (Quantity
Relationships; Counting; Operational Sense), Measurement (Attributes, Units, and
Measurement Sense; Measurement Relationships), Geometry and Spatial Sense
(Geometric Properties; Geometric Relationships; Location and Movement), Patterning
(Patterns and Relationships), Data Management and Probability (Collection and 18
Organization of Data; Data Relationships; Probability).
Kindergarten Dance
Dance is found along with Visual Arts, Music, and Drama under The Arts' heading in the Ontario Kindergarten Curriculum Revised, 2006. It is stated that: "The arts in their many forms provide a natural vehicle through which children express their interpretation of our world. Therefore, the arts play an important role in the development of children's communication and thinking skills" (p. 56). It is important to note that communication and thinking skills are also intimately connected with the process of mathematics, and mathematics involves communicating about and interpretation of the world.
In the Drama and Dance section of the Ontario Kindergarten Curriculum, children are encouraged to: move in ways of their choice in free dance; use a variety of tools to create dance; understand vocabulary related to dance (e.g. words or hand signals related to props, speed, directions, and levels); use imagination and problem solving to create a dance/sequence of movements; express and respond to dance by making connections to their own experiences; as well as communicate their understanding about something through dance. Carefully planned activities provide daily and weekly opportunities for children to explore creative dance. Children can apply and extend their understanding at the centers provided.
Teaching Movement in Alternative Schools
I decided to look into alternative schools to see how flexibly they linked creative movement and/or creative dance to the curriculum. In this section, I will discuss the 19
Montessori, Waldorf, and Arrowsmith schools as well as Roger William Middle School,
Linwood A+ Elementary School, and Whittier Community School for the Arts relative to their curriculum offerings in creative movement/creative dance. All of the schools integrate creative movement and/or creative dance into the curriculum in one way or another.
Montessori Education
Montessori education has movement embedded into nearly all activities in the classroom. As Angelline Stoll Lillard states in her book Montessori: The Science Behind the Genius, "The possible examples are endless: In Montessori classrooms, learning is accomplished through movement" (p. 38). There are eight principles of Montessori education and I focus on the first one in the list: "That movement and cognition are closely entwined, and movement can enhance thinking and learning" (p. 29). This first principle is based on the observation that our brains are developing in a world in which our doing is the learning. This is significant for my study because my study is based on the integration of dance into the mathematics curriculum. Montessori believed that thinking is expressed in the hand before it is expressed in words. She believed that sitting in a static position did not promote learning. Montessori also believed that movement and thinking are the same process in small children. She states:
One of the greatest mistakes of our day is to think of movement by itself,
as something apart from the higher functions . . . Mental development
must be connected with movement and be dependent on it. It is vital that 20
educational theory and practice should become informed by this idea.
(Montessori, 1967, p. 141-142)
Based on this insight, she developed methods to educate young children whereby object manipulation is the central focus. Since students in my study are required to manipulate objects as well as move their bodies, Montessori's theories on movement and thinking are building blocks for this research.
Waldorf Education
In his book, Waldorf Education Theory and Practice, Richard Blunt writes about the theories and practices developed by Rudolf Steiner and how they are used in today's
Waldorf classroom. Steiner believed that the needs of every child can only be addressed by finding artistic ways of presenting every subject in the curriculum. In Waldorf education, mathematics is divided into Arithmetic and Geometry. Arithmetic is taught in wholes and with the number one first. Numbers are learned by pointing to the self as number one; the number two is learned when putting two hands together; to learn the number three, children stand together to form a trio of bodies; this process goes on and on with reference to the body parts of humans and animals and groups of students. After concrete pictures have been formed of the numbers, then rhythm and physical activity are introduced.
Physical activity and rhythm can be brought into counting by having the
children stamp when the final number of a sequence is counted (1, 2, 3,
4...1, 2, 3, 4...). The rhythm and stamping for each sequence strengthens
the idea of the whole. The child should be made to count while he is 21
moving because in this way what takes place in the inner being is also
brought to expression in the outer. (Blunt, 1995, p. 133)
This is just one example of the way that movement is incorporated in the Waldorf mathematics curriculum. All arithmetic problems are approached in the concrete form and a constant connection to the everyday world is made. Abstract numbers are introduced when the child is nine years old.
Rudolf Steiner believed Geometry to be in the origin of the human being's spiritual essence. He claimed that geometrical forms such as the cube and the icosahedron aren't intellectual inventions; rather, they are laws of the cosmos.
These bodies are not invented; they are reality, but unconscious reality. In
these and other geometrical solids lies a remarkable harmony with the
subconscious knowledge which man has. This is due to the fact that our
bone system has an essential knowledge; but your consciousness does not
reach down into the bone system. The consciousness of it dies, and it is
only reflected back in the geometrical image which man carries out in
figures. Man is an intrinsic part of the universe. In evolving geometry he is
copying something that he himself does in the cosmos. (Steiner, 1919, p.
53)
This quote captures the deep thinking and reflective nature of Rudolf Steiner's philosophy and ideas surrounding education. Steiner goes on to explain: "Thus you call forth the form from your whole body by appealing to the sense of movement which extends throughout your body" (Steiner, 1919, p. 121). Following from Steiner's 22
philosophy, Geometry lessons involve students in using a rod and moving through various exercises to complete geometric forms with the body.
Arrowsmith School
Norman Doidge, M.D. is the author of The Brain That Changes Itself as well as a psychoanalyst, psychiatrist, and researcher, who divides his services and time between
Columbia University, in New York, NY, and the University of Toronto, in Toronto,
Ontario. In his text, he comments on exercises used at the Arrowsmith School, a school in
Toronto for students with learning or attention-deficit disorders. This school was founded by Barbara Arrowsmith Young. Barbara was able to acknowledge her own learning disabilities as a graduate student at OISE and with the help of a colleague was able to build a program for herself to overcome her obstacles. Tracing exercises were developed by Barbara to stimulate neurons that have been weakened in the pre-motor area of the brain. Barbara claims that tracing exercises improve children's ability to read, write, and speak. Although this is not directly connected to math and dance, the learning of concepts through movement activities is part of daily lessons at the school. Barbara believes that
"When weak links in the chain are strengthened, people gain access to skills whose development was formerly blocked" (p. 41). She claims that this experience can be tremendously liberating for the student.
Roger Williams Middle School
Roger Williams Middle School is located in Providence, Rhode Island. This school received a grant to bring in mathematics and kinesthetic specialist Galeet
Westreich, PhD, to work with special education and sixth grade students in their math 23
and dance classes. According to McGreevy-Nichols (2002):
Westreich has developed what has become known as the Kinematics
Teaching Strategy (KTS), a method for teaching mathematics through
dance. KTS was tested in second-, third- and fourth-grade math classes in
Washington, DC, as well as in Orange County, Florida, and the results
indicate that students taught through KTS demonstrate greater math
aptitude than those who are not. (McGreevy-Nichols, 2002, p. 82)
KTS, like other kinesthetic teaching methods, is based on problem solving.
McGreevy-Nichols (2002) states that according to Westreich, abstract mathematical concepts that are being learned through dance need to follow the four-step model, which includes:
1. Verbal explanation of the concept
2. Kinesthetic exploration of the concept, in which students decide which dance
movements should be performed in what sequence for each part of the concept
3. A combined verbal exploration and discussion of how the concept was
represented
4. A written summary of the solution, rendered mathematically (p. 82)
The above scaffolding is a clear way for students to embrace new ways to think about mathematics; it also offers students a chance to work in groups, present their work, evaluate and revise work if necessary. A few of the mathematical concepts taught through the KTS model to the students at Roger Williams included exploring simple fractions, multiplication of single and two digit numbers, and statistics concepts. 24
Linwood A+ Elementary School
The A+ in this school's name stands for A+ academics. The school is based on a model that has school-wide support behind an arts infused dance curriculum. The school is located in St. Paul, Minnesota. At Linwood A+ there were four goals for the 2004-2005 academic year which included: "Having the A+ Model reflected throughout the school; improving achievement in reading and writing; improving achievement in math; and maintaining a respectful and caring school community" (Mohn, 2004, p. 122). The dance teacher, Katherine D. Mohn is the Dance Specialist at Linwood A+ and she embraces these four goals in her classroom: "For the most part, Linwood A+ students dance about
'academic' themes. How we dance about them is the dance curriculum . . . Math concepts
(e.g. geometric shapes, points, line segments, radius, perimeter, and so forth) lend themselves to movement" (p. 122). Katherine notes that in dance class, movement supports both higher-level thinking processes and comprehension.
Whittier Community School for the Arts
Whittier Community School for the Arts is a K-5 arts-focused school in the
Whittier community of Minneapolis. This school was involved for three years (1998-
2001) in an innovative arts integration, which included an intense partnership and collaboration between individual classroom teachers and dance artists from the area to use concepts central to mathematics and dance to teach curriculum. Linnette Werner
(2001), who analysed the integration, expresses the central question in her paper: "How does integrating dance and math in an intense co-teaching model of integration affect student attitudes toward learning math?" (p. 3). The goals of the integration project were 25
to engage students in mathematics in ways that reached students' multiple intelligences and encouraged them to make complex connections and experiment with new problem solving techniques. Through a combination of data collection methods including observations, student surveys and teacher interviews, Werner drew her conclusions. The findings concluded that there was a significant difference between the dance/math students' attitude toward math compared to the non-dance/math students' attitudes toward math. The dance/math teachers also commented on notable differences with their students. The dance/math teachers stated: "Students were better able to make connections among diverse subjects and pieces of knowledge than they were before the project, which made learning math more interesting and applicable to everyday life" (p. 3).
Werner contends that "the literature surrounding student attitude toward learning suggests a strong link between positive attitude and student achievement scores."
(Gottfried, 1990, p. 8).
Learning Theories
I decided to look into learning theories to see how they linked movement to learning new material. In the following section, I will examine Jean Piaget's cognitive- developmental theory, Howard Gardner's theory of multiple intelligences, Norman
Doidge's understandings of the workings of the inner brain and neuroplasticity. All of the theories recognize movement as enabling and enhancing the learning process.
Piaget's Cognitive-Developmental Theory
Jean Piaget was a Swiss psychologist and philosopher and is well known for his studies in pedagogy. As mentioned in the text Life Span Development, Piaget's question: 26
"How does thinking develop?" is posed. In his studies, he was struck by the fact that all children go through the same sequence of discoveries about their world, make the same mistakes and reach the same solutions.
Piaget developed the model of a scheme. A scheme is an internal cognitive structure that gives an individual a procedure to follow in a specific circumstance. Piaget proposed that each of us start life with a small record of sensory and motor schemes. This includes looking, tasting, touching, hearing, and reaching schemes. As we grow, our schemes become better adapted to our world and inevitably work better.
Through the processes of assimilation, accommodation and equilibration, a young child with built in schemes begins to develop mental schemes. Mental schemes help us to use symbols and think logically. Assimilation is the process of using schemes to make sense of experiences. The complimentary process is accommodation, which involves altering the scheme as an outcome of some new information attained through assimilation. The process of accommodation is crucial for developmental change.
Equilibration is the process of maintaining the balance of assimilation and accommodation to create schemes that will fit the environment that surrounds us.
Piaget also suggests that logical thinking evolves in four stages. The stages include: the sensorimotor stage, the preoperational stage, the concrete operational stage and the formal operational stage. The children that I worked with were in one of the first two of the stages mentioned, or somewhere in-between. The preoperational stage ranges in children from 18 months to about age 6. This is where children acquire symbolic schemes, such as language and fantasy that they use in thinking and communicating. In 27
the concrete operational stage, 6 to 12 year-old children begin to think logically and become capable of solving problems. Piaget's view was that each stage grows out of the one that precedes it and each stage involves a major restructuring of the child's way of thinking.
Neo-Piagetian theories expand Piaget's theory rather than contradict it. For instance, theorists Juan Pascual-Leone and Robbie Case have found that the ages of the stages are approximate and that each individual processes information differently. These researchers also report that differences in prior knowledge affect memory (Bee, Boyd &
Johnson, 2003, p. 42). Bee, Boyd, and Johnson explain, "in a sense, prior knowledge gives you a set of mental 'hooks' on which to hang new information" (Bee, Boyd &
Johnson, 2003, p. 43). All of the lessons for the case study were designed to activate students' prior knowledge.
Multiple Intelligences
In Intelligence Refrained: Multiple Intelligences for the 21s' Century, the Harvard psychologist Howard Gardner, gives a current report on his theory of multiple intelligences as well as how human development and education are affected by and understood through his theory. Dr. Gardner's book takes a close look at the origin and an identification of intelligence.
Gardner's recent definition of intelligence is conceptualized as follows: "A biopsychological potential to process information that can be activated in a cultural setting to solve problems or create products that are of value in a culture" (Gardner, 1999, p. 14). Gardner emphasizes that intelligence is a neural potential, and that intelligence not 28
only stems from nature but also stems equally from being nurtured. This suggests that intelligence can be encouraged or smothered, depending on what qualities are and are not present in one's cultural upbringings.
Although it is up for dispute, Gardner believes that there are eight-and-a-half intelligences. He originally decided to separate candidate faculties that he thought could be intelligences. Faculties directly tied to sensory modalities provided the starting point and then these were grouped by their disciplinary roots. The current list of intelligences includes: linguistic, logical-mathematical, musical, bodily-kinaesthetic, spatial, intrapersonal, interpersonal, and naturalistic. Existential intelligence only qualifies as half of an intelligence.
The three intelligences that I worked with were the logical-mathematical, the bodily-kinaesthetic, and the spatial. Logical-mathematical and spatial intelligences have traditionally been valued in our school system. Logical-mathematical intelligence is described as: "The capacity to analyze problems logically, carry out mathematical operations, and investigate issues scientifically" (p. 42). Spatial intelligence is: "The potential to recognize and manipulate the patterns of wide space as well as the patterns of more confined areas" (p. 42). The bodily-kinaesthetic is valued to a degree within the school system and is described as: "The potential of using one's whole body or parts of the body to solve problems or fashion products" (p. 42). In the activities that I designed, students had many opportunities to explore connections among these intelligences.
Brain Plasticity
In The Brain That Changes Itself (2007), Doidge uses a layperson's terminology 29
and personal stories to explain the brain's inner workings and to talk about neuroplasticity—the brain's ability to form new connections throughout life.
The word neuroplasticity took a long time to emerge in scientific literature. In the late 1960s and early 1970s, scientists were beginning to make discoveries that were proving that the brain is constantly changing with every activity performed. "If certain
'parts' failed, then other parts could sometimes take over" (Doidge, 2007, p. XV). Paul
Bach-y-Rita, one of the founding pioneers in brain plasticity, describes the process of using a machine with electrodes to reinforce any potentially healthy tissue. He believes that by introducing this machine, new trails for neurological pathways are blazed. "A brain system is made of many neuronal pathways, or neurons that are connected to one another and working together. If certain key pathways are blocked, then the brain uses other pathways to go around" (Doidge, 2007, p. 9).
Bach-y-Rita believes he has the ability to "connect anything to anything" (p. 24).
For example, people who have no facial muscles have little to no control of the movements that occur in the face, including closing their eyes or speaking. Bach-Y-Rita found that by surgically attaching a nerve from the patient's tongue to the facial muscles, a reactivation of the part of the brain that controls the facial muscles occurs.
Susan McGreevy-Nichols' article (2001) states that according to Dr. Hon J.
Ratey, the author of A User's Guide to the Brain (2001, Pantheon Books) and, clinical psychiatry professor from Harvard University, there is evidence that peoples' ability to learn new information and retain old information is enhanced by biological changes within the brain brought on only by physical activity. Our physical movements use many 30
of the same neurons that are used for reading, writing, and mathematics. Dr. Hon J. Ratey states:
Physically active people reported an increase in academic abilities,
memory retrieval and cognitive abilities. What makes us move is also
what makes us think. Certain kinds of exercise can produce chemical
alterations that give us stronger, healthier and happier brains. A better
brain is better equipped to think, remember and learn. (McGreevy-
Nichols, 2001, p. 146) 31
Chapter Three: Methodology and Methods
[Qualitative research] seeks to understand the world through interacting
with, empathizing with and interpreting the actions and perceptions of its
actors . . . [It] tends to collect data in natural settings . . . [and] generate
theory rather than test it. (Brockington and Sullivan, 2003, p. 57)
In this chapter, I will be discussing my methodology, and the materials, participants, and setting of the qualitative case study that I conducted. A case study is my main organizing principle for this research. I also use Aboriginal Research Methodology for the perspective it offers. I collected data through focus groups, interviews, participant observation, digital photographs, and student journaling. After every lesson I taught, the students were asked to write about a journal topic that I had given them. Still digital images, recorded focus group interviews, my field notes of the settings, activities, and students were used as documentation. Quotes, my comments, reactions, and feelings about my initial interpretations that I recorded in my journal are also used.
Aboriginal Research Methodology
Aboriginal Research Methodology's focus of locating oneself is used in the research as a form of respect to myself and those participating. Absolon and Willett
(2005) state: "To locate is to make a claim about who you are and where you come from, your investment and your intent. To put yourself forward means to say who you are, give yourself voice and claim your position" (p. 112). To locate or position myself in this study was and still remains very important to me. Every time I think about the study or the writing, I reflect on my intentions for this thesis. 32
Case Study
Merriam (1988) notes that, "A case study is an examination of specific phenomenon such as a program, an event, a person, a process, an institution or a social group" (p. 9).
I used the case study format for a variety of reasons, but primarily because I was working with a bounded system—a public elementary class. Merriam (2009) also notes
Adelman, Jenkins, and Kemmis' description and examples of bounded systems: "The most straightforward examples of bounded systems are those in which the boundaries have a common sense obviousness, e.g. an individual teacher, a single school, or perhaps an innovatory programme" (Merriam, 2009, p.42). In my study, the system was bounded in terms of student demographics, the number of students that made up the study group, the time I was able to work with the study group, and the material I was to cover.
Due to my desire to create activities and teach students mathematical concepts through creative dance experiences, I was particularly interested in a descriptive case study. Merriam (1988) shares reasons as to why descriptive case studies are useful: "In presenting basic information about an area of education where little research has been conducted, innovative programs and practices are often the focus of descriptive case studies in education. Such studies often form a base for future comparison and theory building" (p. 27). Case study inquiry conforms to many similar principles in an attempt to investigate a phenomenon "within its real-life context; when the boundaries between context and phenomenon are not clearly evident; and in which multiple sources of evidence are used" (Yin, 1989, p. 23). 33
I was the chief investigator, and because this is a qualitative case study, I was the primary instrument for gathering and analyzing the data. A case study is subjective and relies on the researcher's previous experience to interpret the data provided. Through analyzing my notes and reflections, the students' work and comments, photographs, and comments the host teacher made, I sought to provide the reader with clear glimpses of the lessons. As stated by Lincoln and Guba: "Through the use of'thick description,' case studies provide the reader with 'snapshots of reality' and are a powerful means of building intuitive, empirical generalizations" (Lincoln and Guba, 1985, p. 120). My findings will be analyzed, drawing from the collected data and my previous experience.
Study Overview
My data collection was carried out over a period of two weeks in January 2009 with a follow-up lesson in March 2009.1 was the guest teacher in a Kindergarten classroom in a Toronto public school. I taught three math/creative dance lessons per week for two weeks to one Junior/Senior Kindergarten class consisting of twenty students; as well, I taught a follow-up lesson a month-and-a-half after the six lessons were completed.
To prepare the lessons for the research, I gathered and created materials that were to be used. In increments, I introduced movement into the lessons. Some of the creative dance components of my lessons were digitally photographed for the purpose of analysis. In order to ensure triangulation, I conducted two semi-structured focus group interviews with a group of four students, conducted pre- and post-interviews with the teacher of the class, and collected participant observations and student reflections, as well as digital photographs of the lessons. 34
Merriam's (1988) thoughts on the semi-structured interview are as follows:
In the semi-structured interview, certain information is desired from all
respondents. These interviews are guided by a list of questions or issues to
be explored, but neither the exact wording nor the order of the question is
determined ahead of time. This information allows the researcher to
respond to the situation at hand, to the emerging worldview of the
respondent, and the new ideas on the topic. (Merriam, 1998, p. 74)
I wanted each of my interviews to be a dialogue and not to be simply a question and answer period. Thus, I chose to use the semi-structured interview format. Five
Kindergarten students participated in the focus group interviews. Due to the students' ages, the interview length was as long as the students could pay attention—approximately
15 minutes. It was important for me to have students' feedback on what and how they were learning, so I wanted their voices heard within this research. Including children in research that involves them is important in that it recognizes that they, as Prout states, are: "competent commentators on their own lives" (Prout, 2002, p. 68). Moreover, as my research was about student participation and curriculum integration, I considered youth input to be essential. As Vaughn, Schumm, and Sinagub state: "Children are likely to provide more spontaneous responses than some adults" (p. 132).
Before each of the focus group interviews began, I checked the setting in which the interview would take place for its suitability as the focus group site. To ensure comfort of the research participants, I took into consideration the size of the room; I tried to find a room that was not too big or small. I also needed to take into consideration the 35
host teacher's schedule and an appropriate time to withdraw the students from their program. The focus group interviews were digitally tape recorded; as well, I recorded notes in journal fashion.
Participant Observation
I decided to include participant observation as a component of my research methodology, due to my desire to participate in the teaching of mathematics through creative dance. It is considered important to combine formal qualitative research techniques (such as interviews) with participant observation to help triangulate the data
(Hammersley and Atkinson, 1995; Davies, 1999; Fetterman, 1989). Regardless of the struggle participant observation poses, it was necessary for me to include this research tool in my study, due to my involvement in teaching the mathematics/creative dance activities to the students. "Participant observation is a schizophrenic activity in that one usually participates but not to the extent of becoming totally absorbed in the activity. At the same time one is participating, one is trying to stay sufficiently detached to observe and analyze" (Merriam, 1988, p. 94). James Clifford (1988) mentions the need to continuously switch focus as a participant observer: "Participant observation serves as shorthand for a continuous tracking between the 'inside' and 'outside' of events: on the one hand grasping the sense of specific occurrences and gestures empathetically, on the other stepping back to situate these meanings in wider contexts" (p. 34).
It was important for me to get a feel for what it would be like to teach mathematics through creative dance. As a participant, I had a digital camera to record parts of the creative dance component of my teaching sessions. The photographed 36
material acted as another set of eyes that I could rely on to validate my findings.
Participants
The participants were the students and the teacher in the classroom, as well as myself. As mentioned earlier, I was the guest teacher, observer, and researcher.
To select the participants for the study, I approached the principal of the school because I knew of her interest in arts integration and in student academic success. The principal assisted me in finding a host teacher. At a staff function, she summarized my intentions and mentioned that I was looking for a primary or junior classroom with which to conduct my case study. Teachers from a variety of grade levels showed interest in having their class participate. In the end, I decided to work with the Kindergarten teacher of the school because I wanted to have this study done at the most primary level possible in the public educational system. It is my hope that this work can eventually be extended beyond Kindergarten and into the primary, junior, and intermediate grades.
The host teacher had been a Kindergarten teacher for fifteen years and showed interest in integrating the arts with mathematics. The host teacher agreed to a pre- and post-interview with me, and she agreed to attend the math and creative dance lessons that
I would instruct in her class.
The students in the host teacher's class were potential participants. There were twenty students in total, from both junior and senior kindergarten, their ages ranging from four to six years. There were ten boys in the group and ten girls. Eight students were in senior kindergarten and twelve students were in junior kindergarten. The students needed to have parent consent to participate in the pre- and post-audiotaped focus group 37
interviews and to be photographed while they were participating in the lessons; in addition, I needed permission to collect and use images and words that they included in their journal entry after each lesson.
To assist in the process of selecting participants to be involved in the focus group interviews, I asked parents and the teacher to let me know if there.were students who they felt would not work effectively in a focus group interview setting. Vaughn, Schumm, and
Sinagub (1966) note that, "It is beneficial to ask parents or guardians, the teacher, and other relevant individuals about the child's ability to participate with others in a focus group format" (Vaughn, Schumm and Sinagub, 1966, p. 133).
Since the lessons were curriculum based, all students were to participate, but not all of the students were involved in the data collection. Only nineteen of the students brought back permission forms on time, and only fourteen students had permission to participate in all aspects of my research. Three boys and one girl in senior kindergarten were selected to participate in the pre- and post-audiotaped interview. These interviews were used to help me understand the students' background knowledge of mathematics and creative dance, as well as to follow up and see if their understanding changed after the lessons.
Being both a participant and observer, I was involved in a variety of ways. The full-time tasks included being the guest teacher in the class and the researcher at the same time. I also had other part-time job descriptions that included music maker and photographer. Wearing many hats was challenging. A written copy of the thesis will be given to the school board and a summary of the findings will be shared with the principal, 38
teacher, and parents.
Setting
The school in which the research was conducted is located in Toronto, Ontario in an upper/middle class neighborhood. The study setting was in a public school classroom.
It was my intention to conduct the lessons in the classroom because it is important that this work be relevant to teachers. Being a teacher, I know how hard it can be to schedule times to use more open spaces such as the library, gym, or school stage.
Although it was not in my original plan to use a separate room from the students' homeroom classroom, the school had an old kindergarten classroom used as a primary resource/breakout room. Through the school's email, I booked this room for eight days— one of which I used for the purpose of setting up the room. The room had a variety of primary furniture and materials. I spent the day prior to the first lesson preparing the space so that the children could move safely in the room—moving furniture and classroom materials to the edges of the room, wiping down the tables and the chairs, and having custodial help to sweep and mop the space.
Scheduling
My plan was to teach three math and creative dance lessons per week for two weeks. I wanted to be in contact with the host teacher in whose classroom I was to be a guest so that the planning could occur. Ideally, I wanted the case study to begin in
October 2008; however, due to delays in the proposal being accepted by the Board of
Education, the dates and times for data collection were finalized as follows:
Tuesday, January 6, 2009 from 1:15-1:55 PM 39
Thursday, January 8, 2009 from 1:15-1:55 PM
Friday, January 9, 2009 from 1:15-1:55 PM
Tuesday, January 13, 2009 from 1:15-1:55 PM
Thursday, January 15, 2009 from 1:15-1:55 PM
Friday, January 16, 2009 from 1:15-1:55 PM
Tuesday, March 10, 2009 from 1:15-1:55 PM (Follow-up Lesson)
Scheduling was done by meeting with the host teacher and finding dates and times that worked for both of us. During this meeting, the host teacher and I agreed upon the mathematics curriculum to be covered.
Materials
I created or bought the materials to be used as props for the lessons. Lesson plans, designed to be complete and ready to follow, were printed out so that I had them in hard copy in case I needed to refer to them. I had chart paper, markers, and tape on hand to write down the students' ideas. The charts were posted on the walls for student and personal reference. A hand drum was used to make music while the students were moving. I also had a notebook to record my observations and both a dance and mathematics checklist with students' names on it for assessment purposes.
Being the researcher for this project, I needed to mark out the space where children who had permission to be photographed could move freely. This area was taped off using yellow electrical tape. An area taped off with red electrical tape was used for the students who did not have permission to be photographed. It was important for me to have these distinct areas defined to ensure I was following specified permissions. A 40
digital camera was used to take the pictures of the students learning and creating; these would be used as reference and for analysis purposes. I referred to the lesson plans for the questions to ask and for the content being delivered in the class.
Prior to the lessons, I prepared the props that were to be used during the various parts of the lessons. I bought large amounts of sewing elastic to be used for the lessons in which two-dimensional geometry was being explored. Two large elastics were tied together to create one large round elastic. This elastic was used for the whole-class exploration of composing and decomposing two-dimensional shapes. Six single sewing elastics were tied together to create six medium-sized round elastics. These elastics were used by small groups of students exploring two-dimensional shapes.
Two-dimensional plastic shapes, including a circle, a rectangle, a square, and a triangle, were used to show the students what these two-dimensional shapes looked like.
Students had the opportunity to touch these shapes. These manipulatives, along with my drawings of them on chart paper, were used during the lessons. Plastic three-dimensional solids, including a sphere, a cube, a rectangular prism, and a square based pyramid, were used to show the students what the solids looked like. They were passed around for the students to feel. These three dimensional solids were also sketched on chart paper during the lessons.
Ten bamboo sticks, about five feet long, were bought. Four were left long, and the other six were cut in half. All the ends of the bamboo sticks were sanded down with sand paper to remove any sharp edges. These props were placed in a large plastic bucket and were used for groups of students to build large three-dimensional solids. Students were 41
taught about safe prop handling prior to use.
Student materials, which included five table and chair sets, needed to be prepared
in advance, prior to each lesson. The tables lined the perimeter of the room and were used
for the students to write and draw their reflections. Before each lesson, a sharpened pencil
and piece of blank paper was placed at each table for the students' reflection time. A
basket of crayons was at the center of each table. After each lesson, students were
encouraged to write their names at the top of their paper and draw and write about
something that they learned during the lesson. Students were given approximately 10 minutes to complete these tasks.
Curriculum
All lessons were based on curriculum strands mentioned in the literature review. I
implemented lessons that have dance and mathematics expectations integrated into them
as stated in the Ontario Curriculum. Before preparing the lessons, the curriculum
expectations needed to be decided upon. It was important for me to have consensus with the host teacher as to what curriculum expectations were to be covered. All the
curriculum expectations came from The Kindergarten Program, (revised) 2006 under the mathematics and the dance sections. The expectations that were addressed are as follows:
Mathematics Curriculum Expectations
Overall Expectations:
• Describe, sort, classify, and compare two-dimensional shapes and three-
dimensional figures, and describe the location and movement of objects
through investigation. 42
Specific Expectations:
• Identify and describe, using common geometric terms, two-dimensional
shapes (e.g., triangle) and three-dimensional figures (e.g., cone) through
investigations with concrete materials.
• Explore, sort, and compare traditional and non-traditional two-dimensional
shapes and three-dimensional figures.
• Compose pictures and build designs, shapes, and patterns in two-
dimensional shapes, and decompose two-dimensional shapes into smaller
shapes, using various tools or strategies.
• Build three-dimensional structures using a variety of materials, and begin
to recognize the three-dimensional figures that the structure contains.
• Investigate the relationship between two-dimensional shapes and three-
dimensional figures in objects that they have made.
Drama and Dance Curriculum Expectations
Overall Expectations:
• Demonstrate an awareness of themselves as artists through engaging in
activities in visual arts, music, drama, and dance.
• Use problem-solving strategies when experimenting with the skills,
materials, processes, and techniques used in the arts both individually and
with others.
• Communicate their ideas through various art forms.
Specific Expectations: 43
• Use problem-solving skills and their imagination to create drama and
dance.
• Demonstrate an awareness of personal interests and a sense of
accomplishment in drama and dance.
• Express their responses to drama and dance by moving, by making
connections to their own experiences, or by talking about drama and
dance.
Data Collection
To ensure internal validity of my data, I followed a section of Merriam's model that identifies six basic strategies. The strategies are summarized and were addressed as follows:
• Use of triangulation -1 used multiple means of collecting data.
• Use of member checks -1 took the data back to the participants to clarify
accuracy.
• Use of long-term observation -1 repeatedly observed the students over the course
of seven lessons.
• Use of peer examination - my advisors commented on information found as it
emerged. I was open to constructive criticism and insights.
• Use of participatory or collaborative modes of research -1 involved participants in
most phases of the research.
• Use of ongoing clarification of assumptions and worldviews at the outset of the
study -1 have addressed these issues. 44
Reliability and validity are intricately connected. Reliability of my data was ensured because of the steps I took to ensure validity. Guba and Lincoln (1981) note,
"Since it is impossible to have internal validity without reliability, a demonstration of internal validity amounts to a simultaneous demonstration of internal reliability" (Guba and Lincoln, 1981, p. 120).
Photographs
For analysis purposes, I focused on certain students for the photographed documentation. These students were chosen with attention to parent and student written consent.
Student Journals
After the movement component of each lesson, students were invited to share their impressions of the class and what they learned on pieces of paper. I used pieces of paper instead of a journal because the host teacher commented that the students were accustomed to working with pieces of paper during their Writing Workshop time.
Students were encouraged to write their name on the paper as well as to draw and write their understanding of the lesson on the paper. I collected these at the end of each class.
The student reflections, used for analysis purposes, were chosen with attention to parent and student consent.
Data Analysis
To analyze student journals, focus group transcripts, interview transcripts, photographs, and my observations, I used the concept of emergent themes. After I collected the data from the focus group interviews and the interviews with the host 45
teacher, I transcribed this information. I also typed up my journal notes of each lesson.
Next, I printed off the transcripts and notes and photocopied them twice in case one section could fit into more than one theme. Using different colored highlighters, in the column next to the text, I highlighted different sections that seemed to have similar themes. For example, gender was blue, and on any part beside the text that I thought contained information pertaining to gender issues, I put a blue line down the side of this information. After all the data was coded, I cut up the different sections and sorted them into piles by their colors/themes. Each theme was put onto a different colored sheet of
Bristol board. During this stage of analysis, I also included photographs of lessons and selections from student journals and placed them on the appropriate Bristol boards. From this arrangement of the data, I tried to see if there were sub-themes. Many hours were spent trying to find appropriate and relevant themes. The emergent themes that were presented were as follows: a special case, creativity, diverse representations, and emerging competence.
Ethics
Ethical protocols for York University and the School Board in which I conducted my study were followed. All participants were assured of their confidentiality and their rights to withdraw from the study at any given time. 46
Chapter Four: The Study
In this chapter, I describe the study, day by day, to give the reader an idea of what
the sessions were like. My quotes and other data sources are used to paint a clear picture
of the study.
Lesson One
The students came into the designated room and sat in a circle on the carpeted
area of the room. I began by welcoming the students and introducing the study and myself. I moved on to asking each of the students to say their names, one by one around the circle. Although it took me a while to get to know children's names, I tried to put them to memory quickly, as I needed to be able to address each individual student
especially for permission purposes. In hindsight, I should have had the students wear nametags.
I explained the rules for the class (respect yourself and others), as well as the need to stay within the dance space during the creative dance time. I noted where the
observation area was and why it might be used (for time-outs, students needing a break,
students not allowed to participate due to lack of parental consent).
Next, I beat the drum twice very quickly and let the students know that when they heard this sound, it meant to stop, look, and listen. The students were also told that any
other beats that I made with the drum would be the music to which they would
dance/move. Neutral position, whereby the feet and legs are parallel and in line with the hips, the arms are to the side of the body, and the shoulders are back, was demonstrated. I explained that students would be asked to take this position often during the time that I 47
would be with them. I also explained what we would be working on in dance and math.
We next brainstormed about circles, rectangles, and triangles. I asked students to tell me if the shape reminded them of anything or if they could think of any words that described the shape, and then I recorded their responses on the chart paper. As an example of brainstormed words for the. square, students used words such as: four corners, four sides, Wall-E's garbage block, a square jelly, a mouth, and a person. What I noted was that words that I thought would be used, for example, straight or flat, were not.
Words for the circle included: round, head, no sides, no corners, and doesn't move. Due to taking more time than I had anticipated for the brainstorming component, the creative dance section and the reflection time of this first lesson were rushed. I noted to myself that next class I needed to cut back on the introduction of the lesson to leave more time for the body of the lesson.
Students were then led through a five minute warm-up in the dance space. The warm-up included stretching, rotating, and creatively moving the major body parts, starting from the toes and moving up the body, ending at the head. This warm-up was repeated in all of the lessons to follow. The warm-down for this lesson (and the ones to follow) was the reverse of the warm-up.
I then pointed to the first chart, which was the square, and picked out descriptive words that were listed on the chart paper. Rather than the children pretending to be a square, I encouraged students to physically explore the qualities of a square. The concept of moving like the descriptors was a bit complex for the students. Moving akin to what a shape might suggest to them invoked more movement; as well, my guidance appeared to 48
be more understandable when I said move 'like the shape,' for example, move 'like a square.' We moved through all of the shapes that were brainstormed in this way. Anna in
Lesson 1 commented: "I liked moving like a circle because of how it made my arms feel." Gabe in Lesson 1 said: "It was easier to be a triangle than to be a square. The
iigure 2. Lesxui One - slmlent"» e\p!orins> Else concept of Hat shapes.
square hurt my legs." Daniel in Lesson 1 spoke about being a rectangle: "I really had to
stretch my belly with the stretched-out square."
When the students were asked to move like a square, many students copied one
another and lay prone on the floor with their bellies down. As seen in Figure 2, students
explored the concept of flat shapes, in particular the square. As seen in the photograph, 49
Jonathan knelt down in the fetal position. Notice that Clara lay on her back, creating the square with the negative space between her legs. For the flat rectangle exploration, many students got down on their bellies and stretched out their arms and fingers.
For the triangle exploration, Daniel made a triangle with his fingers (see Figure
3). Many students related to the triangle being a rocket ship top and moved fully through the space with this imagery, pointing their arms and fingers. Sounds of pretend rockets zooming about were also heard.
I'h'iisv3. Lesson One Daniel makes a triangle with his firtoets.
I noted that when we started to work with the ideas from the brainstorming
session, it took a lot of creativity on my part to encourage movement exploration from
these brainstormed words. It was an improvisational process for me and was challenging. 50
After this process was complete, I asked: "When you were dancing, were there any shapes that had kind of the same movement or feeling in your body? Which ones?
Why might this be?" Students said that the square and the rectangle were alike because they both had four corners. I then asked them to explore movements of shape descriptors with similar qualities. Next, they explored movements of shape descriptors that created different movement qualities. Students moved through the square and rectangle as well as the triangle and the circle, to feel the similarities and differences among them.
Students were then led in a five minute warm-down. Following this, I invited the students to quietly leave the dance space and encouraged them to draw and use the written word to tell me about what they learned in class. (Paper, pencil, and crayons had been set at each grouping of tables prior to the beginning of the lesson.)
Lesson Two
I began this lesson by explaining what the overall expectations were for the class so students were aware of what they would be learning. I showed large pictures of the cube, the cone, the sphere, and the rectangular prism. I encouraged the children to describe in words and with imagery each solid, as the corresponding three-dimensional model was passed around the class for the students to touch and explore with their hands.
I explained what constituted a face and an edge; we then counted these features for each of these solids. The only solid the students did not know was the rectangular prism.
Then I had the students vote for the most interesting shape between the cube and the cone, the sphere and the rectangular prism. The students voted that they wanted to explore the cone and the sphere. Modifying the lesson to focus on two of the solids 51
(instead of all four) gave more time for in-depth, physical exploration.
During the brainstorming, students described the cone and the sphere; we did not venture into describing the cube or the rectangular prism. Students described the cone as: a carrot, a nose, a hat, an upside down ice cream cone, and a pyramid in Egypt. The sphere was described as follows: a circle, a wheel, the earth, and a ball. Much like the first lesson after the brainstorming session, I led the students in a five minute warm-up.
Again, similar to the first lesson, I then pointed to the first chart (on the cone), and picked out words that were listed. I did not want the students to pretend to be a cone, but rather I wanted the students to physically explore the qualities of a cone. Students explored the round face of the cone with rounded body movements. I encouraged students to show me different ways of interpreting what round meant; this seemed to help with the copying problem from the first lesson.
Next, we moved into exploring the point of the cone at different levels. During this exploration, I stopped the class to see if there were ideas of body parts the children might use to point. Fingers, arms, knees, and pointed toes were all suggested. 1 recommended the pointing of the elbows. More pointed body part exploration occurred
(see Figure 4), including movement through space; during this exploration, I needed to remind the students to avoid bumping into one another. 52
\
Figure /. Lesson 1 wo - a student explores a point.
Next, the children improvised movement as they explored the sphere. I encouraged dancing with no sharp edges. Ball-like movement, including rolling and bouncing, were seen as well as the rounded shapes of body parts (see Figure 5). I noted that some students somersaulted (on the carpeted area). 53
J S5% 4: H #c^ *f
figure 5. Lesson 'Two - Students explore rounded bod> shapes.
I took particular note of how well Andre, a Senior Kindergarten student who was somewhat developmentally delayed, understood the concepts. During the creative dance exploration of the sphere, he was tightly rolled up in a ball with his hands over his knees, as he rolled around.
Students were then led through a creative dance exploration of the similarities and differences between the cone and the sphere. The class was led in a five minute warm- down and then students were invited to draw and write what they learned during this class; see Figure 6 for an example of a student's reflection on this class. 54
tinure 6. Lesson 1 »o - a student's written and drawn reflection.
At the end of the class, the host teacher commented, "Classroom management was good today." She mentioned that she noticed Andre truly understood the concepts and showed this through his creative dance, both during the sphere and cone exploration. She continued on, saying that he had told her that he doesn't like to come to school. On the way out the door, Andre said, "I love this class!" Two other children echoed "me too!"
Lesson Three
I explained what the overall expectations were for the class. We reviewed the charts used in the first lesson with various shapes, including the square, circle, triangle, and rectangle, and their associated words/images.
Students were led through a five minute warm-up. Then, I presented the large 55
piece of elastic and told students that together we would make very large shapes with the use of the elastic. Although I told students not to touch the elastic until I directed them, many students found the elastic far too intriguing to ignore. We began as an entire class with the creation of the circle and then moved on to the square, the rectangle, and then ended with the triangle.
With the circle exploration, 1 invited the students to take hold with both hands and put the elastic band behind their backs and then stand with their arms by their sides.
Many students put the elastic around their backs but some put it around the back of their neck; I reminded them not to do this. The host teacher helped by intervening to deal with this issue; it was good to have the second set of hands. I asked: "What will you all need to do to create a very large circle?" Students responded by taking a step or two backwards.
What I found was that not all students took steps of equal distance; also, some students took more than one step.
I then inquired: "What will you all need to do to create a smaller circle as a group?" I needed to help adjust the students in the class since the circle looked a bit more like an oval. When I asked about a small circle, a child said "step in," so we all stepped in, and again the same stepping problem occurred. I asked how we could make an even smaller circle and the answer was the same as above. I inquired: "How do you know that you have made a circle? What are some things you know about circles?" Students replied with comments such as "circles are round" and that "they do not have corners."
Students put the elastic down and I instructed them to sit around it in the dance space. Most followed my instructions but some were still fascinated with the elastic. We 56
then moved onto the square exploration and I asked: "How might we organize ourselves so that we can create the special rectangle called the square?" A student replied, "We need four things to be corners." I replied, "I see students sitting around the dance space.
Do you think we could use students as the corners?" Students agreed we could. Upon reflection, I would have changed the way I phrased this question so that students could have had the opportunity to come up with the answer to what they could use as corners.
Four students, all of whom had volunteered, were selected to be the corners of the class square. With the elastic around their backs, some of the students (acting as corners), needed adjustment. It proved difficult for the class to form a perfect square. All other children were invited to be the sides and they held onto the elastic with both hands. The elastic was a good guide for the students, acting as the side of the square. I guided students through this process by showing them the picture of a square. When students were in position again, I asked: "How do you know that the class has made a square?
What are some things that you know about squares?" Various students responded that there were four corners and four sides. I stressed that there were also four equal sides.
Students disassembled their square shape and returned to the edge of the dance space.
We moved to the rectangle exploration. I questioned: "What will we need to do now to create a different kind of rectangle?" I guided the students through this process by showing the picture of the rectangle. Students knew that four students were to be corners and that the others were to be the sides. Four students were selected and when students were in position, I asked: "How do we know that the class has made a rectangle? What are some things that you know about rectangles?" One student answered: "Because there 57
are two long sides and two short sides." The students created the rectangle quickly and accurately, and then the students disassembled their rectangle. See Figures 7 and 8 for several representations of creating a class rectangle with the use of elastics, as drawn by two of the participating students.
"*L.
/
Figure 7, Lesson Three - a student's reflection of making a large rectangle with the use of an elastic.
Each member of the class is shown taking his/her place in the group effort, holding on to the elastic 58
Figure 8, Lesson "I hrcc a second student's representation of the class making a large rectangle with
the use of an elastic.
Next, I asked: "What will the class need to do to create a triangle?" Students' responses indicated that they knew that they needed to have three students to be corners. I guided them through this process, including showing them the picture of a triangle. I chose a particularly small Junior Kindergarten student to be one of the corners and he didn't quite know what to do. Other students were busy giving instructions to him and he appeared confused. I needed to guide the students who were directing him. I asked a student who was a corner to give the small Junior Kindergarten student direction and she 59
instructed him to move toward a corner of the dance space. She continued to verbally
guide him as he walked into place. Again, students became the sides. Due to my high
involvement in the teaching process, I was unable to take any photographs of this
exploration.
After the creation of the triangle was completed, I asked that the students sit down
in the dance space. I asked them to take the elastic from behind them and place it in front
of them. I gathered up the elastic piece of material. Students were then divided randomly
into groups of four. As there was one student absent this day, the host teacher offered to join the uneven grouping. Each group was given a medium sized elastic. I invited them to
go through the process again of making the shapes we explored as an entire class, except
this time with their small groups. In the small groups, we started with the square; I
needed to go around to the small groups to make sure students understood how to create
it. See Figure 9 for a student's reflection on creating a square with his group and with the
use of an elastic. For the most part, students understood the idea, yet some were confused.
I guided groups into 'approximate' squares. All of the groups were successful; the
rectangle exploration and the rectangle investigation appeared easy for them to
accomplish. 60
\
Figure 9. Lesson Three - a students representation of working in groups of four to create a square
using a medium sized elastic.
Figure ill. f ,es-son 1 hrce - a student's representation of making shapes sn small groups.
Due to time constraints, I gave an option to the groups. They were invited to make a circle or a triangle. Most of the groups chose to create the triangle. While one group chose to have the fourth child stand out, I encouraged this group to find a way to incorporate the fourth student. See Figure 10 for a student's reflection on making the rectangle and the triangle with her group.
Another group chose to have one student stand out; again, while I posed the same 61
question as above, the girl standing out didn't want to be any other part of the triangle. As seen in Figure 11, a student standing in the corner of the triangle generously offered her corner to this girl and chose to stand in the middle of the triangle.
•<* T
Figure 11. Lesion Three - u student chouses to stand in tic middle of her group's triangk*.
One group chose to create a circle. They were getting aggravated. Daniel tried to twist the elastic, "This is too hard!" he said. I asked: "Why do you think it is too hard?"
He responded, "We don't have enough people!" I stopped the class and asked, "Why might it be hard to create a circle with four people?" Clara answered, "Because they don't have enough people and it looks like it has cracks in it." After the exploration, students drew very expressive pictures of the lesson, even though they did not have much time.
Lesson Four
Students were led through a five minute warm-up of the major body parts, starting from the toes and moving up though the body, ending at the head. Students were then 62
separated into pairs, and partnered-up, based on permissions to have photos taken. It was
good to have pre-partnering. I challenged the students to create the shapes that I called
out with both body parts and their whole bodies. I called out and pointed to pictures of the circle and the square. See Figure 12 for an example of two students creating a square with the use of their legs and feet. Numerous times during the class, I drew students'
attention to examples in the class.
We had only explored the circle and the square shapes before students needed reminding that this was a group shape, not an individual shape. Some students were
confused and some students did not make good partners. At one point, I needed to stop the class to give some examples of what body parts they could use for this study.
! i 1 • ••::..-• •i^>^ «j*'^s \ y.- .-^r d
i.- ---•
figure 12. Lcisym Four - iv-n ^tudeoti create A square l>> skiing on the iioor, extending iiicir iejjs and
connectiuu tlicir k-et together. 63
I then asked the students to work individually and to generate the shapes with their bodies (all or part) that I would call out. I called out and pointed to pictures of the circle, square, rectangle, and triangle. Again, students needed to be guided individually as to what body parts they could use. Students created very recognizable shapes.
Students were then told to create a creative dance. The creative dance needed to have a good, clear beginning, middle, and end. I let them know that one way this could be done would be by starting in a very still position to show that the dance was going to begin. I also explained that the middle part needed lots of movement and when they finished, they needed to be very still again to signal the ending of the creative dance. We rehearsed three times and then dance as a performing art was explained.
The students were instructed that each one would present his/her shape dance. The creative dance had to have a beginning, middle, and an ending, as well as having three shapes in it. I encouraged students to think of the three shapes about which they wanted to creatively dance, and then share this with the student sitting next to him or her. Next, I invited students to explore their first movement, then their second movement, and then their third movement. I encouraged them to move these shapes and use levels (high, medium, and low).
I then said, "It is now time to put your creative dance together! When I double beat the drum, this means to begin in neutral position." I began to play the drum and did not stop until I saw that everyone had ended in the neutral position. Next I said, "When I double beat again, it means that everyone is done and students may sit down." 64
Subsequently, I invited half of the students to sit in the observation area and the other half to spread out around the dance space. I encouraged observation from the audience to view with no judgment and asked the children to show their appreciation with applause at the end. I let the students present their work to one another and then to have the observing and dancing groups switch. The performers needed to be guided through the changing of the shapes. I have to admit, I thought that they would have understood from the practice in this lesson how to move from one shape to the next on their own, but this was not the case; I needed to guide them through this part of the presentation. I noted that the audience was very attentive. Both groups performed and students clapped after each presentation.
When both groups finished, I added a component to the lesson. We sat and I asked, "Did anyone see anybody do something awesome or amazing?" A student acknowledged her peer for her shapes. This student shared her sequence with the class again. Another girl acknowledged a girl in the class for her circle shape. The student being acknowledged showed this shape as well. A little boy was acknowledged for his levels. He again presented his low rectangle, medium level circle, and his high triangle.
Daniel said that he thought he was doing something amazing and I asked again, "Did anyone see anybody do something awesome or amazing?" Daniel then responded, "It was all pretty amazing!" 65
.^h oh ^
Figure S3. Lesson l-'otir - "Make Your Own Dance." A student's written reflection and chosen shapes to dance dining the shape dance.
£•:•—•.-„. r\
/ \ \~\ \ *- » S -v
Hs>tirv 14. Lesson i-our - a student's three chos.cn shapes for the shape dance 66
A,
Figure IS. Lesson Four - a student draws, herself being the shapes she has chosen for the shape dance.
After this class, the host teacher came right up to me and expressed how pleased she was with the way that the class had gone. It was really supportive to have another teacher in the room to help with complimenting, correcting, and guiding the students.
This was especially true when I had many hats to wear, and roles to play. Unfortunately, no pictures were taken during this time because I needed to play the drum. See Figure 13,
14, and 15 for student reflections on this class and the shapes they chose to incorporate into their dances.
After both groups shared their work and responses, we began the warm-down.
Then students were invited to quietly leave the dance space and draw and write what they learned in mathematics.
Lesson Five
Going into this lesson, I was wondering if it would be too hard or too much for the students. I began the class thinking that I might need to modify on the spot. One of the modifications I had on the backburner was that I was not going to introduce too many 67
solids, because we only had time to explore a few in each of the previous sessions.
Consequently, I decided again to give the students a choice about what solid they wanted to investigate.
The choice was between the cube, the rectangular prism, the cylinder, and the pyramid which we had not yet had time to investigate. The students selected the cube. We brainstormed ideas about the cube; I said that if we had the time we would construct the others. We didn't brainstorm ideas or words about the other solids; in the end, this proved to be a mistake in my opinion. During the brainstorming section we counted the edges and described the cube. I invited the students to describe the cube and tell me what it suggested to them. Some of the descriptive words included: a box, a rectangular prism, square faces, and a diamond. We counted the edges and I recorded that there were twelve.
We then had a warm-up and the rules governing the use of the bamboo sticks were made clear. I then pointed to the picture of the cube and asked, "If each student is allowed to hold only one bamboo stick, how many students will it take to create a cube?
How do you know this?" I asked! Clara answered, "It would take four students." I asked if she was sure. Gabe spoke up and said, "Twelve!" "How do you know this?" I inquired.
"Because we counted how many edges are on a cube and there are 12." I asked, "Who would like to try to be an edge?" Twelve students were chosen and they began creatively dancing through the space with intention. I put the three-dimensional model on the floor in the middle for them to use as a visual reminder. This was a change from the original lesson.
It was interesting to observe the twelve students clustered into smaller groups. 68
They began making squares amongst themselves in groups of four, including one group making a square in the air. After a minute or so Gabe said, "I think that we should start on the floor." A few students were working alone. I needed to remind them and the whole
group that they were not working by themselves or in small groups but as a whole and
entire group. After I mentioned this, the cube was assembled almost instantly (see Figure
Figure Hi. Lesion I He - students work together wilh tSic use of bamboo Miclo> lo create a cube.
The students followed the little boy's comment about building from the ground up.
Altogether, the construction of the large cube took about two minutes. I was in awe! Both
the host teacher and I were impressed. I noted how invested the students were in the
activity.
During this lesson, no drumbeats were occurring. The students sitting around
watching the creation process were invited to explore the solid once it was built. Some of 69
them went into the solid, others touched the edges, and some simply walked around the cube.
The cube was disassembled and I noted how much time we had. At the beginning of the lesson, I had intentionally cut back to allow for more exploration time during the movement component of the lesson. I accidentally forgot to ask how the students knew that this was a cube.
We then moved on to the exploration of the rectangular prism. I held the model and we counted the edges. Students were able to tell me that it would take 12 students to make the prism. We did not brainstorm this solid as we had done with the cube. The creation began. It appeared to be harder for the students. I wondered if this was because there were more Junior Kindergarten students involved in the creation of this solid, or, was it because they were again trying to build the solid from the top and work down?
The first rectangle was made in the air. It appeared harder for the students to build from the top down as opposed to from the bottom up. An interesting thing began to happen as I watched the creation unfold. Students who were sitting but began to direct students moving with bamboo sticks in the center of the dance space. Eventually, I needed to stop everyone and direct the conversation. Through this directed conversation, the solid was made. The creation of this solid took longer than I had anticipated. Once more, the model of the solid was in the center of the space for the students to use as a reference.
Due to time and lack of activity for the on-looking students, I noted that I was losing some of them. This was problematic. They needed something to do rather than 70
simply watch. Could they play maracas, draw, build with straws? Perhaps I should have had another center open for them to be more involved. Unfortunately, I don't have a photo of the rectangle creation because many students who were building were not allowed to have their photo taken. I invited the students to allow the model to melt away and for the students to return the bamboo sticks to the proper place. The rectangular prism disassembled smoothly.
We had time for the construction of the pyramid. We counted the edges on the model, yet we did not brainstorm this solid as we did the cube. The students began the building of this solid (see Figure 17). Students started from the ground up without having to be reminded. This really helped. Pictures of the creation of the square-based pyramid were taken (see Figure 18).
Figure I'7. I.c-son Fhe - Student* work togethei to create A square-based p>ramid.
I noted that the students around the perimeter of the dance space were a bit lost and appeared tired. Students were not as interested in watching other students move while they were sitting. Perhaps I should have given the observers some pre-assigned focus for 71
giving feedback to the movers. During this class, since the students were working from models, there was no discussion of what characteristics they needed to consider when creating a cube, a rectangular prism, or a pyramid. In hindsight, such a discussion might best be connected to having each student hold and examine a 3D model.
Figure IS. Le«son i'he - a square based p\rnmid h formed!
Students were led in a warm-down and were encouraged to write or draw reflections on the class. The writing and drawing component of the lesson had all of the students focused. Andre, the little boy mentioned earlier, had always drawn circles up to this lesson; this time he drew what looked more like a rectangle. Also, one Junior
Kindergarten boy who, to date, had drawn scribbles, drew his idea of a triangle.
Lesson Six
This lesson began with students joining me in a sitting circle in the dance space. I 72
reviewed the charts posted on the wall of the classroom with various solids and words on them. Students were then led through a five minute warm-up.
I told the class, "We will be creating three of the solids that we created last class including the cube, the rectangular prism, and the pyramid." I reviewed the solids taught.
This took a while. I asked, "How many students will it take to create a cube the way we did last class?" Some students still answered it would take four students. The students were able to create the cube, although not as perfectly or as quickly as the last lesson.
When I asked the class, "Is this a cube?" One little boy answered, "It is because it looks the same as the block (the model that was shown earlier). The rectangular prism evolved in a similar way to how the cube was constructed in this session.
After thinking about the last lesson and the problem of the on-looking students, I was going to have these students use instruments to accompany the creative dancers with musical accompaniment; however, I changed my mind at the last minute and suggested hand clapping. I noted that the instrumental/musical accompaniment needed to be ironed out and asked myself how I might deal with musical accompaniment in the future.
When it was time to build the pyramid, the students were so caught up with the point at the top that some didn't see the total solid; everyone wanted to be an edge that formed a vertex at the top. There were too many edges merging at the top and there was no base. I purposely tried not to intervene but decided to freeze the students due to the time constraints. Finally I asked, "Is this a pyramid?" It was obviously not a pyramid.
The students unanimously said NO due to it not looking like the solid/model that I had provided for them. The students asked if I could help them. I helped direct the students 73
into their respective places. A square based pyramid evolved.
When this was completed, I led the class in a solids dance. Like the shape dance, all students were encouraged to begin and end the creative dance in neutral position. I started with a double drumbeat and verbally guided them through a creative dance exploration. Students were asked to imagine being inside a cube into which only they could fit. I encouraged the students to feel the faces and vertices. I then guided them to imagining the cube expanding and growing and encouraged the students to explore this expanded space. I moved from the cube onto the pyramid, the cylinder, and the sphere in both close and far proximity to the student dancing within it. Then, the creative dance ended and I invited students to find neutral and stillness, as I concluded with a double drumbeat.
I asked students to return to the circle and share their experience of creatively dancing inside the imaginary solids. From the post-interview with some of the students, this conversation was recorded:
C. I liked when we were feeling how we would be if we were inside the
shapes.
M. You liked that part? Why did you like that?
C. Because . . . (pause)
M. When you were imagining that you were inside a solid, why did you like
that?
C. Because I was trying to get into the shape (laughing).
M. You were trying to get into it. Was it hard or easy? 74
C. Hard.
M. Did it help you understand the shape better?
C. Yeah.
J. And getting inside it was hard, I had to find, I had to find a door and the
door was the same color as the shape.
M. Yeah?
G. Well, when I got inside the can, when the can shrinked, I thought I can't
believe I am going inside that mouse door.
M. So you were imagining you were going in a little door?
G. Yes, because when the can was big I opened a big door, when the door
was small, I opened the little teeny tiny mouse door.
M. So you had to go in really tiny?
In the solids dance, the students were "really into it." Some students needed guidance regarding what to do. I demonstrated some examples. The movement component of the class ended as always with a warm-down and students were invited to write and draw their thoughts of the class.
Students were also guided through a post survey of the classes that they had had so far. The survey had four questions for the students to answer. The students were to color one of five faces to represent their answer to three of the four questions. The faces ranged in emotions from being a very unhappy crying face to a very happy smiling face.
Only one face per answer was requested. During the survey, the students were quite 75
confused about all of the faces on the paper. Some colored more than one in a line while others answered all the questions on one line. This made less clear data for analysis purposes. I wondered if perhaps the survey was too advanced for the students.
Lesson Seven
I invited students to join me in a sitting circle in the dance space and let them know that this would be a follow-up to the work we did back in January. I reviewed the rules of the class and the appropriate use of props. I asked the students to name the various two-dimensional shapes and three-dimensional figures they knew. Students were able to recall the square, the rectangle, the circle, and the triangle. I did the same for three-dimensional figures and students recalled the sphere, the cube, the cone, the rectangular prism, and the pyramid. I asked students to describe them to me and I drew them. I asked, "What three things does every creative dance need to have?" Students needed to be reminded of these requirements.
Students were led through a five minute warm-up. Then I said, "We will be creating three solids that we have created in the past: the cube, the rectangular prism, and the pyramid." All of the solids were created with relative ease and students were able to answer how many students were needed to create each of the solids. All the solids were created from the ground up and once the solids were built, I asked the class: "Thumbs up if you agree, thumbs down if you disagree." All students gave thumbs up. See Figure 19 and 20 for students' reflections on this class. 76
Figure 19. i ,esson Seven - student draws the square, the circle, the triangle, and the rectangle. Student attempts to draw the three-dimensional version of the cube and the rectangular prism. 1 he number
twelve h written to indicate how nianj edges the cube and rectangular prism have. 77
Figure 20. Lessoii Seven - a studvnt's reflection on the ciitNS.
Next, I presented the large piece of elastic and told the students that we were going to be making very large shapes together. We started with the circle, then moved onto the square, the rectangle, and ended with the triangle. I put the elastic circle in the center of the group and invited the students to take hold with both hands and put the elastic band behind their backs. I then invited students to stand and place their arms at their sides. I asked students, "What will you all need to do to create a very large circle?"
Showing me, students took steps backwards. I reminded each student to take two medium sized steps.
I then asked, "What will you all need to do to create a smaller circle as a group?" I invited students to show me. Students took steps in. When the students were in position, I asked, "How do you know that you have made a circle? What are some things you know 78
about circles?" Students were able to answer that circles are round. Students were able to move through the square, rectangle, and triangle exploration, during this class. Since the students were working from models, there was no discussion of the characteristics they needed to consider when creating a cube, a rectangular prism, or a pyramid. In hindsight, such a discussion might best be connected to having each student hold and examine a 3D model.
Figure 21. Lesson Scsen - two bo\s ttj to create a ciicle using a varict) of body parts. 79
v.: ^*- -
Figure 22. Lesson SCUMI - two bo%s use their arms and hands to trv to create a triangle.
: T.~W-W™'GT~'
t*l§
•^Vjj.-i.^v '^^^
-• v.
:-^«\*f?,c ? ^
figure23. Lesson SeM'H - two bo}« cieate a triangle with their lc«s.
Next, I paired students together and invited them to create a shape dance together.
Students were asked to find a beginning position. I then ask them to show me the 80
following: a square, rectangle, circle, and triangle. I called each of these shapes out and the students were encouraged to use any part of their body or their whole body to create the shapes, then to find an ending (see Figures 21, 22, and 23 for examples of shapes created in pairs). After a few practices, the students were divided into two groups; half the class was to start as observers and the other half were presenters. Both groups took turns in each of the roles.
After the presentation, I invited students to share their observations and comments. I asked, "How are rectangles and squares the same? How are they different?
A student responded, "A rectangle and a square have four sides." Another student responded, "A rectangle is a stretched-out square." What two shapes are the most different to you? Why?" Many students felt that the circle was the most different from the other shapes because it is round.
Students were then led in a warm-down and were asked to draw and write their reflections. Because they had only a little time to do this, they were rushed with their responses. 81
Chapter Five: Emergent Themes
The emergent themes that surfaced within my data analysis and that will be
discussed in this chapter are: a special case, creativity, diverse representations, and
emerging competence. The following section will provide my definitions of the emergent
themes, examples of these themes from my case study, and reasons why I chose to
categorize each example as I have.
A Special Case
In the pre-interview, the host teacher mentioned that she had 20 students in her
class; one boy was an English Second Language student who was just learning English
and another boy in Senior Kindergarten was quite developmentally delayed. I will call this developmentally delayed student Andre.
After the first class, the host teacher mentioned to me that Andre had been saying he didn't like to come to school. What touched me was, on the way out the door after the
second class, Andre said with exuberance, "I love this class!" A few students echoed him.
I felt so happy. During the second lesson, I took note of Andre and how well he
understood the concepts. During the exploration of the sphere, he was tightly rolled up in
a fetal position with his hands over his knees and he was rolling around.
Andre also clearly demonstrated curved movement while physically exploring the
circle. During the post-interview, held after the study, the host teacher commented:
Ms T. I saw Andre, who tends to be very passive in class and often uninvolved
because he doesn't really understand what's happening, seemed very
comfortable with the movement and was able to move in very creative and 82
appropriate ways and really follow the directions well. It seemed that he
was feeling pretty confident about that. That was neat.
Andre was able to physically communicate his understanding of the mathematics being taught. During reflection time, Andre would always draw one large blue circle that would be colored in. On the seventh day of lessons, he drew what looked to me to be more of a rectangle than a circle. After this class, the host teacher mentioned that she had noticed this as well. See Figures 24-27 for the subtle development in Andre's in-class reflections.
figure 25. VndreS representation of a circle: (a) 1 esson Three; (b) I.csson Four. 83
lh 4.V..
i -.* ft iri v»"« (a;! (b)
Figure 26. Andre's representation of a circle: (a) Lesson Fhe; (b) Lesson Six.
# & --'\-' 'C-^.'-h -^ "^-'' " V
Fieurc 27. Andre's representation of a circle: Lesson Se\en.
Teachers who used the Math in Motion curriculum activities in the classroom
reported that the students expressed a joy of learning that was not present when they were
taught in more traditional ways (Goodway, Rudisill, Hamilton & Hart, 1999). The
teachers also reported that the children seemed to engage in the activities at a deeper
conceptual level. In a similar way, I found that Andre expressed joy for learning as well
as a deeper understanding of concepts taught. Also, Barbara Arrowsmith Young states,
"When weak links in the chain are strengthened, people gain access to skills whose 84
development was formerly blocked" (Doidge, 2007, p. 41). She claims that this experience can be tremendously liberating for the student. I labeled this 'a special case' because Andre's joy of learning and understanding of the material being taught was exceptional in comparison to what the teacher had mentioned to me. I propose that the mathematics and creative dance experiences did begin to strengthen some of the weak links in Andre's brain, enabling him to feel liberated and successful as a student.
Creativity
I labeled a number of actions and approaches as creative. Creativity to me means original thinking and/or expression. I called a number of actions of the students creative if they produced a meaningful and original interpretation/representation of what was being asked of them.
During the third lesson, when students were working in groups of four, they were given the choice to create a small triangle or a circle with an elastic. Most groups chose to create a triangle. I used the label 'creative' in the instance where a child decided to hold onto the outside portion of the elastic and became a side of the triangle to join the others who were the corners. This was labeled as creative because the child was able to use her imagination and think outside the box as to how she could become part of a triangle. Her idea to become a side, transcended perceived boundaries that it would take only three students and an elastic to create a triangle. See Figure 28 for a student's representation of the creation of a triangle. 85
Figure 28, Lesson I hree-student's drawing showing hov* four students can create a triangle*.
Also, during this lesson another group had a student standing out as the group created a triangle. I encouraged the group to think of different ways for her to be involved. She didn't feel like being any other part of the triangle. A student, who was already a corner, offered her spot to the student standing out. The student who gave up her spot chose to stand in the middle of the triangle (see Figure 11). I labeled this as creative because the student who gave up her spot as a corner was able to express herself more originally than the others. Unlike the others, she had a different perspective and experimented with how she could become part of the group triangle.
In the post-interview with the students, I asked them about the elastic activity and 86
encouraged them to share their thoughts on this. The following conversation was recorded:
M: I noticed that many of you liked that activity; remember when you had the
elastics?
Dan: I liked that too.
M: A lot of you liked that activity. Why did you like that activity?
Chloe: Because the elastics stretched and everybody wanted it, we wouldn't want
to stop touching the elastics.
M: Was it because the elastic was stretchy?
Chloe: U-hum!
Gabe: I love the elastic because I have a whole bunch of elastics and a whole
bunch of toys. And, I love elastics because you can do lots of different things with
them.
Gabe's prior knowledge of elastics enabled him to expand on his understanding of how elastics can be used. This is connected to what Bee, Boyd, and Johnson state, "In a sense, prior knowledge gives you a set of mental 'hooks' on which to hang new information" (p. 43).
In the post-interview, the host teacher commented on the leadership and creativity she noticed in connection with the activities that were part of the lessons.
Ms T. I saw some really interesting problem solving happening, some good
teamwork and it was also interesting to see who took on leadership roles, 87
which we will talk about in a minute, I guess. And, some fascinating
creative movement and it is always interesting to see kids in a different
kind of situation than we sometimes have in the classroom. See which kids
shine in a different kind of way or which kids feel comfortable doing
something that I might not see them doing much in the regular classroom
program. So, there were kids who surprised me, at how creative and
comfortable and enthusiastic they were with the activity.
As noted earlier, the bamboo sticks were needed for the fifth lesson. Each bamboo stick was used as an edge of a solid. During this lesson we counted the edges of the solids to be explored and described them. I asked, "How many students will it take to create a cube?" Chloe answered four. I asked if she was sure and she said she was not. A Senior
Kindergarten boy named Dan said it would take twelve students to create a cube. I asked how he knew this. He replied, "Because we counted how many edges are on a cube and there are twelve." I labeled Chloe's response to my question as creative because her answer was original, and not at all what I was expecting; she was going beyond my imposed rule of the one to one student bamboo/stick ratio. Perhaps groups of four could have worked on building the cube and other solids. As will be mentioned in Chapter 6, smaller group work in building the solids may have helped engage all the students right away. As seen in Figure 15, Chloe is not holding onto any of the bamboo sticks and two children appear to be holding on to two.
As implied in the Ontario Kindergarten Mathematics Curriculum, each mathematical problem has a variety of ways of being approached as well as answered. 88
Perhaps I should have given the opportunity for Chloe to explore her idea when she suggested that it would take four students to create a cube. Thanks to the help of
Professor Walter Whiteley, I suggest that it would have been possible for four students to build a cube. This would mean that each student would be required to manage three sticks. I didn't give this answer a chance to be explored because earlier on in that lesson, I had mentioned that each bamboo stick would represent an edge of the solid and each child would hold one stick. In addition, Chloe's idea that it could take four students to create a cube was creative because, as stated earlier, she went beyond my proposed rule of the one to one student/bamboo stick ratio.
Diverse Representations
I labeled a number of actions and approaches as diverse. A diverse representation to me means expression, ideas or actions that are different or unlike others, or an assortment or range of embodied understanding. This differs slightly from creativity because I defined creativity as actions or approaches of the students if they produced a meaningful and original interpretation/representation of what was being asked of them.
During the first lesson, the students explored two-dimensional geometry. One of the ways students did this was by exploring the concept of flat. Many students lay down on their bellies while exploring the qualities of a flat rectangle. The stretching out of arms, fingers, legs, and toes was witnessed. Figure 2 illustrates the diverse ways students explored the flatness of two dimensions. In this lesson, one student knelt in the fetal position for the exploration of a square and a few students made a square with their legs 89
bent at the knees, with the soles of their feet touching. One child sat and formed the square with her legs while another child lay on his back and made a square. Some children also explored making shapes with their fingers (see Figures 3 and 29). I labeled the above examples as diverse because a medley of responses to the creative dance exploration was presented. Some students used their fingers to make two-dimensional shapes while others used their arms, legs, or whole body.
1.; "38 * **}*?
I ; i !
Figure 29. Lesson One - Mudent explore creating a shape vnlh his fingers.
Also during this lesson, as seen earlier in Figures 22 and 23, and in Figures 30 to
33, students displayed a range of ways to demonstrate their understanding of how to create two dimensional shapes with a partner by using a variety of body parts including their legs, fingers, arms, and their bodies. Multiple representations were witnessed during this lesson. Imitation of some of the partners was witnessed, and although I was encouraging creative and different responses and trying to dissuade students from 90
imitating, imitation could be considered a gain. At times, students did pick up and explore additional representations from other students. This imitation could be considered positive, especially when the goal is a connected network of representations.
x
i-v.
(a) (b)
Hgure 30. Students' pinsicai representation of a tectangle. In (a) two students bend ttieii legs and join their feet together to form a rectangle. In (I)) two students join fingers and thumbs to form a rectangle. 91
(? ife'*^
••••LA
figure 31. Students lie on the floor head to toe and try to create a rectangle.
•' <*
"^•T
fignre32. Le^on I-our - a siodent draws herwU nut king ;i triangle »uh another vtudcnr. 92
In the second lesson, the exploration of the sphere took place. Students explored trying to creatively dance with totally rounded (no sharp) body parts. Students moved their interpretations of a sphere; for example, they moved like a ball via bouncing and rolling. A variety of rounded body shapes were witnessed including rounded torsos, arms, and fingers (see Figure 5). Some students were more static and knelt in the fetal position or sat with their legs crossed, curling their spines towards their legs. One student somersaulted across the carpet as he was exploring the rounded body shape of the sphere.
As seen in Figures 5 and 33, a heterogeneity of responses is included that illustrates the concept of round. I chose to categorize all of these examples as diverse because each one is similar yet different from the next.
(a) _ v^_ f" (b) . il#(cK'lfc' "It W^^JS
figure 33. Reflections In three different -students representing rounded both ilmpe^a). {!>) and (c).
Emerging Competence
Finally, I labeled a number of approaches or actions as emerging competence. My understanding of emerging competence is that students' understandings and knowledge is 93
coming forth and that this understanding was clear for me to see.
As the lessons proceeded, the students became more aware of the range of responses that was appropriate to the variety of cues I was giving. In each subsequent lesson, I made the warm-ups a bit quicker than in the previous lesson. Cutting back on the warm-up and working on transitions (i.e. from sitting to standing, stillness to movement, movement to stillness), helped keep the body of the lesson unaltered. Cutting back on the warm-ups was a strategy of mine and students demonstrated emergent competence by how smoothly they transitioned through the activities in the lessons. Many activities were repeated in each lesson, from exploring the concept of flat, creating shapes with the body both independently and with others, to creating shapes with the elastic, and building three dimensional solids. As the students repeated activities, it appeared that it was easier for them.
In the first lesson, students were getting used to what was expected of them. A lot of new information as to how the class was run, how they were expected to behave, where and when they were and were not allowed to move, as well as who was in charge was set into the structure of the lesson. Behavior was the worst on the first day. During this class, I needed to guide the students with terminology surrounding the shapes. When students explored moving in line with the descriptors of the shapes, they appeared lost.
They connected a lot more with the imagery suggested by the shapes (see Figure 34). 94
(a). JL in;
Figure 34. Lesson One: Examples - Students connecting with the imagery of the circle.
By the second lesson, the students had had the opportunity of having at least one lesson with me, and knew what to anticipate in regards to my expectations. The students felt more confident and were better behaved than in the first lesson. At the beginning of the second lesson, large pieces of chart paper with solids drawn on them were posted on the classroom wall. The solids represented included the cube, cone, sphere, rectangular prism, and square-based pyramid. Students were invited to tell me the names of the solids. The only solid the students didn't know the name of was the rectangular prism. At the end of this class the host teacher commented, "Classroom management was good today." She commented on Andre's participation in the class, suggesting that Andre really understood the movement, both during the sphere and cone exploration. The quality of Andre's representation developed as well (note the difference in Figure 24 between lesson one and lesson two). I would label the overall tone of the second lesson as emerging competence because the students' behavior clearly changed, and I concur it is because they had a better understanding of the project, my expectations of them, and the 95
routine of the class than they did in the first lesson. I also label Andre's representation as emerging competence due to clearly more effort and intention that went into creating his reflection on that class.
In the third lesson, students were able to tell that there were four corners and four sides that make up a square. I had to stress that there were also four equal sides. A student was able to answer that a rectangle has two long sides and two short sides. Also, students let me know that they needed to have three students to be corners to create a triangle.
Students were beginning to use mathematical terminology when describing shapes throughout the lesson. In the post-interview with the host teacher, she commented:
Ms T. I began to see them using some language, some more appropriate names
for the shapes or the parts of the shapes, not necessarily all of them but
they were becoming more familiar with all that.
See Figures 7 to 10 for examples of students' understandings of large and small shapes that were created during this lesson. I label the fact that students were beginning to use mathematical terminology when describing shapes as emerging competence because they needed more prompting in the first and the second lesson to do this.
During lesson four, students worked well in pairs to create the circle, square, rectangle, and triangle. Some students still needed to be guided individually as to what body parts they could use. Once some ideas were presented, they completed this part of the lesson well, and demonstrated understanding. See Figure 12 for a pair of students creating a square with their legs. Little copying, that is where students connected with other students' representations, was seen during this lesson as opposed to the previous 96
lessons where more of this was evident. Students were more confident making shapes in pairs with little copying, and student reflections were clear and accurate (see Figures 13 to 15). I label this as emerging competence because many students were able to clearly demonstrate their knowledge of the characteristics of shapes better than in the previous lessons.
During the fifth lesson, students engaged in building the cube. Some proper math language was used, although some students were confusing the square and the cube.
During this lesson, the cube was the only solid for which we brainstormed ideas and descriptors. While some students wanted to build from the top down, this confused other students. A SK boy, named Gabe, took a leadership role and said, "I think we all should start together on the floor." Ms. T. comments on her observations of the cube being made in this lesson:
Ms. T. I noticed Gabe who is very bright and interested and focused and all but
usually fairly passive in a social situation, was really taking on a
leadership role. When they were trying to work together in a group to
create a three dimensional solid, and he obviously had a very clear idea of
how it should be done but he was able, in a very positive and proactive
way, to try and direct the other kids and it was really nice to see him
taking on that role and seeming quite comfortable with it.
I supported Gabe's ideas and with his guidance, the students built the cube almost instantly (see Figure 16). Without brainstorming the other solids, the assembly of the solids took a lot longer and the students lacked vision and clarity in the building phase. I 97
label Gabe's leadership and understanding in this lesson as emerging competence because he clearly understood how the cube could be built and his confidence to direct his peers also came forth during this lesson.
During the sixth lesson, the students seemed to lack some direction in building the cube and needed more guidance. I propose that this was the case because there were more
JK students building the solid during lesson six. Students showed a great deal of interest when they creatively danced inside the imaginary solids. They moved their hands along the flat faces of the cube, the rectangular prism, and the square-based pyramid. They responded with curved movements when inside the sphere. A student commented on this experience during the post-interview:
M: What do you like about math? Do you like anything different about math
that you haven't told me before?
Chloe: I liked on our last day in January, um when we were feeling how we
would be if we were inside the shapes.
M: You liked that part? Why did you like that?
Chloe: Because . . . (Pause)
M: When you were imagining that you were inside a solid, why did you like
that?
Chloe: Because I was trying to get into the shape (laughing).
M: You were trying to get into it. Was it hard or easy?
Chloe: Hard.
M: Did it help you understand the shape better? 98
Chloe: Yeah.
I label the creative dance that the students were doing inside the imaginary three- dimensional solids as emerging competence because the movements that the students were doing inside the imaginary solids were appropriate and directly related to the solid they were imagining being inside.
In lesson seven, students were able to create the shapes and the solids more easily, followed my instructions more closely, and worked best in their groups. In the post- interview with the host teacher, comments on how students gradually became more comfortable and competent with the expectations were expressed. The following dialogue was shared:
M. Could you share your observations with me of the Math and Motion
classes?
Ms T. The kids um seemed as the lessons went on they got more interested, more
comfortable with the expectations and you know the parameters around
what was happening so they at the beginning would say "When do we get
to play?" or "Why don't we get to play?" but by the end they seemed to be
eager to go to the classes. Within that there were some exceptions.
From the post-interview, the host teacher continues to comment on participation and group problem solving:
Ms T. But generally, they seemed to be excited about it; I think they liked it; they
certainly liked the movement part of it and a lot of them seemed to like the
opportunity to work together in a group and do some problem solving 99
around that. They became more comfortable doing the tasks. Where some
of them would watch initially you know it seemed like a lot of them
loosened up and became a bit more involved as it went along. I saw some
really interesting problem solving happening, some good teamwork!
Students were engaged in solo, small group, and class problem solving activities and continued to become more confident with what was expected of them. See Figures 21 to
23 for examples of students working appropriately and collaboratively in groups, making shapes with parts of their body. Also, student reflections developed from previous lessons
(see Figures 19, 20 and 27).
In the post-focus group interview, students shared their new understanding of what math is. The following dialogue emerged:
M: What is math? Do you have a different idea of what math is now?
Gabe: Seeing how shapes are all different.
M: They are different. What are some ways that shapes are different?
Chloe: Some shapes don't have corners and some do have corners.
M: You are right. What shapes do have corners?
Jonah: Some have three corners and some have four.
M: What shapes are shapes that have four corners?
Gabe: Square!
M: A square has four corners, yes.
Jonah: A rectangular prism.
M: A rectangular prism is a solid. Remember that is three-dimensional 100
geometry.
Dan: Oh yeah!
Jonah: Oh yeah, I forgot.
M: A rectangle has four corners.
Dan: You can use math to measure how far the sun is.
Jonah: I forgot that solids are full shapes.
M: What is that?
Jonah: I forgot that solids are shapes that aren't flat.
M: Right, you forgot that but now you remembered?
Jonah: Yeah.
The students' ability to create shapes and the solids more easily, and to follow my instructions are labeled as emerging competence. Their work in pairs and small groups as well as their more complex drawings are also labeled as emerging competence because in all of these areas the students gradually improved. Also, the post interview with some of the students illustrates that some of the mathematical material and language taught was understood and retained.
The students in this study did evolve in their problem solving abilities. When the students realized that their bodies could be used in ways that they had not experienced before, this knowledge became a tool for the students and the teacher witnessing, to be used in other classes. The host teacher's experience also connects with Melanie Layne's experience, a teacher who participated in workshops at the Kennedy Centre. Among many things, she gained the confidence to experiment. She comments:
Due to the teaching artists' leadership, instruction and inspiration during
the workshops, I now am able to use dance to teach vocabulary, patterns
and cycles in science, imagery in poetry and geometry in math. (Elder,
2007, p. 21)
The host teacher mentioned to me that the class I was teaching was also working on the theme of space; after one of the lessons that I taught, she transferred creative dance exploration to other lessons with the same class. She invited her students in her Drama and Dance class to explore the way an alien would move through creative dance. Also, the host teacher transferred some ideas of mathematics and creative dance to another class she taught. She borrowed the elastics and tried a version of the elastics' lesson with the students in her morning
JK/K class, which was a different class than the one I taught. I consider these above examples to be emerging competence both for the students and the host teacher because the understanding of creative dance integration is being extended beyond my lessons and into other lessons as well as with other students. 102
Chapter Six: Answers to Questions and Conclusions
In this section, I will discuss the research questions that are stated at the beginning of this document as well as discuss my conclusions. The questions are as follows:
1. Are the students engaged in the learning of mathematics when it is learned
through creative dance?
2. How might student attitudes and feelings towards mathematics change when
learned through creative dance activities?
3. Will student self-confidence in mathematics class be improved when it is learned
through creative dance?
4. What sequence of activities supports students' learning?
First Question
In response to the first question: Are the students engaged in the learning of mathematics when it is learned through creative dance? it was my hypothesis that students would become more engaged in the learning of mathematics, especially when learning with their bodies (parts or whole). What I found was that all students did become more engaged in the learning of mathematics when it was learned through creative dance.
In the post-interview, the host teacher commented:
Ms. T. It is always interesting to see kids in a different kind of situation
than we sometimes have in the classroom. See which kids shine in
a different kind of way or which kids feel comfortable doing 103
something that I might not see them doing much in the regular
classroom program. So there were kids who surprised me, at how
creative and comfortable and enthusiastic they were in the activity.
I began to see them using some language, some more appropriate
names for the shapes or the parts of the shapes.
In the post interview with the students, Jonah shares his accurate understanding of three- dimensional solids:
Jonah: I forgot that solids are full shapes.
M: What is that?
Jonah: I forgot that solids are shapes that aren't flat.
Certain students took on leadership roles without being directed or instructed to do so. I was not anticipating this result as the literature did not mention that random students might take on leadership roles. As mentioned in Chapter 5, Gabe took on a leadership role, a responsible position that the host teacher mentioned she was surprised to see. Commenting on Gabe's engagement in class, the host teacher stated:
Ms. T. He was able, in a very positive and proactive way, to try and direct the
other kids and it was really nice to see him taking on that role and seeming
quite comfortable with it.
See Figure 16 for students following Gabe's direction to build the cube from the ground up. The host teacher spoke about a Junior Kindergarten student named Chloe and a
Senior Kindergarten student named Ava who became more engaged in the lessons:
Ms. T. Chloe too, tends to be quite passive and uninvolved in class, and often was 104
also acting much more as a leader and trying to direct the other kids and
how they could work together or what was needed. There were some other
kids like Ava who also became somewhat of a leader although she wasn't
as confident sticking with it. She could identify problems and try to direct
people, but then if they didn't listen to her, she kind of backed off.
For some students, their engagement in the mathematics and creative dance lessons manifested itself as assuming a leadership role within the classroom environment.
During the brainstorming session for the two dimensional solids, one of the numerous brainstormed words for the triangle, was a rocket ship top. Once we moved into the dance space and I said this word, many of the students related so much to this word that it triggered them to really begin creatively dancing through the space at a variety of levels with a few pointed body parts (see Figure 4).
In math education, the word 'retain' is typically used relative to the brain holding onto information. While using slightly different terminology, Griss, coming from the standpoint of dance education, comments on how physicality helps to internalize, interpret, and maintain information: "Interpreting a concept through physical means [that it ] helps children—especially those at the elementary level—to grasp, internalize, and maintain abstract information" (Griss, 1994, p. 79).
As mentioned in Chapter 5,1 noted in my journal how Andre was appropriately exploring the sphere. During the post-interview, the host teacher commented:
Ms T. I saw that Andre, who tends to be very passive in class and often
uninvolved because he doesn't really understand what's happening, 105
seemed very comfortable with the movement and was able to move in
very creative and appropriate ways and really follow the directions well.
Andre was able to physically communicate his understanding of the mathematics being taught. The host teacher and I noted his progress in his reflections throughout the course of the seven lessons that I was teaching (se.e Figures 24 to 27).
Every student participated in the lessons that I presented. As the lessons progressed, the students grew increasingly comfortable with the creative dance. Coates and Franco state the importance of the role of movement (and in this case study, creative dance) in a child's education:
Although most parents can appreciate the role of movement in a child's
development, many do not realize that it contributes significantly to a
child's learning and understanding of academic content. Research places
movement at the core of intellectual function. It is not only a manifestation
of physical well-being but also our first form of communication before
speech development. (Coates and Franco, 1999, p. 170)
One of my primary conclusions is that there is great importance in providing children with opportunities to engage in creative dance. I would like to elaborate on this and mention that when creative dance opportunities are presented, leadership qualities can emerge in unsuspected ways with students who we might least expect to take on such a role. Math in Motion as Goodway, Rudisill, Hamilton, and Hart mention, was developed as a result of the concern expressed by early childhood educators who found it difficult to keep "at risk" and delayed children interested 106
and engaged in traditional mathematics activities. As stated by Goodway, Rudisill,
Hamilton, and Hart, 1999:
It is often said that movement is the language of early childhood, yet we seem
to spend more time resisting children's desire to move rather than taking
advantage of this valuable resource. We believe that physical activity is a
wonderful medium in which to teach mathematics concepts, (p. 181)
My findings in regards to Andre match these findings in that I conclude that creative movement has an important place in the teaching and learning of mathematics for "at risk" and delayed children.
The philosophy behind the Ontario Kindergarten program is based on the premise that children's early learning experiences are paramount to their future well- being. As stated in the Ontario Kindergarten Curriculum Revised, 2006: "The arts in their many forms provide a natural vehicle through which children express their interpretation of our world. Therefore, the arts play an important role in the development of children's communication and thinking skills" (p. 56).
Experience is vital to the young student. For the young, experience also has a
variety of synonyms including action, doing, or involvement. This is part of the
philosophy behind the Ontario Kindergarten program and its acknowledgement that
learning experiences need to have an action/doing component in order to ensure students'
future well-being. Impact grows through regular use of movement, sequences, and lesson
structure. Creative dance is an integral part of development and needs to be part of
regular activities in the classroom. 107
Second Question
In response to the second question, "How might student's attitudes and feelings towards mathematics change when learned through creative dance activities?" my findings have shown that students enjoyed the lessons and that they were positively
affected by the mathematics and creative dance lessons.
In the post-focus group interview with the students, this conversation between
Chloe and me unfolded as follows:
M: What do you like about math? Do you like anything different about math
that you haven't told me before?
Chloe: I liked on our last day in January, when we were feeling how we would be
if we were inside the shapes.
M: You liked that part? Why did you like that?
Chloe: Because . . . (Pause).
M: When you were imagining that you were inside a solid, why did you like
that?
Chloe: Because I was trying to get into the shape (laughing).
M: You were trying to get into it. Was it hard or easy?
Chloe: Hard.
M: Did it help you understand the shape better?
Chloe: Yeah. 108
In this conversation, it is clear that Chloe enjoyed the activity of imagining that
she was inside the solid. She also replied that it helped her understand the shape better. In the post-interview, the host teacher commented on how certain activities appeared fun:
Ms. T. The idea of dance seemed really fun and we do some dance, not as
structured as that but they really seemed to like dancing with a particular
goal or instructions and then having the other kids watch and then
switching and then having the audience respond to what they saw. So that
kept them focused and provided some feedback to the dancers, so I like
that whole idea.
As noted in Chapter 5, after the first class, the host teacher mentioned to me that
Andre's attitude towards school improved dramatically. In the post survey that the
students were given, question two asked 'When I am in math in motion class I feel:' this was followed by five faces ranging from very happy to very sad (see Appendix C: Post
Survey for Students). The survey was read aloud to the students and they colored in the
face they felt most fitting. Of twenty students, eight students colored in the very happy
face, three students colored in the happy face, three students also colored in the not happy
or sad face, one student colored in the sad face, three students colored in the very sad face
and two of the surveys had all the faces colored in. Andre's attitude, student responses to question two, and student responses to the fourth question in the post survey (see Figures
43 and 44) prove that my hypothesis was correct; that is, student attitudes and feelings towards mathematics lessons were positively affected when creative dance activities were integrated into mathematics. 109
4 Would you like to draw a'picture o f tea th m motion or tell me raj
(a) Cb)
Figure 35. Lesson Six - students* responses to the fourth question of the survey (a), (b).
Figures 36 and 37 have shown me that their mathematical knowledge increased as well. 110
1 : V . :..:!:.,•.• :r. •:.• .'i .-., . |. tuicj|-: i- -.:' •: l.i ':••• • •-. '.-'i1 -tihrm
figure 3(). Lesson Sit - student's response to post survcj question #4.
Figure .??. Student's drawing of a square-based pyramid as part of his response to question #4 of the
post-sun e\.
Although I did not compare these participating students with non-creative dance/mathematics students, my conclusions match Werner's conclusions from the study done at Whittier Community School for the Arts in Minneapolis in that "the literature suggests a strong link between positive attitude and student achievement" (Werner as Ill
quoted in Gottfried, 1990, p. 8). Her findings concluded that there was a significant difference between the dance/mathematics students' attitude toward mathematics compared to the non-dance/math students' attitudes toward math. I found that students
greatly enjoyed the integrated mathematics and creative dance lessons. I also conclude that students were representing in a fairly complex way, what was learned in lesson
seven. Given the opportunity to work longer with these students and to track their
success, I contend that overall achievement scores would increase.
Third Question
The third question was: Will students' self confidence in mathematics class be
improved when mathematics is learned through creative dance? I hypothesized that
students would feel more confident about themselves during mathematics class. I also hypothesized that this self-confidence developed in mathematics class would filter into other curricular subjects and interactions with others. What I have found is that self-
confidence for these students is connected to body memory.
My special case, Andre, as noted in Chapter 5, is a clear example that confidence in math class can be improved when mathematics is learned through creative dance. In the post interview, the host teacher commented on Andre's creativity and appropriate ways of expressing himself as well as his ability to follow the directions well, she stated:
"It seemed that he was feeling pretty confident about that. That was neat." The confidence that Andre gained in this class helped him lose his fear and hate of school. As
stated in the literature review, Melanie Layne, an Arts Integration Resource Teacher at 112
Bailey's Elementary School for the Arts and Sciences in Falls Church, VA, notes:
I am convinced that kinesthetic learning is a natural way for students to
engage in learning while building self-control, confidence and community
within the classroom. It is a way of teaching that is especially imperative
for students who are learning disabled or are ESL learners and may have
difficulty learning rational schooling methods. (Elder, 2007, p. 21)
Griss comments that when children learn curriculum in physical ways, they become more aware of their own individual and inherent resources:
You will find that representing academic curriculum in physical ways
makes the learning more accessible and memorable for children, and
fosters creative and dynamic energy in the classroom. Besides learning
specific curricular content from these kinesthetic activities, children
exposed to creative movement as a language for learning are becoming
more aware of their own natural resources. They are expanding their
concepts of creativity and of how they can use their own bodies. They are
learning through their own creations. (Griss, 1994, p. 34)
The above quote also connects to Montessori education. As noted in the review of literature, there are eight principles of Montessori education. I've chosen to focus on the first one in the list: "That movement and cognition are closely entwined, and movement can enhance thinking and learning" (Stoll Lillard, 2005, p. 29). This first principle is based on the observation that our brains are developing in a world in which our doing is 113
the learning. Montessori also believed that movement and thinking were in fact the same process in small children. Also, in the literature review Dr. Hon J. Ratey states:
Physically active people reported an increase in academic abilities,
memory retrieval and cognitive abilities. What makes us move is also
what makes us think. Certain kinds of exercise can produce chemical
alterations that give us stronger, healthier and happier brains. A better
brain is better equipped to think, remember and learn. (McGreevy-
Nichols,2001,p. 146)
My findings connect to this literature and I conclude that by including creative dance as part of the learning process, we are then learning through the body, creating body memory, which results in self-confidence. Body memory connects to retention of material learned. If educators encourage more positive, enriching, body memory experiences, I propose that we are then enabling students to build their self-confidence, learn, and retain curriculum.
Fourth Question
What sequence of activities supports students' learning? It was my hypothesis that the way I sequenced the lessons would support the students' learning of the material presented. The way that the lessons were sequenced did support the students learning.
The evidence supports the conclusion that multiple lessons were essential and that 3x per week was positive to their developing comfort, abilities, and confidence. It worked well to teach two and three-dimensional geometry together and to integrate movement into the 114
lessons, I will come back to this last theme.
However there are changes that I would make if I were to re-teach the following lessons. In this section I will highlight suggestions for changes to each lesson as well as trends that I witnessed. Specifically, the focus will be on issues that arose around the mathematics and creative dance.
In each lesson the following are trends that I witnessed, alterations I made to the teaching process, and/or other suggestions to alter the outcome:
Lesson One
• Moving like the shape/solid descriptors was too complex and abstract for
this age group. Encouraging students to move according to images or
descriptors of the solids enabled more movement.
• The plan included too much. Next time, I would cut back on content, i.e.
pick two solids to be explored rather than six.
Lesson Two
• The plan again included too much. Give students choice as to what two
solids they want to explore; this seemed to work as a modification to this
lesson.
Lesson Three
• During lesson three, I introduced the used of elastics as props. I invited the
students to take hold with both hands and put the elastic band behind their
backs and then stand with their arms by their sides. Many students put the
elastic around their back but some put it around the back of their neck. In 115
future I would physically demonstrate where the elastic is placed on the
body (behind back) prior to creating large class shapes with elastic.
• When I asked: "What will you all need to do to create a very large circle?"
I found that not all student steps were of equal size, which altered the
shape of the circle; also, some students took more than one step. In future,
I would be prepared to adjust students and/or have one student
demonstrate what a step means.
• During this lesson we explored the square with the elastic. At one point I
asked: "How might we organize ourselves so that we can create the special
rectangle called the square?" A student replied, "We need four things to be
corners." I replied, "I see students sitting around the dance space. Do you
think we could use students as the corners?" Students agreed we could. In
future, I would allow students time to come up with the idea that they
could be the corners.
• When creating the triangle, square, or rectangle with the entire class, I
would ask more open-ended questions. For example: "What parts of a
triangle are there and how might we create this as a class?" I would let the
students come up with the ideas rather than giving them the answers as I
did in the instance related above.
• The students were divided into groups of four and given medium sized
elastics. They had the opportunity to create a square and a rectangle, yet
due to time constraints, I gave an option to the groups to either create a 116
circle or a triangle. In future, I would consolidate the rectangle and square
exploration to allow time for all groups to experience exploring all of the
shapes, including the triangle and circle. This could mean continuing this
lesson on another day.
• There was too much content and the students were rushed to reflect on the
lesson. In the future, I would give a good amount of time for discussion
and drawing/reflection.
Lesson Four
• The plan again included too much. I would cut back on content, i.e. pick
two shapes or let the students pick what shapes they want to explore.
• It was overwhelming trying to be the live musical accompaniment and the
teacher at the same time. In future, I would use recorded music.
Lesson Five
• Not all the solids had been brainstormed at the beginning of this lesson. In
future, I would make sure all solids have been brainstormed and prior
knowledge has been activated prior to moving into this lesson.
• Students sitting and observing the creative dancers, at times began to lose
focus. In future, I would give a pre-assigned task for the students who will
observe the builders, i.e. draw, build individual models of what they see,
or watch for what they see the movers doing, and then have the observers
share their observations. I would treat it more like a performance situation
and perhaps switch groups to allow all students the experience of creating 117
each solid.
• Again, it was overwhelming to be the musical accompaniment and the
teacher at the same time. In future, I would use recorded music.
Lesson Six
• For the same reason as noted in lessons four and five, I would use
recorded music.
• Again, students sitting and observing the creative dancers began to lose
focus. In future, I would treat the building of the solids as a mini
performance, encouraging the observing group to give feedback, and then
have the groups switch roles.
Lesson Seven
• For the same reason as was indicated in lessons four to six, I would use
recorded music.
• Transitions from one part of the lesson to the next were not the smoothest.
To allow for smoother transitions in the body of the lesson, I would make
sure all solids and shapes are reviewed from previous lessons.
The lessons were delivered three times per week, over the course of two weeks. I presume that if the lessons were implemented on a daily basis, the students would have retained more information. This could be an area for further research. I also think that the content of each of the lessons could have been cut in half. Regular lessons every week may allow for adding content so that more was covered over the entire year. In other contexts, people find that doing a few lessons requires the instructor to reduce the content 118
and develop the capacity. How much time is needed for students to practice and develop capacity and learn more using the new capacities? This is a question for further research.
In order for me to accurately answer if I think the lessons supported students' learning, I looked to the post-interview with the host teacher and her final comments on the following question I posed to her:
M. Did you want to talk about changes?
Ms T. I think my general feeling overall was that it seemed really appropriate for
the SKs and a few JKs; a few kids got a lot out of it. But for some of the
JKs it was really so advanced that it was hard to keep their attention, keep
them focused, and it disrupted it [the lesson]. I mean they weren't really
disruptive, they were pretty good but still it was hard to pay attention to
them and their needs when they were uninterested or lost the concepts and
stay focused with what the other kids were doing. So I would say in
general it was better for SK, maybe even Grade 1, and also probably better
in an ideal situation to be able to work with a small group at a time rather
than a whole class. So half the class at a time and take them away and do
their thing and have an opportunity for all the kids to be really, really
actively involved. When students were doing the dance and everybody
was dancing, it seemed to work better than when a few kids were coming
up with props or elastics to do something where the other kids watched.
That was more challenging.
M. For each lesson? 119
Ms T. Maybe not for each lesson but for more lessons or even interspersed with
shorter lessons. It seemed a big leap to go from the 2D shapes to the 3D
solids and for them to understand how many edges are there, and what are
the shapes of the faces, and how they interconnect, and it almost seemed
that there could have been a whole other series of lessons in there talking
about just edges or talking about curves you know what's an edge, what's
a face. They were kind of thrown into those concepts kind of fast, and they
didn't all get it right off. Like, we are still at the point of talking about
corners and points. Points in particular is [sic] what they call all these
things. Yeah, I think that some of it was beyond them. I think it would
have been easier for them to make the 3D solids if they had a lot more
work previously really exploring solids individually. You know, holding
them, talking about them, comparing them—all that stuff and then moved
into how can we put these together. They were pretty complex concepts
for a lot of them.
Perhaps the lessons could have been stretched out and built around either two- dimensional geometry or three-dimensional geometry; the host teacher thought it was a big jump to go from two-dimensional geometry to three-dimensional geometry because the students did not have the vocabulary needed for the parts of a three-dimensional solid.
I was trying to work with both areas of geometry simultaneously and this, as the host teacher mentions, may have been too difficult for the students at the Junior Kindergarten level. 120
However, I do not think that this was a major obstacle or leap for the students. I witnessed the students equally engaged and creatively dancing freely with both the two- dimensional shapes and three-dimensional solids. If students were given a bit more time to assimilate the information and language learned on each aspect of geometry, I do not think that the host teacher would have made her comment about the large leap from two- to three-dimensional geometry. Ms T. mentioned the students' lack of prior experience with the solids; if I focused more in depth on one concept rather than on both, then perhaps the younger students would have understood it more. As Professor Whiteley pointed out, prior to starting school, students had experience primarily with three- dimensional solids. As an example, a child experiences three-dimensional geometry when she crawls inside a box. Perhaps as Professor Whiteley suggested, the experience of connecting what they have been doing to a school activity is new.
It is critical to make sure the students who are observing other students have a pre-assigned task or can be occupied in another way and that prior knowledge is activated. Chloe mentioned that it could take four students to create the cube; perhaps groups of four could have worked on building the cube and other solids. Building the solids in smaller groups may have helped engage all the students right away, solving the problem of students who lacked focus.
If I could redo this study, I would give more time to particular lessons. In my opinion, it worked well to teach two- and three-dimensional geometry together and to integrate movement into the lessons.
Sequence is important; but the repetitive creative dance aspects that helped to 121
build the confidence and comfort of the students are even more important. These activities provided a comfortable framing for exploring new mathematics. Part of the comfort with the mathematics might be the security experienced within the sequence of a
'creative dance' lesson. Repetition is of vital importance. As stated earlier in chapter 5, when students repeated actions, it appeared easier for them. As lessons progressed, less prompting on my part was needed for the students to use appropriate mathematical vocabulary and creative dance.
Some options of sequences of the lessons could be as follows:
• Option one: Follow the order of lessons one to seven but choose to either
cut the content from lessons one, two, and four, or complete lessons one,
two, and four over the course of three days (forty minutes per class), or
allot longer periods for them (eighty minutes per class) and complete all
the content in the lessons over the course of three days.
• Option two: Complete all the two-dimensional geometry lessons first.
These include lesson one, three, and four. Then complete the three-
dimensional geometry lessons. These include lesson two, five, six, and
seven. One must take into account the suggestions listed in option one for
lesson one, four, and two.
• Option three: Complete all the three dimensional geometry lessons first.
These include lesson two, five, six, and seven and then move on to the two
dimensional geometry lessons including lesson one, three, and four.
Again, one must take into account the suggestions listed in option one for 122
lesson one, four, and two.
Concluding Remarks
One might think that creative dance training is necessary to do these activities.
This isn't so, but there are certain skills that are essential. These skills include being able to guide students through exploring the traditional elements of creative dance including; body, space, time, force (energy), and relationships. I believe that teachers will gain confidence with the curriculum material I have developed through experiential professional development. Such professional development might be in the form of workshops in which teachers would be introduced to the area of curricular integration. By focusing sessions on learning the elements of creative dance and some key terms and ideas in mathematics topics that lend themselves to exploration through movement, teachers would gain the tools needed to work creatively when teaching mathematics kinesthetically.
Although I chose to integrate creative dance with mathematics around two- dimensional shapes and three-dimensional figures, integrating movement may work just as well or even better with other strands of mathematics. Patterning, mirror actions, rotational actions, translations of actions and measuring using parts of the body, large steps, rolls, and leaps are a few possible topics. A classroom teacher might also try using creative dance in science, e.g. to explore ideas such as falling, rising, balancing, and the topic of life cycles. Many of the mathematics and science ideas could also work well if taught together, e.g. falling and rising temperatures and reflections or balancing and 123
translations. Integrating creative dance with multiple strands and across several subjects should have a cumulative impact, as students transfer their developing comfort with these ways of learning.
Working with two different parts of curriculum is not easy and takes a lot of creative thinking, trial, error, and determination. My research showed that space and shape work in mathematics links well to creative dance. It also revealed that knowing certain specifics is critical, e.g. faces, edges, and vertices of the polyhedral, and that there are dance ideas that are central to this work, e.g. the idea of pretending to be inside an object, the idea of imaging roundness, the idea of encouraging work at different levels, and the idea of having students work individually, in partners, and with a group.
In conclusion, I found that the participating kindergarten students gained a great deal from the mathematics and creative dance lessons that I designed. My research showed that the activities were successful in helping students kinesthetically explore mathematical shapes and solids. Students were engaged and participated in the learning process throughout the lessons. Images and comments revealed that more than half of the class felt positively affected when they were in the mathematics/creative dance class that
I taught them. Evidence showed that confidence for one developmentally low student increased, and that several students grew in mathematical knowledge. I believe that one reason for the success of these activities was the fact that creative dance allowed for individual responses from children.
I am aware that this study has limitations and that there is further work to be done by myself or others. Some of the limitations of the study are the small sample size (only 124
20 students), and limited time (two weeks with one follow-up day). I would be interested to further investigate the student achievement scores and attitudes towards mathematics by observing a number of teachers teach integrated mathematics and creative dance lessons to their students regularly over the course of a year. I would also be interested in finding and comparing the results of the mathematics/creative dance classes with mathematics but non-dance classes. References
Absolon, K. & C. Willett (2005). Putting ourselves forward: location in aboriginal
research. In L. Brown & S. Strega (Eds.), Research as resistance (97-126).
Toronto: Canadian Scholars Press.
Bee, H., Boyd, D., & P. Johnson. (2003). Life span development. Toronto: Pearson
Education Canada Inc.
Blunt, Richard. (1995). Waldorf education theory and practice. Cape Town: Novalis
Press.
Brockington, Dan and Sian Sullivan. (2003). "Qualitative Research." Development
fieldwork: A practical guide. Regina Scheyvens and Donovan Storey, Eds.
Clifford, J. (1988). The predicament of culture: Twentieth century ethnography,
literature, and art. Cambridge, Mass: Harvard University Press.
Coates, G. D. & J. Franco. (1999). Movement, mathematics, and learning experiences using a family learning model. In J. V. Copley (Editor), Mathematics in the early years
(pp. 169-174). United States of America: The National Council of Teachers of
Mathematics, INC.
Davies, Charlotte. (1999). Reflexive ethnography: A guide to researching selves and
others. London and New York: Routledge.
Doidge, N. (2007). The brain that changes itself. New York: Penguin Group.
Elder, L. E. (2007). "Dance teacher" recognizes: Perpich center for arts education's arts
courses for educators in dance. Dance Teacher, 29(10), 48.
Fetterman, David M. (1989). Ethnography: Step by step. Newbury Park, London and 126
New Delhi: Sage Publications. London, Thousand Oaks and New Delhi: Sage
Publications.
Flender, N. (2000). It's academic: Leading the way. Dance Teacher, 22(8), 68, 70-71.
Fraser, Diane L. (1991). Playdancing. Pennington, NJ: Princeton Book Company.
Gardner, H. (1999). Intelligence reframed. New York: Basic Books.
Goodway, J. D., M.E. Rudisill, M.L. Hamilton & M.A. Hart. (1999). Math in motion. In
J. V. Copley (Editor), Mathematics in the early years, (pp. 175-181). United
States of America: The National Council of Teachers of Mathematics, INC.
Kassing, Gayle and Danielle M. Jay. (2003). Dance Teaching Methods and Curriculum
Design. Champaign, IL: Human Kinetics.
Gilbert, A. G. (1992). Creative dance for all ages: A conceptual approach. Reston, VA:
American Alliance for Health, Physical Education, Recreation, and Dance.
Griss, S. (1994). Creative movement: A physical language for learning. Educational
Leadership Journal.
Hammersley, Martyn and Paul Atkinson. (1995). Ethnography: Principles in practice.
London and New York: Routledge.
Klotz, K. (Producer). (1991). The Platonic solids (DVD). Berkeley, CA: Key Curriculum
Press.
Lillard, A. (2005). Montessori the science behind the genius. New York, New York:
Oxford.
Lincoln, Yvonna and Egon Guba. (1985). Naturalistic inquiry. Newbury Park, London
and New Delhi: Sage Publications. University Press. 127
McGreevy-Nichols, S. (2002). Dance in K-12: An equation for success: A teaching
strategy that encourages creativity and dancemaking while helping students grasp
math concepts. Dance Teacher, 24(5), 82-83.
McGreevy-Nichols, S. (2001). Dance in K-12: Thinking on your feet. Dance Teacher,
25(10), 146.
Merriam, Sharan, B. (2009). Qualitative research: a guide to design and implementation.
San Francisco: Jossey-Bass
Merriam, S. (1988). Case study research in education: A qualitative approach. San
Francisco: Jossey-Bass.
Mohn, K. (2004). Rocking the curriculum dance and academics at an a+ school. Journal
of Dance Education, 4(4), 121-123.
Montessori, M. (1967). The absorbent mind. (C. A. Claremont, Trans.). New York:
Henry Holt.
Ontario Kindergarten Curriculum Revised, 2006.
Platonic Solid, [n.d.] In The Internet Encyclopedia of Science. Retrieved from
Pierce, R.L. and CM. Adams (2004). "Tiered Lessons: One Way to Differentiate
Mathematics Instruction." Gifted Child Today, 27 (2), 58-66.
Prout, A. (2002). "Researching children as social actors: an introduction to the children 5-
16 programme." Children & Society. Vol. 16, No. 2: 67-76.
Rhodes, A. M. (2006). "Dance in interdisciplinary teaching and learning." Journal
of Dance Education. Vol. 6, No. 2: 48-56. Sterner, R. (1966) (From lectures in 1919): Study of man. Fourteen lectures given in
Stuttgart in Aug/Sept 1919. Translated by Daphne Harwood and Helen Fox,
revised by A.C. Harwood. London: Rudolf Steiner Press.
Vaughn, S., Schumm, J. S., & Sinagub, J. (1966). Focus group interviews in education
and psychology. Thousand Oaks, CA: Sage Publications Inc.
Werner, L. (2001). Arts for academic achievement. Changing student attitudes toward
math: using dance to teach math. University of Minnesota.
Yin, Robert K. (1989). Case Study Research: Design and Methods. Newbury Park,
London and New Delhi: Sage Publications. Appendix A: Lesson Plans 1-7
Lesson 1 Researcher: Megan Johnston Subject: Mathematics and Dance Grade: Kindergarten (JK/K) Date: January 6, 2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations Overall Expectations: Describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation.
Specific Expectations: Identify and describe, using common geometric terms, two-dimensional shapes (e.g., triangle) and three-dimensional figures (e.g., cone) through investigations with concrete materials.
Explore, sort, and compare traditional and non-traditional two-dimensional shapes and three- dimensional figures.
Drama and Dance Curriculum Expectations
Overall Expectations: Demonstrate an awareness of themselves as artists through engaging in activities in visual arts, music, drama, and dance. Specific Expectations: Demonstrate an awareness of personal interests and a sense of accomplishment in drama and dance.
Materials: List with students' names on it to track students familiarity with both two and three dimensional objects and shapes, a hand drum, masking tape, pictures of 2D shapes including a circle, square, rectangle, and triangle drawn onto chart paper, a marker, blank papers with students' names on them and marker bins on tables.
Set up: Prior to the lesson, ensure that all materials are in place and the dance space has been marked off with masking tape and that space will be taped out for students to observe others working.
Lesson 1 of 7: (5 min) I will begin by welcoming students and invite the students to enter the dance space and sit in a circle. I will introduce myself and review the rules of the class (respect yourself and others), as well as the need to stay within the dance space. I will note where the observation area is and why it is used (for time outs, students needing a break, students not allowed to participate due to no parent consent).
I will bang the drum twice very quickly and let the students know that when they hear this sound it means to stop, look, and listen. Any other beats that I make with the 130
drum will be the music to which they will dance. I will demonstrate what neutral position is. Feet and legs parallel and in line with the hips, arms to the sides, shoulders back, and taking deep breaths. Students will be encouraged to enter this position often during the time I am with them. Explain what the overall expectations are for the class so students are aware of what they will be learning.
(7 min) We will then move on to the brainstorm and I will show each chart paper that has the picture of the shapes. Later in the lesson, I will ask for words that describe each shape. I will show what a side and a corner is on the shapes and I will record these and the brainstormed words below the picture. I will ask: "Do any of these pictures remind you of anything that you know?" I will record their responses and if necessary draw a picture. Note the same words that are used for different objects. I will give counter examples of each shape. All of the charts will be posted up so students can see the charts.
Warm-up: (5 min) Students will be lead through a five minute warm-up of the major body parts starting from the toes and moving up the body, ending at the head.
(10 min) I will then point to the first chart and pick out descriptive words that are listed on the chart paper. Rather than the children pretending to be a square, I will encourage students to explore the qualities of a square physically. For example, if students used the words 'straight or flat' or that a square reminds them of a window, I would encourage movements that make the body straight and flat both still and moving through space. I will progress through this movement exploration with the circle, square, rectangle, and triangle.
Ask: "When you were dancing, were there any shapes that had kind of the same movement or feeling in your body when you were dancing? Which ones? Why might this be?
I will then ask them to explore movements of shape descriptors with similar qualities and movements of shape descriptors that create different movement qualities. I will call out the shape one after the other.
Warm-down: (5 min) I will lead the class in a five-minute warm-down, which will be the reverse of the warm-up.
(7 min) I will then invite the students to quietly leave the dance space and encourage them to draw what they learned in math today and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: I will be sensitive to the students who may need more time to explore the concept. I will select a certain number of shapes to be explored (not all of them), and describe for students if necessary. 131
Accommodations: For the students who are not allowed to participate or do not wish to, I will have them work with Mrs. Titus on the lesson entitled "Feel the Shapes" pg. 352 from Nelson Mathematics K Teacher's Guide.
Assessment: I will use both the dance and math checklist to assess that students can do the following:
-Student can identify and describe 2D shapes. -Student can explore, sort, and compare 2D shapes. -Student physically accomplishes exploring the qualities of 2D shapes.
Extensions: Please refer to the "Keep Going" section on pg. 364-365 to be used for continued learning at the various centers in the classroom.
Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work.
End. Lesson 2 Researcher: Megan Johnston Subject: Mathematics and Dance Grade: Kindergarten (JK/K) Date: January 8, 2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations Overall Expectations: Describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation.
Specific Expectations: Identify and describe, using common geometric terms, two-dimensional shapes (e.g., triangle) and three-dimensional figures (e.g., cone) through investigations with concrete materials.
Explore, sort, and compare traditional and non-traditional two-dimensional shapes and three- dimensional figures.
Drama and Dance Curriculum Expectations:
Overall Expectations: Demonstrate an awareness of themselves as artists through engaging in activities in visual arts, music, drama, and dance.
Specific Expectations: Demonstrate an awareness of personal interests and a sense of accomplishment in drama and dance
Materials: List with students' names on it to track students' familiarity with three dimensional objects, a hand drum, masking tape, geometric 3D solids drawn on chart paper, various three dimensional models including the cone, cube, cylinder, rectangular 132
prism, sphere and the pyramid, a marker, blank papers with their names on it and marker bins at the tables.
Set up: Prior to the lesson, ensure that all materials are in place and the dance space has been marked off with masking tape, and that space will be taped out for students to observe others working.
Lesson 2 of 7: (10 min) Explain what the overall expectations are for the class so students are aware of what they will be learning. We will begin with the brainstorm and I will show each chart paper that has the picture of the solids on it. While we are describing each word I will pass around the corresponding 3D model. I will explain what a face and an edge is and we will count these. I will record this onto the chart paper and give counter examples of each figure. We will come up with words that describe each solid. I will record these words below the picture.
I will ask: "Do any of these pictures remind you of anything that you know?" I will record their responses and draw a picture. I will ask if we used the same words for any of the objects, note the same words that are used for different objects. All of the charts will be posted up so students can see the charts.
Warm-up: (5 min) Students will be led through a five-minute warm-up of the major body parts starting from the toes and moving up the body ending at the head.
(15 min) I will then point to the first chart and pick out descriptive words that are listed on the chart paper. Rather than the children pretending to be a sphere I will encourage students to explore the qualities of a sphere physically. For example, if students used the words 'rounded or like a ball' to describe a sphere, I would encourage finding a variety of ways to create rounded body shapes and roll or bounce like a ball. I will accompany this work with a simple drumbeat.
I will do an exploration similar to that stated above with all of the three dimensional solids. When this is done, I will bang the drum twice.
Ask: "When you were dancing were there any 3D objects that had kind of the same movement or feeling in your body? Which ones? Why might this be?"
I will then ask them to explore movements of the solid with similar qualities and movements of the solids that create different movements. I will call out the solids one after the other and encourage exploration.
Warm-down: (3 min) I will lead the class in a warm-down, which will be the reverse of the warm-up.
(6 min) I will then invite the students to quietly leave the dance space and 133
encourage them to draw what they learned in math today and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: I will be sensitive to the students who may need more time to explore the concepts. I will select a certain number of solids to be explored (not all of them), and describe for students if necessary or help them brainstorm when it is writing time.
Accommodations: For the students who are not allowed to participate or do.not wish to, I will have them work with Mrs. Titus on the lesson entitled "Let's Be Detectives," pg. 336 from Nelson Mathematics K Teacher's Guide.
Assessment: I will listen to the vocabulary that the students have used in this activity and model the correct geometric terms. I will use both the math and dance checklist to track the following: - Student can identify and describe 3D figures. - Student can explore, sort, and compare 3D figures. - Student physically accomplishes exploring the qualities of 3D solids.
Extensions: Please refer to the Keep Going activities found on pg. 344-345 in Nelson Mathematics K Teacher's Guide for continued work on three-dimensional geometry to be done at the centers.
Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work.
End.
Lesson Three
Researcher: Megan Johnston Subject: Mathematics and Dance Grade: Kindergarten (JK/K) Date: January 9, 2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations
Overall Expectations: Describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation.
Specific Expectations: Compose pictures and build designs, shapes, and patterns in two-dimensional shapes, and decompose two-dimensional shapes into smaller shapes, using various tools or strategies.
Drama and Dance Curriculum Expectations 134
Overall Expectations: Use problem-solving strategies when experimenting with the skills, materials, processes, and techniques used in the arts both individually and with others. Specific Expectations: Use problem-solving skills and their imagination to create drama and dance.
Materials: Assessment checklist, a hand drum, masking tape, chart paper from first lesson taped up, a marker, blank papers with students names on them and marker bins on tables, five medium elastic pieces of material sewn together to make circles, and one large elastic piece of material sewn together to form one large circle.
Set up: Prior to the lesson, ensure that all materials are in place and the dance space has been marked off with masking tape and that space will be taped out for students to observe others working.
Lesson 3 of 6: (2 min) I will invite the students to join me in a sitting circle in the dance space. Explain what the overall expectations are for the class so students are aware of what they will be learning. I will review the charts posted up with various shapes and words on them.
Warm-up: (5 min) Students will be lead through a five minute warm-up of the major body parts starting from the toes and moving up the body, ending at the head.
(15 min) I will present the large piece of elastic and tell students that we will be making very large shapes together. We will start with the circle then move onto the square, rectangle, and end with the triangle. I will put the elastic circle in the center of the group and invite the students to take hold with both hands and put the elastic band behind their backs. I will then invite students to stand and place their arms at their sides. I will ask: "What will you all need to do to create a very large circle?" I will invite answers and experimentation.
I will then ask: "What will you all need to do to create a smaller circle as a group?" I will invite answers again and encourage experimentation. When the students are in position I will ask: "How do you know that you have made a circle? What are some things you know about circles?"
I will ask: "How might you organize yourself so that you can create the special rectangle called the square?" I will again invite answers and experimentation. Some children will need to act as the vertices and others as the edges. I will guide students through this process by showing them the picture of a square. When students are in position again I will ask: "How do you know that the class has made a square? What are some things that you know about squares?"
I will ask: "What will you need to do now to create a different kind of rectangle?" Again I will guide the students through this process by showing the picture and verbally 135
guiding when necessary. When students are in position, I will ask again: "How might you know that the class has made a rectangle? What are some things that you know about rectangles?"
I will ask: "What will the class need to do to create a triangle?" Listening to responses, I will guide them through this process as well as showing them the picture. When students are in position, I will ask: "How might you know that you have made a rectangle? What are some things that you know about rectangles?"
When this has been completed, I will ask that the students sit down. I will ask them to take the elastic from behind them and place it in front of them. I will gather up the elastic piece of material.
(8 min) I will then divide the students into groups of four and give them medium sized elastics. I will invite them to go through the process again of making the shapes that I call out. I will call out and show the picture of those shapes mentioned above. I will use the drum and create a simple drumbeat, and the students will be encouraged to explore. I will ask: "How do you know that your group made a circle? A square? A rectangle? A triangle?" After each of these explorations, I will encourage students to share what they know about these shapes.
Warm-down: (3 min) I will reverse the warm-up that was stated earlier.
(6 min) I will then invite the students to quietly leave the dance space and encourage them to draw what they learned in math today, and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: For students who are having trouble creating the shapes, I will guide them one-on-one through the process.
Accommodations: For the students who are not allowed to participate, they will work with Mrs. Titus on the tabletop activity entitled "Creating Shapes," pg. 361 Nelson Mathematics K.
Assessment: I will watch and listen to see how the students plan to make their shape(s). Are they emphasizing the attributes that make each shape different? I will use both the dance and math checklist to assess the following: - Student can compose a picture, build a 2D shape and decompose a 2D shape using a variety of tools. - Student uses problem solving and uses imagination to create a dance.
Extensions: Please refer to the "Keep Going" section on pg. 364-365 in Nelson Mathematics K, to be used for continued learning at the various centers in the classroom. 136
Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work.
End.
Lesson Four
Researcher: Megan Johnston Subject: Mathematics and Dance Grade: Kindergarten (JK/K) Date: January 13,2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations Overall Expectations: Describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation.
Specific Expectations: Compose pictures and build designs, shapes, and patterns in two-dimensional shapes, and decompose two-dimensional shapes into smaller shapes, using various tools or strategies.
Drama and Dance Curriculum Expectations
Overall Expectations: Use problem-solving strategies when experimenting with the skills, materials, processes, and techniques used in the arts both individually and with others.
Specific Expectations: Use problem-solving skills and their imagination to create drama and dance.
Materials: Assessment checklist, a hand drum, masking tape, chart paper from first lesson taped up, a marker, blank papers with students' names on them, and marker bins on tables.
Set up: Prior to the lesson, ensure that all materials are in place and the dance space has been marked off with masking tape and that space will be taped out for students to observe others working.
Lesson 4 of 7: (2 min) I will invite the students to join me in a sitting circle in the dance space. Explain what the overall expectations are for the class so students are aware of what they will be learning. I will review the charts posted up with various shapes and words on them.
Warm-up: (5 min) Students will be lead through a five-minute warm-up of the major body parts, starting from the toes and moving up, ending at the head.
(5 min) I will then divide students into pairs and challenge the students to create the shapes that I call out with both of their bodies (all or part). I will call out and point to 137
pictures of the circle, square, rectangle, and triangle. I will use the drum to create a beat for the students. I will double beat the drum to indicate that students need to freeze in a shape. I will draw students' attention to examples in the class.
(3 min) I will then ask the students to work individually and to create the shapes with their bodies (all or part) that I call out. I will call out and point to pictures of the circle, square, rectangle, and triangle. I will use the drum to create a beat for the students. I will double beat the drum to indicate that students need to freeze in a shape and again I will draw students' attention to examples in the class.
(2 min) I will let the students know that when we create a dance, we need to have a good beginning, a middle, and ending to the dance. I will let them know that one way this can be done is by starting in a very still position to show that the dance is going to begin. The middle part has lots of movement and when they are done, they need to be very still to indicate the ending of the dance.
(8 min) I will say that the students are going to each create a shape dance. The dance has to have a beginning, middle, and an ending as well as having three shapes in it. I will invite students to think of the three shapes they want to dance about, and then to share this with the student sitting next him or her. I will then invite students to explore their first movement, then their second movement, and then their third movement.
(2 min) I will then say: "It is now time to put your dance together! When I double beat the drum, this means to begin in neutral position." I will begin to play the drum and not stop until I see that everyone has ended in the neutral position. When I double beat again, it means that everyone is done and students may sit down.
(4 min) I will invite half of the students to sit in the observation area and the other half to spread out around the dance space. I will let the students present their work to one another and then have the observing and dancing group switch. I will follow the same drumbeats as stated above. After each group has finished performing, I will invite the students who were observing to share observations with, or address questions they have to the dancers.
Warm-down: (3 min) When both groups have shared their work and responses, we will begin the warm-down. I will reverse the warm-up that was stated earlier.
(5 min) I will then invite the students to quietly leave the dance space and encourage them to draw what they learned in math today and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: For students who are not confident to work on their own, I will allow partner work. 138
Accommodations: Students that are not participating in the group that I am working with will be working with Mrs. Titus on the lesson entitled "On the Lookout for Shapes," pg. 355 in Nelson Mathematics K.
Assessment: I will watch, listen, and note to see how the students plan to make their shape(s). Are they emphasizing the attributes that make each shape different? Can students create a dance that has a beginning, middle, and end? I will use the math and dance checklists to assess the following:
- Student can compose a picture, build a 2D shape and decompose a 2D shape using a variety of tools. - Student uses problem solving and uses imagination to create a dance.
Extensions: Please refer to the "Keep Going" section on pg. 364-365 in the Nelson K Teacher Guide to be used for continued learning at the various centers in the classroom.
Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work.
End.
Lesson Five
Researcher: Megan Johnston Subject: Mathematics & Dance Grade: Kindergarten (JK/K) Date: January 15,2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations Overall Expectations: Describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation. Specific Expectations: Build three-dimensional structures using a variety of materials, and begin to recognize the three-dimensional figures that the structure contains.
Drama and Dance Curriculum Expectations Overall Expectations: Communicate their ideas through various art forms. Specific Expectations: Express their responses to drama and dance by moving, by making connections to their own experiences, or by talking about drama and dance.
Materials: Assessment checklist, a hand drum, masking tape, chart paper from second lesson taped up, a marker, blank papers with students names on them and marker bins on tables, 20 medium bamboo sticks, 4 long bamboo sticks, and two hula hoops. 139
Set up: Prior to the lesson, ensure that all materials are in place and the dance space has been marked off with masking tape, and that space will be taped out for students to observe others working.
Lesson 5 of 7: (2 min) I will invite the students to join me in a sitting circle in the dance space. Explain what the overall expectations are for the class so students are aware of what they will be learning. I will review the charts posted up with various solids and words on them. I will emphasize the number of edges of each solid.
(3 min) I will introduce the props that will be used in this class and how to properly handle them and where they belong after use. I will let students know that sometimes props are used in dances. The props that will be used are both medium and long bamboo sticks. I will emphasize that these are not toys and will be used to help the class create 3D models. I will say that each bamboo stick will represent an edge.
Warm-up: (5 min) Students will be lead through a five-minute warm-up of the major body parts starting from the toes and moving up, ending at the head. The warm-up will end with all students sitting in a circle.
(20 min) I will point to the picture of the cube and ask: "If each student is allowed to hold only one bamboo stick, how many students will it take to create a cube? How do you know this?"
I will ask for 12 volunteers to get a bamboo stick and enter the circle. I will encourage them to try to figure out how to create the cube on their own and if they need assistance, then I will give it. I will accompany this movement/math exploration with a simple drumbeat. When the students are in position, I will ask all the students, including the ones observing: "How do you know this is a cube? What are some things you know about a cube?"
I will point to the picture of the rectangular prism and ask: "How many students with bamboo sticks will be needed to create a rectangular prism? What could we do to this cube to create a rectangular prism?" I will invite responses and either allow four students holding medium sized bamboo sticks to melt away and four other students holding long bamboo sticks to take their place, or have four students with a medium sized bamboo stick to enter the circle and connect to an existing edge.
When students are in position I will ask all: "How do you all know this is a rectangular prism? What do you know about rectangular prisms?"
When this has been completed, I will ask all students in the circle to allow the model to melt away and for the students to return the bamboo sticks to the proper place.
I will point to the pyramid picture and ask: "How many students will be needed to 140
create a pyramid? How do you know this?" I will then invite eight students to get bamboo sticks to enter the circle and to work together to create the pyramid. When the students are in position, I will ask the students: "How do you know that this is a pyramid? What are some things you know about a pyramid?" I will ask the students to allow the pyramid model to melt and for the students to return the bamboo to the correct location.
Next, I will point to the cylinder picture and ask: "How might we create the cylinder using the materials here?" I will invite responses and encourage their exploration. If necessary, I will invite two students to take the hula hoops and enter the circle and then one by one I will invite all the students to get a bamboo stick and connect it to both of the hoops. I will ask: "Is this a cylinder? Why or why not? What do you know about cylinders?"
I will invite students to allow the cylinder to melt and the students to return bamboo sticks, and sit in a circle. Next, I will point to the picture of the cone. I will ask: "How might we make a cone using the materials provided?" I will encourage answers and explorations. When students feel they are in position, I will ask: "Is this a cone? Why or why not? What do you know about a cone?"
This solid will melt and students will return to the circle. I will then ask how we might build a sphere. "Is this possible with the material we have? Why or why not?" I will allow for exploration and then encourage students to use their hands to show me the shape of a sphere.
Warm-Down: (3 min) I will reverse the warm-up that was described earlier.
(6 min) I will then invite the students to quietly leave the dance space and encourage them to draw what they learned in math today and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: I will not use the drumbeat if some students need more one-on-one guidance with this activity.
Accommodations: Students that are not allowed to participate in this movement activity will work with Mrs. Titus on the activity entitled "Tall Toppling Towers," pg. 342 in Nelson Mathematics K.
Assessment: I will listen to the vocabulary that the students use around this activity and model the correct geometric terms. I will guide students through the movement activity making sure they are correctly using the props given and participating in group dance. The math and dance checklists will be used to assess the following: - Student can build three-dimensional structures using a variety of materials. - Student can investigate the relationship between 2D shapes and 3D figures in 141
objects that they have made. - Student expresses his/her ideas about dance through movement and talking.
Extensions: Please refer to the "Keep Going" activities found on pg. 344-345 in Nelson Mathematics K for continued work on three-dimensional geometry to be done at the centers.
Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work.
End.
Lesson Six
Researcher: Megan Johnston Subject: Mathematics and Dance Grade: Kindergarten (JK/K) Date: January 16,2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations Overall Expectations: describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation.
Specific Expectations: Investigate the relationship between two-dimensional shapes and three- dimensional figures in objects that they have made.
Drama and Dance Curriculum Expectations
Overall Expectations: Communicate their ideas through various art forms.
Specific Expectations: Express their responses to drama and dance by moving, by making connections to their own experiences, or by talking about drama and dance.
Materials: Assessment checklist, a hand drum, masking tape, chart paper from second lesson taped up, a marker, blank papers with students' names on them, marker bins on tables, 20 medium bamboo sticks, 4 long bamboo sticks, and two hula hoops.
Set up: Prior to the lesson, ensure that all materials are in place and the dance space has been marked off with masking tape and that space will be taped out for students to observe others working.
Lesson 6 of 7: (2 min) I will invite the students to join me in a sitting circle in the dance space. Explain what the overall expectations are for the class so students are aware of what they will be learning. I will review the charts posted up with various solids and 142
words on them.
Warm-up: (5 min) Students will be lead through a five minute warm-up of the major body parts starting from the toes and moving up the body ending at the head.
Say: (15 min) "We will be creating three of the solids that we created yesterday. The cube, the rectangular prism and the pyramid."
I will point to the picture of the. cube and ask: "How many students will it take to create a cube? How do you know this?" I will ask for 12 volunteers to get one bamboo stick each and enter the circle. I will invite them to create a cube. When the students are in position, I will ask all the students, including the ones observing: "How do you know this is a cube? What are some things you know about a cube? What is the shape of each face of the cube? How many squares do you see in this cube?" I will allow this shape to dissolve and the bamboo sticks to be returned.
I will point to the picture of the rectangular prism and ask: "How many students with bamboo sticks will be needed to create a rectangular prism?" I will invite responses and allow for 12 students to enter the circle and to create the rectangular prism. When students are in position, I will ask: "How do you know this is a rectangular prism? What are some things you now about a rectangular prism? What shapes do you see in the rectangular prism? How many of each?"
When this has been completed, I will ask that all students in the circle allow the model to melt away and that the students return the bamboo sticks to the proper place.
I will point to the pyramid picture and ask: "How many students will be needed to create a pyramid? How do you know this?" I will then invite eight students to get bamboo sticks to enter the circle and to work together to create the pyramid. When the students are in position, I will ask the students: "How do you know that this is a pyramid? What are some things you know about a pyramid? What shapes do you see within the pyramid? How many of each?" I will ask the students to allow the pyramid model to melt and for the students to return the bamboo sticks to the correct location.
(8 min) When this has been completed, I will lead the class in a 'solids dance.' All students will be encouraged to begin and end the dance in neutral position. I will start with a double drumbeat and verbally guide them through a dance exploration. Students will be asked to imagine being inside a cube into which only they can fit. I will encourage the students to feel the faces and vertices. I will then guide them to imagine the cube expanding and growing and encourage the students to explore this expanded space. I will move onto the pyramid, the cylinder, and the sphere in both close and far proximity and then end the dance, inviting students to find neutral and end with a double drumbeat. I will ask students to return to the circle and share their experience dancing inside the imaginary solids. I will ask: "Did you have different feelings dancing inside the different solids? Why might this be? Did dancing inside the shapes remind you of anything? How did it feel to dance with the shape very close to you? How did it feel to dance with the shape farther away from you? Did your movements change? How? Is there anything else you want to share?"
Warm-down: (3 min) I will reverse the warm-up. that was described earlier.
(6 min) I will then invite the students to quietly leave the dance space and encourage them to draw what they learned in math today, and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: Work in partners when doing the shape dance. Accommodations: Students that are not allowed to participate will work with Mrs. Titus on the activity entitled "Shaping 3-D objects," pg. 343 in Nelson Mathematics K Teacher's Guide.
Assessment: Watch to see if students are correctly exploring the interior of the solids that I name. Use the math and dance checklists to record the following:
- Student can investigate the relationship between 2D shapes and 3D figures in objects that they have made. - Students express his/her ideas about dance through movement and talking.
Extensions: Please refer to the "Keep Going" activities found on pg. 344-345 in Nelson Mathematics K for continued work on three-dimensional geometry to be done at the centers.
Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work.
End.
Lesson Seven
Researcher: Megan Johnston Subject: Mathematics and Dance Grade: Kindergarten (JK/K) Date: March 10 2009 Time: 1:15-1:55 PM
Mathematics Curriculum Expectations 144
Overall Expectations: describe, sort, classify, and compare two-dimensional shapes and three- dimensional figures, and describe the location and movement of objects through investigation. See specific expectations on lesson plans 1-6 Drama and Dance Curriculum Expectations Overall Expectations: Communicate their ideas through various art forms. Overall Expectations: Use problem-solving strategies when experimenting with the skills, materials, processes, and techniques used in the arts both individually and with others. Overall Expectations: Demonstrate an awareness of themselves as artists through engaging in activities in visual arts, music, drama, and dance. See specific expectations on lesson plans 1 -6
Materials: Assessment checklists, a hand drum, chart paper and a marker, masking tape, blank papers with students' names on them and marker bins on tables, 20 medium bamboo sticks, 4 long bamboo sticks and two hula hoops, five medium elastic pieces of material sewn together to make circles and one large elastic piece of material sewn together to form one large circle.
Set up: Prior to the lesson ensure that all materials are in place and the dance space has been marked off with masking tape and that space will be taped out for students to observe others working.
(5 min) I will invite the students to join me in a sitting circle in the dance space and let them know that this is a follow-up to the work we did back in January. I will review the rules of the class and the proper use of props. I will ask students to name the various two-dimensional shapes and three-dimensional figures they know. I will ask students to describe them to me and I will draw them and write the descriptors down. I will do the same for three-dimensional figures. I will ask: "What three things does every dance need to have?" I will say: "Thumbs up if you agree, thumbs down if you disagree."
Lesson 7 of 7: Warm-up: (5 min) Students will be lead through a five-minute warm-up of the major body parts, starting from the toes and moving up, ending at the head.
(10 min) Say: "We will be creating three solids that we have created in the past: the cube, the rectangular prism, and the pyramid."
1 will point to the picture of the cube and ask: "How many students will it take to create a cube? How do you know this?" I will say: "Thumbs up if you agree, thumbs down if you disagree."
I will ask for 12 volunteers to get a bamboo stick and enter the circle. I will invite the 12 volunteers to create a cube. When the students are in position, I will ask all the students including the ones observing: "How do you know this is a cube? What are some things you know about a cube? What two dimensional shapes do you see in the cube? 145
How many squares do you see in this cube?" I will say: "Thumbs up if you agree, thumbs down if you disagree." I will allow this shape to dissolve and the students to return the bamboo sticks.
I will point to the picture of the rectangular prism and ask: "How many students with bamboo sticks will be needed to create a rectangular prism?" I will say: "Thumbs up if you agree, thumbs down if you disagree." I will invite responses and allow for 12 students to enter the circle and to create the rectangular prism. When students are in position, I will ask: "How do you know this is a rectangular prism? What are some things . you now about a rectangular prism? What shapes do you see in the rectangular prism? How many of each? What is different about a rectangular prism and a cube? What is the same?"
When this has been completed, I will ask for all students in the circle to allow the model to melt away and for the students to return the bamboo sticks to the proper place. I will point to the pyramid picture and ask: "How many students will be needed to create a pyramid?" I will say: "Thumbs up if you agree, thumbs down if you disagree. How do you know this?"
I will then invite eight students to get bamboo sticks to enter the circle and to work together to create the pyramid. When the students are in position, I will ask the students: "How do you know that this is a pyramid? What are some things you know about a pyramid? What shapes do you see within the pyramid? How many of each?" I will ask the students to allow the pyramid model to melt and for the students to return the bamboo to the correct location.
(10 min) Next I will present the large piece of elastic and tell students that we will be making very large shapes together. We will start with the circle, then move onto the square, rectangle, and end with the triangle. I will put the elastic circle in the center of the group and invite the students to take hold with both hands and put the elastic band behind their backs. I will then invite students to stand and place their arms at their sides. I will ask: "What will you all need to do to create a very large circle?" I will ask students to show me.
I will then ask: "What will you all need to do to create a smaller circle as a group?" I will invite students to show me. When the students are in position I will ask: "How do you know that you have made a circle? What are some things you know about circles?"
I will ask: "How might you organize yourself so that you can create the special rectangle called the square?" I will again invite answers and experimentation. Some children will need to act as the vertices and others as the edges. When students are in position, again I will ask: "How do you know that the class has made a square? What are some things that you know about squares?" 146
I will ask: "What will you need to do now to create a different kind of rectangle?" When students are in position, I will ask again: "How might you know that the class has made a rectangle? What are some things that you know about rectangles?"
I will ask: "What will the class need to do to create a triangle?" Listening to responses I will guide them through this process as well as showing them the picture. When students are in position, I will ask: "How might you know that you have made a rectangle? What are some things that you know about rectangles?" I will collect the large piece of elastic.
(8 min) Next, I will pair students together and invite them to create a shape dance together. Students will be asked to find a beginning position. I will then ask them to show me the following: A square, a rectangle, a circle, and a triangle. I will call each of these shapes out and the students will be encouraged to use any part of their body or their whole body to create the shapes, then to find an ending. After a few practices, the students will be divided so that half the class will be observers and the other half presenters. The group roles will switch. I will play the drum for the presentations.
(3 min) After the presentation I will invite students to share their observations and comments. I will ask: "How are rectangles and squares the same? How are they different? What two shapes are the most different to you? Why?"
Warm-down: (3 min) I will lead students through the reverse of the previously described warm-up.
(5 min) I will then invite the students to quietly leave the dance space and encourage them to draw what they learned in math today, and encourage the use of written words. I will hand out paper and have markers ready at the tables.
Modifications: Work solo when doing the shape dance.
Accommodations: Students who are not allowed to participate will work at the centers, set up as described in the extensions section of this lesson.
Assessment: I will use new (not checked) dance and math checklists to see if students were able to retain the following information:
- Student can identify and describe 2D shapes - Student can explore, sort and compare 2D shapes - Student can identify and describe 3D figures - Student can explore, sort and compare 3D figures - Student physically accomplishes exploring 3D solids - Student can compose a picture, build a 2D shape and decompose a 2D shape using a variety of tools - Student can build three-dimensional structures using a variety of materials - Student can investigate the relationship between 2D shapes and 3D figures in objects that s/he has made -Student physically accomplishes exploring 2D shapes - Student physically accomplishes exploring 3D solids - Student uses problem solving and uses imagination to create a dance - Student expresses his/her ideas about dance through movement and talking
Extensions: Please refer to the "Keep Going" activities found on pg. 344-345 in Nelson Mathematics K for continued work on three-dimensional geometry to be done at the centers. Closure: (1 min) Ask students to put markers away, tuck in chairs, and hand papers to me. Thank the students for their hard work. End. 148
Appendix B: Checklists
Kinesthetic Mathematics-Dance Checklist 2009
_. _ ,„______«„_ .,_™ _.. _i„_«
Students' Student Student Student Student Student Student Names physically physically problem problem expresses expresses accomplishes accomplishes solves and solves and his/her his/her exploring the exploring the uses uses ideas ideas qualities of qualities of imagination imagination about about 2D shapes. 3D solids. to create a to create a dance dance dance. dance. through through movement movement and and talking. talking. 149
Kinesthetic Mathematics-Mathematics 2009
Checklist
i
Students' Student can Student can Student can Student can Student can Names identify explore, identify explore, compose a and sort, and and sort, and picture, describe compare describe compare build a 2D 2D shapes. 2D shapes. 3D figures. 3D figures. shape, and decompose a 2D shape using a variety of tools. Appendix C: Post Survey for Students
Post Survey for Students
1. The activity of creating a Shape Dance made me feel:
2. When I am in math in motion class I feel:
3. This is how I feel right now:
4. Would you like to draw a picture of math in motion or tell me more?