A Case Study of Teaching to Multiple Intelligences-- Music and Mathematics
by Scott Beall
From the moment I set foot into the classroom to begin
my teaching career I have been drawn to explore the potential, value
and complexities of interdisciplinary curricula. As a young teacher
I was struck by a fundamental and basic observation; students too often
receive an anemic, sterilized and disjointed view of the world through
their schooling, partially as the result of knowledge and skills being
differentiated into discreet subject areas. Content is rarely, or often
inadequately, re integrated into its original, authentic and whole state,
rendering meaning and relevance scarce. I looked further. I became intrigued
with how interdisciplinary instruction might in fact be able to open
access to students of diverse "abilities" or "affinities" and enhance
understanding for all students. Inspired by the work of Howard Gardner
and his theory of multiple intelligences, my questions took on more
focus: Could one discipline be used as a medium to teach another discipline
in real practice? To what extent could real instructional experiences
be created that would allow transfer to happen on a substantive level?
And might it be true that in some instances a principle may be understood
more deeply when taught in a medium not typically associated with it,
as opposed to its "native" discipline?
My insights as a musician and mathematician suggested
an obvious context for my work. Music and mathematics integrate naturally,
yet in practice speak to very different parts of a students' experience.
For a large majority of adolescents, music is an ultimate expression
of their identity, associated with passion, mood and emotional release,
while mathematics is a tool one must learn to use to survive and get
by, often difficult and boring, even scary. As a high school mathematics
and music teacher, I created integrated math and music activities which,
among other objectives, spoke to my questions of interdisciplinary transfer
and multiple intelligences teaching. These efforts are now in the form
of a supplemental curricular publication (Functional Melodies--Finding
Mathematical Relationships In Music, Key Curriculum Press). The work
is designed to accommodate the constraints of course definition and
content coverage found in high schools, and to a lesser extent, middle
schools.
In June of 1999 I was invited to the University of Connecticut
in Storrs to participate in "Music and Minds," a 9-day special learning
program for students with Williams Syndrome. Williams Syndrome is a
relatively rare genetic defect affecting a wide assortment of cognitive
functioning. Of particular interest and relevance for my work was the
Williams subjects' marked affinity for and skill with music, and deficiency
in spatial reasoning and mathematics. The combination of these characteristics
made them ideal candidates to explore the potential of transfer in teaching
to a weakness though the medium of a strength, specifically in this
case, the extent to which mathematical concepts could be taught to the
students through the medium of music. I was excited that absent the
constraints of the high school setting, I could explore the principle
of interdisciplinary transfer more completely and in a pure form. Over
the nine day program, I could first teach a concept in a musical context
and then, incrementally, through a carefully planned sequence of activities,
the students would find themselves expressing the same ideas mathematically.
The goal would be to make the transfer as seamless, organic, and intuitive
as possible, so that no great leap would be involved in the transfer
of the musical to the mathematical expression and application of the
concept.
Music and Minds was primarily a music and performance
program with one class of mathematics, which I designed and taught.
I was unsure of what to expect going into this project. I had never
met Williams students before, but had read assorted literature describing
their "roughly second grade mathematical abilities" and "strong musical
abilities, or at least, affinities." Thus it was clear that while I
could map out a prospective curricular plan and establish a unit problem
(goal) for the 9-day course, a great deal of improvisation would be
inevitable as I got to know the students and their specific abilities
and needs. Many of the activities were adaptations from my book, Functional
Melodies--Finding Mathematical Relationships In Music. In the end, some
completely new activities were developed that have become the basis
for a series of complete units based on music and math integration.
Math Objective
The specific target outcome in math skill and understanding
was for the students to be able to make and interpret a graph on two
axes, the specific application being the daily temperature at UConn
for 7 days of the program. Each day of the class the daily temperature
would be noted and entered into a data table on the board. The data
table would be visible each day, and its creation over time would provide
a great deal of familiarity with the idea of ordered pairs of numbers
by the last day of the program when they would actually plot the points
and make the graph. The graph's creation on the last day would be their "exhibition" of sorts--the performance assessment for the program.
Math anxiety was considered as a significant factor in
the entire process. Without exception, the students suffered from acute
low self esteem regarding math and associated it with traumatic experiences.
Hence, particularly in the early class sessions, most of the emphasis
was placed on musical activities with the mathematical aspects kept
to a minimum. From the students' perspective, noting the temperature
each day was just a fun conversation piece that preceded each lesson.
They were specifically not told why they were observing temperature.
I did not want the students' thinking about the temperature chart clouded
by the fear of what they may be asked to do with it at this point.
The Program
The instructional strategy for the program fell into three
stages. Stage 1 engaged students in activities that focused on time
values in rhythm and their representations by a horizontal position
(horizontal axis). Stage 2, developed the notion of musical pitch and
its representation by a vertical position (vertical axis) and combined
this with the notion of the horizontal representation of time. Stage
3 applied the concepts used in the musical representations to a mathematical
application.
Stage 1, The Horizontal Axis As Time, Days 1-4
Our first activity, "Multiples of Drummers" involved clapping
various rhythmic patterns in groups and representing the claps on horizontal,
scaled lines. The class was divided into two groups, each group simultaneously
clapping at different beat intervals. The objective was to determine
the beat number where the patterns would sound simultaneously by observing
both aurally and graphically. The beat number of the simultaneous clap
would be noted as the least common multiple of the two primary rhythms,
and the resulting combined rhythm noted to be a "polyrhythm." For more
advanced students, various polyrhythms would be charted, multiples observed
and a general rule conjectured for finding least common multiples. This
objective was clearly beyond the level of the Williams students. Many
students were unable to accurately represent the clapping patterns graphically.
They had great difficulty tracking along a single line and placing dots
at the intersection of lines. In many cases they lacked the ability
to place dots in a consistent pattern, such as a dot every two scale
units. It was difficult to tell at this point if it was a problem with
hand/eye motor coordination of the physical placement of the dot, or
inability to conceptualize the pattern and transfer the idea of claps
to dots. The days to follow lent some insight into this. In any event,
I became wary at this point as to whether or not the temperature graph
was a realistic goal for these students. I seriously considered abandoning
it, but decided to persevere with the original plan.
In days 2-4 I kept an open mind as to whether or not
the temperature graph was a realistic goal. We continued with clapping
various rhythmic cycles, representing the claps with dots. More simultaneous
patterns were clapped and noted as polyrhythms. Student observers noted
where cycles overlapped (multiples of the primary rhythms). These exercises
were great fun with the students, and they improved on their ability
to represent the clapping patterns with dots.
After the polyrhythms, a rap game was conducted where
students had to determine an entrance point for a lyric line in relation
to an existing drum part. This required the use of graphic materials
as well, representing dots as beats and counting, adding and subtracting
beats (dots) to determine the correct entrance beat. The performance
nature of this activity was great fun, and continued to build on the
skill of using graphic representation to represent auditory patterns.
The students were having a wonderful time in class at this point, commenting
that "this is the best math class I've ever had," and cheering wildly
at each success. Classes would commonly begin and end with songs.
Each day a real world "problem of the day"(POD) was presented
at the beginning of class. PODs were analogous to the music problems
mathematically, and served as a warm-up for the class, while presenting
another context for the math they would be doing with the music. At
the end of the class the POD was revisited, solved, and connected to
the music. This was something of an experiment, and not the primary
focus of the lessons so I did not expect many of the students to make
the connection, though some did on various levels. A POD example would
be the following: Candy bars are 2 for $3. If you purchase 6 candy bars
and give the cashier a $20 bill, what will be your change? The corresponding
musical activity involved finding the entrance beat in a 25 beat drum
passage for a 4-beat phrase, repeated 3 times, which must end on the
last beat.
Stage 2, The Vertical Axis As Pitch, Day 5-8
The balance of the program followed a carefully designed
sequence of activities that used melody to establish the concept of
a graph on two axes. The following steps occurred over the 4 day period.
1) Qualitative Pitch Graph: This sequence began qualitatively,
establishing a pitch as "high" or "low", and connecting this idea to
the high and low area of a picture frame (graph without grid lines).
Melodies were sung and drawn. A vertical placement on the graph was
naturally associated with pitch value through their experience (high
placement-high pitch, low placement-low pitch). The horizontal axis
evolved to be the order of the note, which would later be associated
with time.
2) Introduce Grid Lines: Numbers were then assigned to
the pitches and a grid was added to the picture (graph). The C major
scale was used and each note was assigned the corresponding integer,
1-8. "Name that graph" games followed where a melody was played and
students were shown three graphs, one of which accurately represented
the melody.
3) Data Tables From Existing Graphs: The next day students
began analyzing melody graphs by locating the notes, determining the
ordered pair that represented them and entering the pairs on a table.
4) Graphs From Data Tables: Students reversed the process
in step 3 above and plotted melody graphs from given data tables.
5) Enrichment/Variation: To build on this skill in an
interesting musical context, the melody graphs were altered by the students
using the mathematical operations of adding 2 to each data value (pitch)
and then multiplying each pitch value by 2. This process created transformed
data tables which the students graphed and a musical composition was
played that utilized the transformed melodies with the original.
Stage 3, Transfer to Math--The Temperature Graph
On the last day the students were presented with the task
of graphing the temperatures that had been accumulating in the data
table on the board all week. The temperature data were very similar
to the melody graphs (by design) in that the melodies had 7 notes, and
the temperature table had 7 days. The only leap for the students at
this point was to work with different units; days and degrees (temperature)
as opposed to note sequence and pitch values The students performed
famously. Nearly half of the class had the graph accurately created
within 10 minutes with no assistance from the staff. Most others finished
with minimal assistance. This was a remarkable moment, considering our
doubts about accomplishing this at the outset of the program. We were
done with the lesson 20 minutes before the end of class!
Extension: To finish off the day we viewed a temperature
graph and a stock market graph from the newspaper. Maggie had brought
these graphs to our attention at lunch the day before, actually giving
us some advise on when it would be appropriate to buy stocks. She clearly
understood the graphs. The balance of the class faltered somewhat in
the ability to interpret the newspaper graphs, but I left convinced
that if the program were to continue, that our perceived limits of what
these students could accomplish would continue to be re drawn given
appropriate pacing, context, environment and support. I do not have
any hard data regarding their attitudes toward math, but all indications
are that the class was a great success in this area. The students loved
the experience and consistently commented (emphatically) how different
and fun this math class was. How deep and lasting it may be remains
to be seen, and will be born out over time. The curriculum created at
Music and Minds is the beginning of a new book I am developing targeting
elementary level students.
