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Math Elements in the
PianoKids® Method
by Peter Taussig
Math and Music
The link between math and music goes back to Pythagoras who in
the 6th century BC discovered the mathematical explanation for why
certain note combinations are more pleasing than others. That link
between math and music has endured for much of our history ever since.
During the Renaissance math was grouped with the sciences in European
universities. Only in the past 300 years has the link between music and
math been weakened as music came to be viewed more and more as an art,
not a science. Even so, many people have observed that mathematicians,
scientists, and doctors tended to engage in music at an unusually high
rate, and that children who took music lessons did better in math and
science in school. This anecdotal connection was validated in the 1990s
through a ground breaking study at UCLA by Gordon Shaw and his
associates. Their work demonstrated how music lessons enhanced
spatial-temporal skills in children and consequently improved their
math performance. The study spawned a vast array of research that
further corroborated this link between music and math.
Math and Piano
Of all musical instruments, the piano is the ideal beginner's
instrument. It is also the most effective tool to combine music
literacy with the development of higher brain functions. For one, it is
a machine with an obvious logical layout and simple sound production
interface. It takes no time for a beginner to find the notes and
produce the sounds. More importantly, we use both hands to play the
piano but require them to move independently of each other. This
ambidextrous quality contributes to the beneficial effect that piano
playing has on brain development. A keyboard instrument also offers the
best way to communicate with computers, which enables PianoKids
students to use technology to make the process more fun and more
effective.
PianoKids® - Piano Lessons for the Computer Age
The PianoKids method teaches children music literacy and piano
skills through computers and keyboards. It uses problem solving as a
technique for learning to read and play music, and frequently employs
math concepts to accomplish the solutions. The introduction and
reinforcement of math terminology and process in the context of a
fun-filled music lesson with computers makes it easier for student to
assimilate both the musical and mathematical concepts. Consequently,
math helps us teach music, and the music lessons help improve the
students' math performance. It is my hope that further studies will
prove conclusively what we now know only through observation and
parents' reports.
Following are some examples of how PianoKids employs math concepts in
its music literacy curriculum. Bear in mind that these complex tasks
are routinely performed by students as young as 6 years old.
Reading Numbers
In PianoKids we use electronic keyboards and software to learn
to read music and play songs. Various keyboard functions are involved
in the process and students learn how to control them. Function values
are displayed in a window with 3-digit numerical parameters (001
instead of 1). Controlling these functions requires that students be
comfortable with numbers from 0 to 999. This includes knowing the
hierarchy of numbers (e.g. 109 is higher than 98; 064 is lower than
164) and differentiating between similar numbers (e.g. 27, 207, and
270).
Examples of keyboard functions requiring number reading:
a. Song numbers, Voice numbers
Our pre-recorded songs on the keyboard correspond to page numbers in
the book. Young students initially have difficulty finding their page
and then accessing the corresponding file number on their keyboard.
This daily exercise helps them integrate the order of numbers.
The same goes for selecting the particular voice they want to play the
song with. There are over 600 voices on the keyboard. We ask students
to write their selected voice number in their song book and then find
it again. This requires reading and writing 3 digit numbers.
b. Tempo function
Songs are learned using the tempo function of the keyboard. The student
listens to the guide track and adjusts the tempo up or down. The
teacher will often tell a student to adjust the tempo to a particular
value. Once the song has been learned at that starting tempo, we start
working on speed, incrementally increasing the tempo "in 5s" or " in
10s". Going from 68 to 73 for instance teaches the student
experientially that 68 plus 5 equals 73. Tempo values range from 32 to
280.
c. Accompaniment volume
Students can adjust the loudness of the guide track and accompaniment
of their song. This parameter is expressed as a range of 1-127.
Timing
1. Fitting a pattern into a regular grid
Music requires adhering to a auditory grid we call "beat". Like a
visual grid, a beat is regular. In PianoKids this beat is provided
either by a guide track on the keyboard, a cursor and a click track in
the interactive software, or in live exercises, through students'
clapping. Against this steady backdrop the student must play different
rhythmic patterns that are of various length in duration. The task is
similar to fitting different bricks of equal length into a specified
space:
2. Fitting two patterns into a grid - duet or two-handed
playing
Students often play songs as a duet. This is a practical way of
learning to fit two patterns into a single grid. Students train to be
able to juggle three elements at once: 1. listen to a regular beat, 2.
listen to another pattern adhering to that beat, 3. fit their own
pattern to both the beat and the other pattern.
2. Fitting two patterns into a grid - duet or two-handed
playing
Converting length (space) into beats (time) is a challenge to most
students, even once they have understood the relative length of
rhythmic units. A software cursor that clicks on the beat while
advancing on a time line (or a teacher pointing and calling out the
beats) usually solves the problem and illustrates experientially
conversion from space to time.
Addition & Subtraction
1. Time Signature
- a. In a song, students are asked to add up the rhythmic
values of notes in each bar.
-
- b. We use a puzzle in which students have to add or
subtract values to achieve a total indicated in the time signature. In
the example below the total should be 4:
-
c. Students are asked to find as many different combinations of notes
that add up to the total number of beats according to the time
signature. -
d. We show students how changing the order of the notes counts as a
separate combination, even though the elements are the same and the
total is the same, because when you clap or play it, it sounds
different. -
e. Students are asked to come up with deliberately "incorrect"
combinations that violate the time signature. Making mistakes on
purpose is a core technique used in many aspects of PianoKids. Next,
the students find the "wrong" totals and then devise the most
economical way to fix it, by adding a note, deleting a note, or
changing a note. -
2. Ties
We create puzzles of various tied notes asking what the total
number of beats (counts) of the extended note are. These complex ties
are then played on the piano with students clapping the beats.
3. Scale fingering
We teach the scale fingering of C to C based on the
observation that there are 8 notes to be covered with 5 fingers. How
many fingers are we missing? (8-?=5 or 5+?=8). Once we have the answer
3, we play the scale as equations:
- Right hand upwards is 3+5=8
- Right hand downwards is 5+3=8
- Left hand downwards is 3+5=8
- and Left hand upwards is 5+3=8
Notice that RH upwards is the same as LH
downwards. "Can you guess why?" (our hands are designed backwards from
each other.) Such supplementary questions are part of pattern
recognition exercises (see below).
Multiplication, Division, and Fractions
1. Count-in clicks
In our pre-recorded guide tracks on the keyboard, each song is preceded
by two bars of count-in clicks. Students learn that songs in 4/4 time
have 8 clicks and in 3/4 time have 6. We explain it by drawing
"imaginary bars" before the first note of the song and then counting
the "imaginary beats" in those bars. 4x2=8, 3x2=6. Students understand
that there are always twice as many clicks as there are beats per bar!
2. Incomplete bars
Incomplete bars (pick-up bars) combine multiplication and subtraction.
We first figure out the number of clicks for a regular song according
to the time signature (see above). We then explain that the incomplete
bar is part of the clicks and comes before the first note of the song.
What this means is that the incomplete bar notes reduce the number of
clicks. But by how much? A common puzzle is: "How many clicks will you
hear before playing the first note?". The answer requires the student
to subtract the note(s) in the incomplete bar from the usual number of
clicks, e.g. in a 4/4 time song with an incomplete pickup bar of two
quarter notes (below), we subtract the number of beats in the
incomplete bar (2) from the usual number of clicks (8) to get 6 clicks
before starting to play (8-2=6).
3. Rhythmic values
Quarter notes, half notes, and whole notes, teach children the concept
of fractions. Our method also helps them remember the names of the
rhythmic values. We start with a whole note, equating it with a whole
pizza:
We then ask how we could split it up so that two kids would
get an equal share, and explain the concept of "half" a pizza:
What happens if there are 4 kids sharing the pizza? We explain
the concept of splitting a whole into 4 equal parts, each part being a
"quarter of a whole" (we also mention that the coin called quarter is a
quarter of one whole dollar, i.e. 4 quarters make a whole dollar):
4. Time signature
Students are often confused about the two numbers in a time signature.
When asked how many beats there are in 3/4 time for instance, the
answer is often 7 (adding 3+4). To explain time signature we replace
the bottom number with the symbol for quarter note:
This makes it clearer that the top number tells us how many
beats there are in the bar and the bottom number only tells us that we
are dealing with quarter notes.
A logical question then is: "Why do we write a 4 instead of the quarter
note?" The answer introduces students to the conventional notation of
fractions, as follows:
"In math we write 1/4 for "quarter" and it's the same in
music. If we had only one quarter note in each bar the time signature
would be: 1/4 (one quarter). If we had 2 quarter notes in a bar the
time signature would be 2/4 (2 quarters), and so on: 3/4 for 3 quarter
notes in each bar, and 4/4 for 4 quarter notes in each bar."
5. Dotted notes
Dotted notes help illustrate the concept of a value plus its half, e.g.
2 beats plus 1 beat (1 being half of 2). We start with a dotted half
note, explaining that the dot represents half of the note that it is
attached to. Because a half note has 2 beats, the dot represents half
of 2 which equals 1. "What note has just one beat? - A quarter note."
This dot represents a little quarter note that lost its stick.
When we add up the beats of a dotted half note we get a total
of 3, meaning that a dotted half note is held for 3 counts (or beats):
6. 8th notes
Learning 8th notes introduces simple ratios. We remind students of the
process we used to get from a whole note to a quarter note. We cut each
large note in half and get twice as many smaller notes, each smaller
note however is only half the value of the original note. Continuing
the process beyond quarter notes yields 8th notes. An 8th note has the
value of half of a quarter note. There are two 8th notes in each
quarter note. They are called 8th notes because there are 8 of them in
a whole note:
Pattern recognition, differentiation,
and modification
1. X-Y axis
Musical notation represents the relative pitch of notes on a vertical
axis (up is higher, down is lower), while the keyboard represents the
same relative pitch of notes on a horizontal axis (up is to the right,
down is to the left). The ability to translate in our mind the position
of notes on the page into their position on the keyboard is one of the
greatest obstacles a student must overcome. It is however also one of
the most useful skills that will improve performance in geometry,
drafting, and design. We use the display window on the keyboard as a
visual aid (it contains both the notation and keyboard position). An
even more useful aid is the interactive software we use, which prompts
students visually on the correct or incorrect of every note they just
played.
2. Sequencing
Memorizing patterns is the subject of many of our activities. We stress
the importance of remembering not just which notes make up the pattern
but their correct order. Accurate sequencing is fundamental to math,
biology, and chemistry, and useful in memorizing music. We have also
noticed its beneficial effect on mildly dyslexic children and children
with developmental deficiencies. Identifying and memorizing sequences
is also the first step in teaching students to write their own music.
3. Comparing patterns
Almost from the beginning we stress the need to scan each song for
recurring patterns before starting to decipher the notes. Not having to
figure out each individual note in a song is both smart and time
saving. Grouping single events (notes) into larger aggregates (note
patterns) and then identifying similarities and differences between
such aggregate patterns are important skills needed in many math and
science related subjects.
4. Modifying patterns
Our composition exercises rely heavily on variations, i.e. modifying a
pattern so that it will be different but still be recognizable as a
derivative of the original. Learning to identify the characteristics
that "mark" a pattern, and gauging at what point does a variant lose
its connection to the original pattern is a sophisticated skill used in
such diverse fields as statistical analysis and genetics.
5. Graphing inversions
One of the most common pattern modifications is inverting a pattern. To
help students recognize such inversions and later use them in their
compositions we sometimes ask students to graph the direction of the
melody. This helps in memorizing the piece but also introduces students
to the concept of graphing.
6. Intervals and chords - measuring distances between
pitches
A special type of pattern is a 2-note interval (two notes set at a
given pitch distance from each other) or a 3-note or 4-note chord
(several intervals stacked on top of each other). When played on the
piano, these pitch distances are easily identified as physical
distances between the keys.
7. Transpositions
Transposing a scale means to shift its beginning to another note while
maintaining the relative intervals (distances) between the notes.
Because the keyboard layout is not uniform (the black keys are not
distributed evenly but in alternating groups of 2 and 3), half steps
and whole steps are sometimes white-black key combinations and at other
times white-white key combinations (a half step is going from a key to
its nearest neighbor, a whole step is skipping the neighbor and going
to the next key after it):
Students must first analyze the structure of the original
scale by noting the exact sequence of half steps and whole steps, and
then reproduce the sequence accurately in the new (transposed) scale.
We combine this very complex mental process with listening exercises,
and train students' ears to be the final arbiter of whether their
transposition was successful.
8. Mental imaging
A different type of transposition is related to something called "chord
inversions", where we generate different sounding chords from the same
basic 3 or 4 notes. It is done by moving the top note of a chord to the
bottom, or vice versa, and requires students to memorize which notes
make up the chord and then mentally visualize their different
permutations before playing them.
Conclusion
To some musicians and music educators all this may sound dry
and not to the point. A Country-and-Western singer for instance may not
be aware of any connection between math and his music. In fact he may
have flunked his math class in school. And yet what he does is pure
audible math. The rhythmic relations between the notes, the pitches,
the way the string vibrates, the chordal intervals, are all phenomena
based on math and physics. You can arrive at the correct way of playing
without being aware of the underlying logic, but relying on intuition
to the exclusion of knowledge is limiting. That is why at PianoKids we
strive to combine musical intuition with musical understanding, our
ears with our mind. Our mission is to allow children's natural
curiosity and intelligence to explore the scientific underpinnings of
music and use them in acquiring their musical skills. It is remarkable
to see children as young as 6 being able to master these most complex
processes. Far from trying to replace the intuitive nature of music
with cold analysis and calculations, we harness the mind to enable
students to explore their creativity and musical intuition more fully.
It helps to remember that Pythagoras was not only a mathematician but
also a musician and a mystic.
©Copyright Peter Taussig,
2006
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