Introduction/Overview

In the
spring of 2002, Scott Beall received an email from Nigmet Ibadildin
of Zhania Aubakirova's College in Almaty, Kazakhstan, inviting
Mr. Beall to visit the college and hold workshops on the topic
of integrated music and mathematics curriculum and teaching
pedagogy. The college was interested in Mr. Beall's curricular
publication, Functional MelodiesFinding Mathematical Relationships
In Music (Key Curriculum Press, İ2000). The interest of the
college stemmed from the desire to extend and inform their own
independent project, Music and Mathematics: Learning and Mastering.
The college discovered Mr. Beallıs work in a Key Curriculum
Press catalog brought to the school in 2001 by David Rusin,
a visiting mathematics professor from Northern Illinois University.
The trip took place February 6-23, 2003. During the visit Mr.
Beall,

- taught
classes daily to grades 3-11
- led
teacher workshops daily

observed instruction by teachers at
the college
- held interviews with staff, students, and
Zhania Aubakirova
- attended special musical performances by
students and Zhania Aubakirova
- reviewed assorted pieces of student work
and special projects
- arranged student exchanges with his school
in the United States
- gave a lecture to the staff of the Conservatory
of Music in Almaty
- met with Stephen Guice, the Cultural Affairs
Officer from the U.S. Embassy
- attended special student assemblies and
gatherings
- compiled 8 hours of video tape of the visit

This document presents an account of the visit.
A general narrative is followed by
a summary/reflection and recommendations for further collaboration.

Narrative of Events

Orientation and Setting the Agenda

My first day at the school focused
on general orientation and establishing goals, priorities, and
a plan for the work of my 2-week visit. I expressed the desire
to observe a variety of classroom lessons as taught by the teachers
of the school, meet students, see facilities, and end the day
with a meeting with teachers and administrators so that we might
best assess their needs and set an agenda for the coming weeks.
The end of the day also included a meeting with Zhania Aubakirova.
This meeting proved very useful in focusing the coming work.
I first observed a 5th grade and 9th grade math class. The younger
students were diligently working on arithmetic; working series
of problems multiplying decimals with paper and pencil. My interpreter
mentioned that they "never" use calculators here, only maybe in
the later grades, 10 and 11. The class environment was warm, supportive,
safe and friendly, with students presenting solutions on the board
while some watched, and others continued work at their desks.
The teacher and students obviously had a mutually respectful and
nurturing relationship. There were about 10 students in the class,
which was generally the average class size for all classes at
the college. (The term "college" is used to refer to
K-12 schools in Kazakhstan).

The 9th grade curriculum content
involved solving quadratic equations by the quadratic formula
and completing the square. The classroom environment was the same
as that of the 5th graders, with a similar process of individual
work mixed with board presentations and teacher prompting and
questioning. Teachers used inquiry based questioning approaches,
leading students to correct errors on their own through thoughtful
questioning. No technology was present in either classroom. Overhead
projectors, manipulatives and use of any special handouts was
not present. The text, the teacher, and fellow students were the
resources. From these observations I was able to begin setting
a plan to determine which music and mathematics activities would
be appropriate for various grade levels. I also learned something
about the teaching style that the students were accustomed to.
The balance of the day included listening to various student musical
performances, a teacher meeting, and a meeting with Zhania Aubakirova.

In my meeting with Zhania and the
teachers I discussed the philosophy and methodologies of my integrated
music and mathematics curriculum work. I then explained how Functional
Melodies is a specific application of the work that targets teaching
mathematics through the medium of music. We explored the various
reasons to integrate disciplines. I explained learning transfer
theories and ways that integrated music and math curricula can
raise all studentsı understanding of the disciplines and open
access for students of diverse learning styles and intelligences.
I then emphasized how integrating music and math inspires curiosity
and imagination, and generates an enhanced sense of meaning and
relevance for mathematics. Finally we acknowledged the integration
as a model and metaphor to teach interdependence and unity within
diversity in many realms, including culture, language, and nations.
The meeting was very inspiring and we left with a strong sense
of connection and shared purpose regarding the importance of this
integrated vision for teaching. It appeared clear at this time
that the international, cross cultural dimension of this consult
( American--Kazakhstani) was metaphorical to the integrated music
and math work. We acknowledged that the theme of exploring commonalties
between cultures and the conceptual nature of boundaries and divisions
was as a meaningful and important metaphor/extension of integrating
disciplines. This became more significant than we originally expected
as the work progressed, and is currently evolving as students
pursue their exchanges and everyone involved reflects on the experiences
of the visit. More on this aspect is discussed in the summary
section of this document.

It was ultimately decided that the
primary goal for my work of the visit should be to train the teachers
in the College to teach the integrated music and math curriculum
contained in my book, Functional Melodies. A further goal was
that they might eventually be able to share the knowledge with
other teachers in Almaty. I suggested to
the team that the most effective means to accomplish this would
be through two primary strands: 1) modeling instruction with students
and 2) direct teacher workshops. I would teach an assortment of
the lessons to students of various levels in the first part of
each day with various teachers observing and taking notes. Each
afternoon a teacher workshop would be held. Due to the busy schedules
of teachers, it happened that not all teachers could attend all
workshops, nor would the same teachers who observed the modeled
teaching necessarily be present in that dayıs teacher workshop.
Given this situation, it was not necessary for me to structure
days such that the modeled lessons of the day were the topics
in the following teacher sessions, so these two strands followed
their own sequences unto themselves. I planned the various student
classes based on the needs of the students using a sequence of
lessons that would be most coherent for the grade level and the
number of sessions I would meet any specific grade level. This
varied for each grade level. For the teacher sessions I began
with a theoretical overview of integrated curriculum, worked through
a series of activities, and concluded with a general lecture on
various pedagogical methods of progressive education that have
evolved in practice the United States over the past 40 years.

Student Classes

Instruction delivered to students
over the subsequent 9 working days was scheduled as follows:

Grades 3/4 1 class, 6 sessions

Grade 5 1 class, 6 sessions

Grade 7 1 class, 2 sessions

Grade 8 1 class, 3 sessions

Grade 9 2 classes, 3 sessions each

Grade 11 1 class, 4 sessions

My decisions regarding which activities
to do with each grade level included, but were not limited, to
the following considerations:

- Studentsı prior knowledge and pedagogical
conditioning
- Relevance to the mathematical content of
the studentsı primary curriculum
- Logistical practicality of the setting:
facilities, available technology, language, and time
- Action research: To gain insights into interdisciplinary
transfer and general student response at various grade levels
to various activities.

The goals of the student sessions
were (in order of priority):

1. to model the teaching methods
for teachers

2. to expose the students to new ideas, generate new questions
and open their minds to the underlying complexity and interconnectedness
of the universe

3. to enhance their knowledge and understanding in mathematics
and music

Grades 3/4

Lessons for the 3/4 grade students
began with Multiples of Drummers, and progressed to an adapted
and supplemented version of Functional Composer. I was unsure
if they would be able to grasp the concepts of Functional Composer,
but after some careful reflection and consideration of the research
nature of this project I decided to include that activity.

All of the activities with these
students proceeded very smoothly. I made adaptations in pacing
and structure as the lessons progressed and as I got to know the
studentsı abilities, prior knowledge and learning styles. With
this age group I usually do not attempt to establish the general
rule for finding least common multiples, however I decided to
push on with these kids to see if it was possible. Many seemed
to grasp the idea, and I reviewed it upon several meetings. Their
ability surprised me. Many retained the concepts and were able
to complete the final problems at the end of the lesson. These
problems have been known to cause 9th graders difficulty. This
class proudly determined the LCM of 2, 3, 4, 5, 6, and 7 to be
420, using the general rule in many cases!

To introduce the concepts of Functional
Composer I began singing graphs of scales and various melodic
patterns. We then played "name that graph," a multiple choice
melody-graph matching exercise requiring students to hear melodic
intervals and match them to an appropriate graphic representation.
These students performed famously (obviously due to their extensive
musical training and talent) and we moved on to dictation of the
song, "A Lion Sleeps Tonight," where the students made graphs
from the sung melody. Again, the students musical prowess was
impressive, as they finished the graph very quickly and more efficiently
than many students twice their age, based on my teaching experiences
in the US. We then moved on to Functional Composer.

Again, in the Functional Composer
activity the studentsı performance was beyond my expectations.
It was especially significant considering that they had had little
experience working with graphs in the previous schooling. From
what I could tell, this activity served as an introduction to
graphing for the students. More is discussed regarding the significance
of these studentsı performances in the summary section of this
report.

In many cases, the observing teachers
would roam and assist students, further involving themselves beyond
mere observation.

Grade 5

This grade did three activities:
Multiples of Drummers, Record Producer Algebra, and Functional
Composer. We were able to complete all of them, with the exception
of one problem at the end of Record Producer Algebra.

In Multiples of Drummers students
had some difficulty grasping the general rule for LCM. Some of
this may have been a communication issue, as this was the first
group I taught it to at the college. I was adjusting to the process
of working through the interpreter was and learning the extent
of the studentsı vocabulary in English, as well as how to articulate
instruction in ways that would translate well. In Functional Composer
a significant number of students were visibly touched by the higher
message of that activity. They considered the limits of rational
thought, the essence of creativity, and how subjective and rational
ways of knowing interrelate. They also considered the cognitive
complexity of the process of musical composition. I considered
this a significant accomplishment for 5th graders, especially
under the circumstances of such a short instructional sequence.

Grade 7

This class was very adept, and we
were able to do both Record Producer Algebra and Functional Composer
in two sessions. In Record Producer Algebra, students were initially
a little stiff, as it was obvious that they had never engaged
an activity quite like it before. Also, they were inevitably adjusting
to the presence of an American teacher in their class. They quickly
warmed up however. There were marked differences between studentsı
involvement within the class, as is often the case with this activity.
Some remained silent and seemed uninvolved and others calculated
and discussed with each other. Only two groups actively performed
their solutions to the class. It was clear that the others were
considering the mathematics and musical problems of the activity
as well, but were too shy to rap the English phrases in front
of the whole class. I was originally planning to do Multiples
of Drummers with this group, but on Nataliaıs suggestion I did
Functional Composer. The time was short, so it amounted to more
of a presentation than an interactive activity, but students were
attentive and very interested. By all measures the ideas were
new to them, and the activity raised their level of awareness,
curiosity, interest, and understanding of functions and transformations.

Grade 8

This class did Functional Composer,
Name That Function, and Record Producer Algebra. The class was
relatively small and possibly less interactive than some of the
others, though they did excellent work. We were able to explore
some math applications to the transformation concepts in Functional
Composer with this class since their math background included
graphs of quadratics. It presented an interesting opportunity
to introduce the graph of the sine function, a topic they had
not yet studied in their regular classes. I demonstrated to them
that even though they did not yet completely understand the function
itself, the principles learned in Functional Composer would enable
them to identify the formulas for various transformed trigonometric
graphs when referencing the parent graph of y=sin x. After Functional
Composer, we spent a significant amount of time working through
two games of Name That Function. This became very challenging
(as it is for most students) and those with the most musically
developed ears clearly had the advantage. I could see studentsı
frustration here when confronted with a non routine problem with
no previously studied method available. This problem solving temperament
of the students also surfaced with the 9th graders in another
context.

Grade 9

I began these classes with a special
problem unrelated to music, and followed it with Functional Composer,
Name That Function, and then a condensed, adapted version of Scaling
the Scale, parts I and II. The special problem, "Corey the Camel,"
is based on strategic thinking and logic, and asks students to
determine the maximum number of bananas that a camel can get to
a market, given: 1) the market is 1000 miles away, 2) the camel
must eat 1 banana per mile of walking, 3) the camel can carry
only 1000 bananas at a time, 4) there are 3000 bananas that need
to be transported. I chose to begin my work with this problem
for two reasons: 1) to learn about the studentıs problem solving
style and 2) to establish a classroom culture of inquiry, cooperation,
and teacher as facilitator. The problem was borrowed from "The
Interactive Mathematics Program," a progressive integrated math
curriculum published by Key Curriculum Press. Corey the Camel
is one of genera of math problems referred to as "non routine"
and open ended problems that are given to students to solve over
time, often as a "Problem of the Week." In this case, we devoted
one class session to work on the problem, completed it at the
beginning of the following session. It revealed a great deal to
me regarding the studentsı temperament and approach to problem
solving (which had also surfaced the 8th graders). In many ways
they reacted as students in the US would, eager to solve it, and
frustrated that it did not fall into some previously studied formula.
Since the studentıs experience with this problem was condensed
in time, I gave hints more quickly than I normally would have.
Some students almost solved the problem on the first day. They
continued to ask me in the halls about the solution that afternoon.
They had obviously discussed with their peers, as other students
asked me about the problem. It frustrated them, but intrigued
them. At the end I suggested the possible extensions of generalizing
the problem for all parameters and creating a formula.

It is very difficult to make a generalization,
but these students, clearly very adept in their usual math studies,
felt less comfortable with this problem than the American students
that I teach. I expected this. It is no mystery that confronting
non routine problem situations requires practice with non routine
problem situations. This was not the type of practice these students
had had. More is discussed on this topic in the summary to follow.

Functional Composer was very effective
and the mathematical knowledge and ability of these students allowed
me to make connections to specific mathematical applications they
have studied in their texts. These connections were not as obvious
for the students as I expected however. Their performance led
me to believe that the math curriculum in their schooling did
not emphasize graphing to the extent of many curricula in the
US. In any event, we applied the music composition concepts to
quadratic functions and trigonometric functions.

Originally designed as a directed
discovery activity to last 100 minutes, I conducted Scaling the
Scale Parts I and II as a presentation, and condensed it into
45 minutes. Many of the ideas were new to the students, however
not to all, but the context was new to all. I used my guitar to
demonstrate harmonics, discussed timbre, frequency ratios, Pythagorası
method to determine scale tones, and the development of the tempered
scale and the logic and mathematics therein.

Grade 11

This class did Corey the Camel, Functional
Composer, (Movement I and II), Name That Function, and Scaling
the Scale Parts I and II. I also briefly demonstrated the activity,
Inside Out. Each activity moved more quickly than other classes
due to the advanced level of these students.

Teacher Workshops

With the expressed goal of preparing
the teachers to teach as much of my curriculum as possible, I
began the first day's session with a workshop based on a variety
of topics from the introduction of Functional Melodies. This included
general theoretical and philosophical points that my work is based
on as well as methodologies, learning pathways and expected learning
outcomes for students.

On the days that followed I conducted
a series of Functional Melodies activities with the teachers.
These were structured to treat the teachers as though they were
a class of students. I paused for discussion points along the
way to elaborate on pedagogical considerations and issues presented
in the opening session. In side discussions we considered such
topics as how to adapt to different grade levels or music/math
expertise on the part of the students, as well as connections
to the primary math or music curriculum that teachers could capitalize
on. While many of these topics are covered in the teacher text
of Functional Melodies, some are not, particularly the adaptations
for elementary grades. These adaptations included "name that graph"
activities, variations of the worksheet formats to larger sizes,
and ways to change language and pacing. All together, the activities
we covered were Sound Shapes, Multiples of Drummers, Functional
Composer, Name That Function, Scaling the Scale Part 1 and 2,
Record Producer Algebra, and a brief overview of Measures of Time
and Inside Out.

The final session was directed toward
general theory and methods of interdisciplinary project based
curriculum design and implementation. The curricular examples
used did not involve music and mathematics. I used a variety of
curricular examples that I developed Homestead High School in
Cupertino, California for the Foundation Integrated Studies Program
(FISP) under a special Annenberg school reform grant. These were
integrated units of geometry, physics, algebra and biology for
9th and 10th grade students. General pedagogical topics covered
or introduced in this final session included:

1. Discovery based learning and constructivism

2. Multiple contexts for access and understanding

3. Multiple intelligences theory and Howard Gardner

4. Strategies and benefits of cooperative group work

5. Role of the teacher as facilitator and coach

6. Student status relationships in the classroom

7. Problem and project based learning

8. Interdisciplinary transfer

9. Thematic based curriculum and essential questions

10. The role of culminating projects

11. Curriculum spiraling and layered mastery

12. Interdisciplinary curriculum designs

As previously mentioned, the teachers
who attended any given workshop varied, however several music
and math teachers were consistently present for most. My interpreters
were very effective, and for the most part teachers were very
attentive, taking notes and participating in the activities. I
did my best to pause for side discussions as the activities progressed
to address teaching issues.

Classroom Observations and Collaborations

In addition to a busy schedule of
teacher workshops and student classes throughout my visit I was
able to observe a variety of classes taught by the teachers at
the college. The following observations occurred on the days following
my initial observations of the mathematics classes.

I first observed a 3rd/4th grade
class taught by Olga, the school psychologist. Students were involved
in an interesting spatial reasoning type of puzzle activity. They
were given a colored graphic design and the task was to recreate
the design with a set of cubes that had various colors and shapes
on each surface. I joined in with the class. We then did some
tangram puzzles, some of which were very challenging. Olga gave
me a set of the tangrams, which she said were created by a famous
Russian family many decades ago that traveled around the Soviet
Union providing educational materials to children. I plan to incorporate
these into a group dynamics activity I use in my school in the
U.S.

I also visited the computer lab and
had several discussions with Martina, the computer instructor.
I had to use an interpreter here, and the communication was a
little difficult. She was very interested in ways that the computers
could be used in integrated music-math activities. I first attempted
to discuss MIDI (Musical Instrument Digital Interface) with her
but that was a little difficult, and any fruitful application
would depend on a fair amount of software and some hardware which
did not appear to be present. We then moved to some topics that
actually were more relevant to the type of integration we were
working on. We experimented with the Microsoft Excel program to
find meaningful ways that students could represent melodic graphs.
We programmed a starting motif as a number sequence and applied
a wide variety of mathematical operations on it in the manner
of Functional Melodies. It soon became very apparent that the
computer would allow us to experiment with a vast number of different
functions; natural logs, trigonometric, cubic, etc. In order for
the resulting function values to be meaningful musically (to be
within a playable range) we would adjust coefficient values and
experiment with other operations that would adjust the shape of
the graph in ways that seemed interesting and had the potential
to be musical. We discussed how many composers of "new music"
of the 20th century use such techniques to compose, and that it
could be a very interesting project for her students to work through
the behavior of various mathematical functions in the context
of making musical judgments. I shared with Martina that I have
been working on a project to create a software that combines the
graphing and computational capacity of Excel with a music sequencing
program and tone generator, so students could immediately hear
the music created by their mathematical functions. I will definitely
keep them informed of this as it develops.

I then observed a very interesting
4th grade mathematics class that was integrating music. The teacher
had several large posters with musical motifs written on them.
Some posters were duplicates of others, and others were different.
The teacher would put up a combination of musical motifs, and
the students were asked to draw the geometric figure that represented
the set of motifs. For each exercise a student from the class
was asked to go to the piano and play the motifs so that the musical
"period" could be heard as well as seen in notation. Initially
this appeared similar to my activity, SoundShapes, but it was
distinctly different. The emphasis here was on form. In this activity
four repeated identical motifs (AAAA) were represented by students
as a square, three identical motifs (AAA) as an equilateral triangle.
The exercise then got more complex, with an ABA musical form being
represented by an isosceles triangle, and an ABAB form represented
by a rectangle. Students all seemed to generate the same geometric
figure for each form, as though it was clear to them the type
of interpretation the teacher was seeking. This continued for
some time to include more complex geometric shapes. I was especially
impressed with the level of abstraction that the young students
were working with. The activity then moved to some pure geometrical
area problems.

A music theory class I attended proved
to be very interesting. This teacher was responsible for mentoring
Allehan Aubakirovaıs special music and mathematics project that
had recently won a major award in Almaty. This was an 8th grade
class with only a few students present. After viewing her explanation
of pivot chords between relative major and minor keys, the teacher
and I had an extended discussion on the mathematical relationships
within the cycle of 5ths. She demonstrated a number line method
that she created to help students see and use the mathematical
"distances" of various key centers on the cycle of 5ths. She gave
me a poster that demonstrated her work.

Interviews

In addition to ongoing conversations
throughout the visit, several more formal interviews were held
during my visit, both of and by the staff and students of the
school. I conducted two interviews, one with Aina, Natalia (administrators)
and music teacher and another with Zhania Aubakirova at the conclusion
of the trip. I was interviewed by Zhania at the beginning of the
trip, by the students in a town hall meeting, and again informally
by teachers in various meetings. Most of these interviews were
video taped.

My interview with Aina, Natalia and
the music teacher followed several performances by some students.
It focused on some general aspects of the school and how students
are identified for special music training. I learned that the
college has a strong reputation in Almaty for being a high quality
school with a nurturing and intellectual atmosphere. The low student
teacher ratio and overall numbers made it a highly desirable environment.
Most students study music, but they are not required to. All of
the students go to a university upon graduation. Every student
begins study of English in 1st grade, in addition to another language,
hence upon graduation they are typically fluent in four languages
that include Kazak, Russian, and English. We also discussed their
use of technology. While I did not observe students using any
technology in the classrooms, a substantive computer lab was available
and Natalia and Nigmet informed me that most students are proficient
in Power Point, Excel and the school emphasizes technological
proficiency in their students.

My final interview with Zhania confirmed
a strong shared vision between us for the common purpose, motivation
and value of the integrated music and math teaching we were pursuing.
She expressed her belief that the "easy, divided world" is far
less interesting to students than the "linked and interconnected
world." The importance of the work was "obvious" to her. She also
acknowledged the challenges of the integrated vision, and how
it was clearly not in the mainstream of thought in Kazakhstan.
We both expressed the desire to stay in touch, and agreed that
visionary work that challenges mainstream views gains credibility
and can progress more rapidly when leaders connect and work together.

I was later interviewed by a large
group of students in a town hall atmosphere. This lasted over
an hour as the students asked me a broad assortment of questions
ranging from my personal life and hobbies to the war on Iraq.
They were fascinated to get to know as much as possible about
me. I took the opportunity to share with the students my philosophical
outlook on many topics, and we gravitated to the importance of
my visit that extended beyond learning music and math. I suggested
how the exploration of boundaries between disciplines shed light
on percieved cultural differences. The students were fascinated
with finding out about the essence of cultural differences. We
moved on to topics of globalism, the phenomena of a shrinking
world given exponential population growth and the internet. We
discussed how humanity is being challenged in unique ways in our
current age as we reach the threshold of a finite earth to absorb
waste and provide resources. Unique (interdisciplinary) ways of
thinking, and indeed, being and acting, will be necessary in the
coming decades to cope with the challenges we face. Music and
math is in some ways a model of that kind of thinking.

My final luncheon with the teachers
was very significant, a wonderful time and very heart warming.
Teachers expressed sincere and heartfelt gratitude and appreciation
for my work with them. It was another confirmation of how our
work together seemed to extend well beyond music and math. A teacher
gave me a toast that said it all in a very touching way. It gave
me the impression my visit had to some extent opened their minds
and provided inspiration. In this way I served as an ambassador
and representative of country through teaching.

Boy's Day Assembly

The students had a tradition to honor
boys on a specific day and girls on another day. Boyıs day included
an academy awards style assembly of award presentations for the
"best boy" in a variety of categories, e.g. kindness, gentleman,
academic, etc. The awards were interspersed with a variety of
performances ranging from modern, traditional, serious and whimsical.
It was marvelously entertaining and very revealing as to the character
of the students, their sense of humor, versatility of style, diversity
of culture, and commitment to high standards and discipline evidenced
by the impressive musical performances.

New York--Kazakhstan Student Exchanges

Before I left the United States for
Kazakhstan I made a proposal to the teachers in C.V. Starr Intermediate
school in Brewster, NY, (a school in the district where I am employed)
to generate an exchange between their students and those in Zhania
Aubakirovaıs College. As a result, forty Brewster fourth graders
created artwork, wrote notes and assembled photographs addressed
to "students of Kazakhstan," a country which at that time they
knew virtually nothing about. The messages were heartwarming,
all expressing the desire for closer connections between peoples
of the world, world peace and specifically, to begin a friendship
with a distant student in Kazakhstan. I hand delivered these packets
to the college when I arrived.

The students in Kazakhstan responded
enthusiastically with forty beautiful pieces of artwork, notes
and photographs of their own including home addresses, each requesting
to be "friends" and issuing messages of world peace and happiness.
The teachers and students in Brewster were extremely excited and
touched by the response. We will be displaying the Kazakhstani
studentsı work in a gallery in our school, and the students will
soon begin correspondence with their new "friends" from Kazakhstan.
This clearly promises to be the beginning of many long and enriching
relationships for years to come.

Students at H.H. Wells Middle School
also embarked on a curricular project which is currently leading
to student exchanges with Kazakhstan. Before leaving I wrote a
"webquest" curriculum for my students to follow during my trip
( see www.brewsterschools.org/cvstarr/sbeall and go to "Kazakhstan
Webquest"). The primary focus of that curriculum is for students
to learn about the nature of knowledge acquisition, specifically
with regard to learning about distant cultures. Students compared
different methods. Knowing virtually nothing about Kazakhstan,
students initially made hypotheses based on inference and their
prior knowledge. This was followed by internet research. They
then referred to a journal I posted on my website while I was
working at the college in Kazakhstan. On my return with I shared
1.5 hours of video and discussed my visit with the students. A
powerful theme that emerged was the realization that many cultural
differences are shallower that we might initially believe, and
that our common humanity is a strong unifying force. These students
have expressed a strong desire to travel and learn more about
the world on a first hand basis. The first step will be to establish
email relationships with students from Zhania Aubakirovaıs College.

Summary/Reflection

Two weeks is just a snapshot into
the life of any school. It is important to emphasize that the
comments that follow in this section are observations and impressions
I received from my visit regarding the culture of teaching and
learning at the college, not conclusions or verified facts. I
also gained insights into the intricacies of interdisciplinary
teaching in general, as well as specific aspects of my and music
and mathematics curriculum. While many of these insights are very
compelling and suggest a focus for further research, they too
are observations and case studies only, (not formal research findings)
and must be considered in that light.

The work begun during my visit to
Kazakstan is still in progress in many respects. As I mentioned
to the teachers and students upon my departure, the end of our
two weeks together was just the beginning of another vital stage
of the work. We now can begin the process of reflection and application
of the skills, knowledge and insights gained. It was obvious to
all involved that a follow up visit in the future would greatly
enhance the effectiveness of this visit toward realizing the goal
of enabling the teachers at the college to confidently teach Functional
Melodies and incorporate some of the pedagogical principles into
their overall practice. I told the teachers I would be happy to
have conversations by email over the coming months to answer questions
and give advise. Given the language barrier however, it is questionable
how realistic or productive that could be. Also, in the final
teacher workshop, many teachers expressed a great deal of interest
in the topics presented. A follow up visit would allow us to pursue
some of those topics in a substantive way, increasing the extent
to which they would significantly affect the teachersı practices,
and the learning experiences of the students.

School Culture

When the students asked me "what
I thought of their school," my first and most direct response
was it was a "dream school." With 10 students per classroom and
a total of approximately 140 students in the entire school of
grades 1-11, the level of personalization was maximized. Myself
included, many would argue that this in itself is one of the most
influential factors for a "successful" school. The school size
and intimacy created a wonderful family atmosphere. Everyone knew
everyone else, and teachers could follow students over their entire
academic life. The benefits of this are immeasurable. This fact
alone makes it very difficult to draw fair or accurate comparisons
to public schools in the United States regarding the effect of
teaching methodology, content and style. The dynamic created by
these numbers affects every aspect of learning. Indeed, many curriculum
designs, programs and classroom strategies studied by teachers
in the U.S. and elsewhere are created to deal with large heterogeneous
classrooms. In a setting like the college in Kazakhstan, many
of these simply do not apply.

The Students

One of the highlights of my visit
was spending time with the wonderful students at the college.
They were extremely respectful, warm, and diligent, and they radiated
a grounded sense of purpose.

In art, music and mathematics, the
student work I observed was of exceptionally high quality for
each grade level. Many of the musicians played with a remarkable
level of maturity and expression. The artwork was equally impressive.
Students at grade 3 and 4 showed an impressive command of watercolors,
that was expressive and thematically mature.

In mathematics studies and their
general classroom demeanor they were highly organized and motivated
to succeed, clearly taking their studies very seriously. In some
of the higher grades I detected some hesitancy from the students
when they faced non routine problems and situations that required
them to take the lead. This was inevitably connected to a bit
of shyness on their part, undoubtedly accentuated by having an
American stranger teaching them in their classroom, however I
believe this characteristic also reflected the culture of the
school pedagogy as well as their society at large. The students
appeared to be accustomed to a very teacher-centered pedagogy
with less emphasis on inductive, discovery based teaching. They
respected teachersı authority as the expert in all cases, eagerly
waiting to be explained to and shown the way. In some respects
this made them less active risk takers in the classroom, less
proactive, and less apt to take ownership of their own ideas and
creative insights. Much can be said on this topic, but it highlights
a cause and effect relationship where overly prescriptive and
formulaic approaches to the learning of mathematics can inhibit
studentsı ability to creatively problem solve and approach non
routine situations with confidence. This is a consideration in
the U.S. that drives many math education reform efforts.

A quality in the students that I
was struck by was their readiness to consider higher connections,
abstractions and philosophical dimensions of the subject matter.
It was as though they did not feel bound to evaluate everything
being learned in terms of its instant and/or obvious practical
application. There seemed to be an atmosphere of intellectual
risk taking and willingness to wonder about interesting ideas
for their own sake. I believe this stemmed from a culture of learning
that was deeply rooted in an appreciation for the virtues of intellectualism.
This was very refreshing, and an interesting counterpart to the
studentsı shyness and reluctance to be proactively creative in
other respects. Their acceptance of intellectual adventure was
tied to their dependence on the teacher however; they appeared
to need the teachersı "permission" and guiding hand to be led
down that path. Once the hand was extended, they surprised me
in the extent to which they engaged and wondered about abstract
ideas, such as the relationship of subjective and rational ways
of knowing that comes to the fore in activities such as Functional
Composer. Likely, because of the studentsı trust and respect of
my role as teacher, I found it easier to bring them to higher
levels of thinking about the subject matter than I have been able
to with the vast majority of my students in the U.S. I also noticed
this quality in many of the classrooms I observed as well, especially
the music-math class where students made abstract geometric models
of musical forms. At the time, I questioned how much of their
engagement was self generated, and how much they really understood
what they were doing. I wondered how much of it simply followed
out of their duty to be good respondents to their teacher. As
they progressed it became evident that they did know what they
were doing and the student-generated aspect became moot. It did
not matter how the ball got rolling, simply that it was rolling.
If it took a strong teacher directed experience initially, should
this matter? I think not.

In the United States it has grown
increasingly "fashionable" in progressive pedagogical thinking
to emphasize the teacher as facilitator vs. "sage," and to reduce
the studentsı overall dependence on the teacher for their learning.
This has noble intent and has become an important evolution from
the almost draconian practices of rote and mechanistic learning
in the 40ıs and 50ıs. My experiences with students at the college
was cause for reflection. I wondered, have we lost something in
the process? Has our pendulum swung so far that we have lost sight
of any value of "teacher as sage?" The Kazakhstani studentsı respect
for their teachers as intellectuals with much to offer them genuinely
facilitated their higher development. This trust and respect of
their teachers made them willing to venture into territories where
the relevance was not immediately apparent. This served them well
and yielded impressive results. Teachers in the U.S. often spend
an inordinate amount of time and creative energy devising ways
to "sell" content to kids. "Successful" curriculum is evaluated
often first and foremost for its ability to motivate students
and "make learning fun." Students, teachers, and even administrators
too often find themselves evaluating teachers and curriculum with
an almost blind emphasis on how student-centered it is, and how
much fun the students are having. Students watch this, and their
values reflect it, and it becomes self-perpetuating.

My experiences in Kazakhstan provided
perspective on what I perceive as an imbalance in our system in
the U.S. In teaching these students I did not get the impression
that they needed to be continually entertained. It was not my
responsibility to make the case for the relevance of the subject
matter at every turn. The students accepted and trusted my judgement.
This context had a liberating affect in some respects. With student
"buy-in" established at the outset, I could devote more time to
probing the subject matter, which in turn allowed me to bring
students to higher levels of thinking.

The Teachers

The discussion on students says a
great deal about the teachers. The teachers were very highly trained
in their disciplines. They were true intellectuals with very high
standards, and levels of commitment to their students and disciplines.
These values transferred to the students.

It was unclear to me initially what
the teacherıs background was with regard to the pedagogy and theory
of progressive education in the U.S. (topics addressed in the
teacher workshop mentioned earlier in this document). I soon realized
that they knew little of these things, at least in the formalized
and researched form that we are accustomed to. But in many ways
it did not seem to matter. These teachers accomplished great things
with their students, which is a testament to the effectiveness
of a strong expertise in a subject, great intuition, compassion
and commitment. Of course the very small class sizes (10 students)
and a self-selected student population (being a relatively expensive
private school) was a strong factor as well. But in any context
these teachers would be highly effective. It would be reasonable
to assume that their teacher training emphasized content knowledge
over pedagogy. And this led to reflect on the emphasis of much
professional development in the U.S. Teachers in the U.S. spend
a large percentage of their time in professional development courses
that emphasize methodology and teaching strategies as opposed
to learning pure content, and improving their expertise in their
discipline(s). This is cause to consider another imbalance in
U.S. educationemphasis on pedagogy over content knowledge and
expertise.

The teachers had other wonderful
qualities that Americans might envy. They expected a great deal
from students. They were not as bound to beliefs in the developmental
limitations of students at various ages. A problem I have run
into with teachers in the U.S. is the firm belief that students
at various ages are incapable of understanding or benefiting from
certain levels of abstractionthat they cannot understand, or
would be scared by, introductions to higher order thinking. The
Kazak teachers were more open to the possibilities of what students
were capable of, as though they naturally intuited Jerome Brunerıs
idea that all topics can be taught at all levels, as long as it
is presented in an intellectually appropriate form. They had no
formal knowledge of Bruner, but appeared to apply his ideas.

Teaching of the arts emphasized expression
and intention as well as technique. This led to students at very
young ages demonstrating highly mature work.

In the midst of all these strong
qualities, there were many ways the teachers could benefit from
what I was offering. While some teachers were involved with interdisciplinary
work in their Learning and Mastering project with music and math,
most were not used to the idea. Their orientation to mastery of
their disciplines made them very curious about how a single teacher
could actually teach two disciplines, especially when they are
integrated in a common curriculum. To summarize, they were challenged
by my work in two primary ways: 1) how to gain adequate confidence
and mastery in two disciplines to teach them in an integrated
way and 2) general confidence and buy in of the various pedagogical
principles as mentioned earlier in the teacher workshop section
of this document (e.g. constructivism, multiple intelligences,
differentiation, etc.).

I left the school reflecting very
deeply on these ideas, and how education in both America and Kazakhstan
seem to be in different phases of development and how each has
so much to offer the other. I strongly believe that the pendulum
needs to swing back a little in the U.S. education reform efforts
to recognize and value the teacher as intellectual and primary
source of knowledge, to place this in balance. In Kazakhstan,
the teachersı high standards and intellectual orientation would
be well complemented by the progressive student centered, authentic
learning pedagogies that I presented. Again, a matter of balance.
Continued collaboration would undeniably improve both educational
systems, and in turn, the education of many children in both countries.

The Conservatory of Music In Almaty

Zhania invited me to give a lecture
at the Conservatory of Music in Almaty. The audience consisted
of the staff of the conservatory and various other colleagues
of theirs. Few spoke English, so the entire lecture was translated,
phrase by phrase.

I spoke about the general intent
of my work, the theoretical basis and the larger implications
for the type of thinking necessary for humanity to solve the problems
of the 21st century. I sensed a mixed reaction of interest and
skepticism from the audience. As Zhania mentioned in an interview
format portion of the talk, they were very steeped in their disciplines
of music, in most cases being very ignorant, if not scared, of
mathematics. Initially I was surprised at the level of traditional
thinking in this group, but on second thought it made perfect
sense. They had been trained at time and in a culture that was
very traditional in its thinking and methods. One gentleman had
written a book on geometry and music and showed great enthusiasm,
however he seemed to be the exception. This group brought to mind
many of the challenges I have faced in the U.S. as well. The educational
system in the U.S. trains specialists, and few educators are actually
trained in several disciplines, or at least enough so as to give
them the necessary insight and confidence to consider an interdisciplinary
approach. This was a reminder of the longitudinal nature of reform,
that to effect significant changes in thinking regarding education,
be it interdisciplinary or some other, it needs to begin with
the early stages of teacher training. Staff development of established
teachers will always be a hit and miss proposition, with mixed
results, though certainly worth the effort. A true transformation
must be considered a long term project, and comprehensive results
cannot be expected sooner than a generation.

Integrated Disciplines and Transfer

Zhania Aubakirova expressed a strong
vision and belief in the value of interdisciplinary teaching,
especially music and mathematics. In my interview with her at
the end of the trip, she was quick to explain that, while not
being a trained educator herself, it was clear through her experience
in musical work that the application of mathematical principles
and laws are ever present. Indeed this is true, and I felt that
I was listening to myself speak during that interview. As a practicing
musician and mathematician myself, this central observation has
been a prime motivator of my work. Each teaching experience sheds
new light on the possibilities for how to utilize the connection
educationally and raises new questions for research.

The relationships between music and
mathematics can be thought of educationally in two fundamental
ways: as a relationship to know, or a relationship to use. The
first approach considers the connection of music and mathematics
as a knowledge base unto itself. Students learn can learn about
the myriad ways music and math interrelate, some more meaningful
than others, some more relevant or practical than others, and
some just interesting curiosities. The second approach utilizes
the connection as a pedagogical tool; using the medium of one
discipline as a vehicle to enhance understanding in the other,
either using music as the vehicle with math understanding as the
goal, or vise versa. Here the question of interdisciplinary transfer
arises, and whether or not it is even possible as many theorists
and researchers debate. In my work with the 3rd grade classes
I gained some insights into this debate. Before discussing that
it would be useful to briefly summarize come concepts of transfer.

I acknowledge three primary ways
that interdisciplinary transfer can happen. I refer to these transfer
types as true transfer, readiness transfer, and indirect transfer.
True and readiness transfer are facilitated by an interdisciplinary
curriculum design where indirect transfer does not. In the case
of music and mathematics, true transfer learning begins initially
in the music medium. A principle is expressed in the medium of
music and learned as a musical experience. Once the principle
is learned in this medium it can be perceived by the learner as
an entity unto itself, expressible in various ways. It is translated
to the mathematical medium and then expressed and understood as
mathematics, in mathematical language. This process allows musically
adept students to more easily access the understanding of the
principle. It is something like teaching a math topic to a person
in their native language and once learned, translating it to expression
in a language that they have less command with. This transfer
depends on utilizing special connection points between the disciplines,
and is not possible for all principles in each discipline, only
for those shared principles. Learning through readiness transfer
occurs primarily as a mathematical experience that is facilitated
by a musical context. Examples of this might be using songs to
remember math formulas, using musical subjects for math problems,
etc. Also, studentsı love and interest in music renders the mathematics
more friendly and lessens the trauma and perceived difficulty
that is often associated with learning math. Here the primary
cognitive function is mathematics, but it is effectively dressed
up in musical clothes. Indirect transfer is not controlled or
directed by a curriculum or program. It happens within the student
by an indirect symbiosis, such as the case where a student studies
music and mathematics independently in school, and the learning
gains in music, by nature of their cognitive similarities to and
overlap with mathematics, have the affect of improving the students
abilities in mathematics. Much research remains to be done to
inform these ideas however some exists, with varying degrees of
conclusivity in the results. The essential idea of true transfer
has been presented by Howard Gardner in *Frames of Mind*.

Utilizing true transfer in the classroom
is no easy task. Functional Composer is an activity that most
directly utilizes true transfer, however I have come to learn
that the extent to which transfer happens can be largely dependent
on the students' strength in the discipline used as a vehicle
(in this case, music). This can present some problems for heterogeneous
classrooms where a significant percentage of the students are
not musical, or have no musical training. For non musical students,
transfer of the mathematical principles in Functional Composer
(e.g. meaning of a function, graphs, and characteristics of various
families of transformations) can occur, but not as predictably.

In common practice most classrooms
are heterogeneous with respect to students' musical strength.
In my action research with Functional Melodies over the past several
years I have had the opportunity to work with two exceptions,
Williams Syndrome students at Music and Minds (University of Connecticut)
and Berkshire Hills Music Academy (Hadley, MA) and now, students
from Zhania Aubakirova's College in Kazakhstan. William's Syndrome
students have a strong sensitivity and natural feeling for music
but are very low functioning mathematically. True transfer with
Functional Composer was very successful with these students (see
"A Case Study of Teaching to Multiple Intelligences," Williams
Syndrome Journal or at www.scottbeall.com). The students at Zhania
Aubakirova's College were high functioning in both disciplines.
This presented a new opportunity to explore these ideas from an
action research point of view.

I gained a some strong insights
on the nature of transfer or what I might call symbiotic curricular
resonance while working with the combined 3rd and 4th grade class.
Initially I doubted that Functional Composer was appropriate for
them. After all, most math teachers would immediately dismiss
the idea of teaching functions and transformations to 3rd and
4th graders. In addition, these students did not appear to have
studied graphing much at all. I decided to proceed however, and
the results were remarkable. All of the students grasped the ideas
very quickly and created perfect graphs. They were able to characterize
the families of transformations (translation, reflection, stretching/shrinking)
and make predictions, and seemed to have no problem with function
notation and used it to make algebraic representations of the
transformations. Their proficiency with these tasks appeared to
follow directly from their musical ability, and by their conceptualization
of a mathematical function as a musical motif. The melodic variations
were natural for them, and their ease in understanding them transferred,
simply and elegantly, to the concepts of mathematical transformations.
This activity was effectively a first time introduction to graphing,
functions and algebraic representations, and the students performed
famously!

This was an exciting moment for me
and cause for reflection. It was clear that the students were
able to grasp advanced concepts and perform at a higher level
by virtue of the integrated aspect of the curriculum. The integrated
approach proved not only more efficient, but may have actually
facilitated learning to a level that separate study could not
have achieved. I received the inescapable impression that a sort
of symbiosis was taking place. And in regard to transfer through
integrated curriculum, this suggests that it can happen most effectively
when students have the prerequisite strengths in both disciplines
separately; that there needs to be some baseline ability in the
disciplines initially for an interdisciplinary experience to generate
transfer. And once the baseline is established through traditional
curriculum, a interdisciplinary curriculum can increase learning
exponentially in ways the separate curriculum cannot. Such gains
are not limited to improved understanding of complex ideas in
the disciplines at an early age, but include an understanding
of a new knowledge base as well--that of relationships and interconnections.

My work to date with music and mathematics
has focused primarily on using music as a medium to teach math,
as opposed to teaching music with math as the medium. This focus
was not chosen because of any indication that it was the "best,"
most efficient, or even the most useful application of integrating
music and math. It has in some senses been a starting point. It
could be that using mathematics to teach music could be more relevant
in traditional school settings by virtue of the fact that all
students have a baseline ability in math, the vehicle medium.
I have already begun this work at C.V. Starr Intermediate school
in Brewster, NY, using mathematical methods to teach young people
music theory and improvisation.

Integrating Cultures,
The World, And The Universe

The meaning and import of this collaboration
extends beyond integrated music and mathematics curriculum. A
broader theme, almost unwittingly, emerged and surrounded all
of the work as it progressed. Questions, ideas and awareness of
boundary constructs and perceived differences between all realms
(e.g. language, culture, ideology, religion, national boarders,
musical genres, etc.,) emerged and became conspicuously present
in the work. Our work with music and mathematics suggested a universal
principle that many of the differences and boundaries we embrace
daily can soften and become less distinct upon deeper study and
contemplation; that there exists a deeper essence to most phenomena
that somehow binds it all together. We considered that many of
these boundaries are in fact human constructs as opposed to some
absolute reality, and are open for reconstruction. As this was
revealed to be true through the study of the interrelationships
between music and math, the work became a catalyst and access
point that raised all the participantsı awareness and curiosity
about these more broad and general ideas. When Zhania and I met
initially we agreed that one of the important benefits of integrating
music and mathematics was to instill in young children this sense
of the unity and interdependence of all phenomena in the universe.
In our work to accomplish these ends, the teachers and myself
found ourselves to be the students of our own teachings. Our awareness
and understanding of these ideas grew through efforts to teach
children, and in this process we recognized that another boundary
had begun to dissolve right in our midst; the boundary and distinction
between "teacher" and "learner."

Consider this: The evolution of human
consciousness from premodern world views to modernity has been
characterized by a differentiation of the value spheres of art,
science, and morals. In our current post modern world these differentiations
have proceeded to complete dissociation, creating an unhealthy
condition; science and religion are at odds, secularism struggles
to come to grips with the human need for meaning, the subjective
fights the empirical, the head fights the heart, and so forth.
Today, the tendency toward interdisciplinary thinking can be seen
as an historical inevitability; as a part of the necessary consciousness
evolution to reintegrate and establish a healthy balance between
art, science and morals; heart, head, meaning; subjective, rational,
and spiritual, etc. The pull for this reintegration is strong
and evidenced in many facets of life including "green" environmental
movements, interfaith societies, holistic medicine, integral business
management systems and integral education, to name a few. In this
context it seems only natural and inevitable that a concert pianist
from one side of the world would share a vision with a music and
mathematics educator from the other side of the world, seek each
other out, and teach, not only the common essence of music and
mathematics, but the universal connectedness of the universe itself.
Inevitable? Maybe. In any event, raising awareness of the illusory
and conceptual nature of human boundaries and differences might
be just what the world needs most right now.

Recommendations For Further Collaboration

A follow-up trip to the college would
be highly effective toward maximizing the benefits from this trip.
In many respects this trip was a first step that served as an
orientation and overview of the possibilities of integrating music
and math while providing an introduction to a host of other progressive
pedagogical principles. After the teachers have worked with the
materials and processed and applied some the ideas presented,
specific areas of need can be targeted to ensure practical implementation.
As previously suggested, the ideas of music and mathematics integration
are very new to many teachers and are counter to many long held
habits and beliefs about teaching and learning. Even for teachers
trained in some of the pedagogical ideas of discovery learning
and project/problem based curriculum, the interdisciplinary piece
is challenging. Change can only expected to be incremental, and
must evolve over time.

A follow-up trip
would include:
- Mr. Beall team teaching with teachers from
the college
- Professional development workshop content
informed and planned from email consults regarding the progress
of implementation of concepts and curriculum from the first
visit.
- Follow-up exchanges and development of co-curricular
student projects between Brewster Schools in New York and
the Zhania Aubakirovaıs College in Almaty.
- Other programs developed through email correspondence
over coming months as the need is expressed by the teacher
and recommended by Mr. Beall.

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