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Jazz and Math: Improvisation Permutations
Estimated Time of Completion: Depending on prior knowledge of musical notation, the lesson should take between 5070 minutes.
Summary:
After an introduction to improvisation through a drama game, a discussion, and a video clip, students explore how many different rhythmic combinations can be improvised in a jazz/blues piece of music. They use trial and error techniques, derive a mathematical formula, and apply the formula to calculate the number of possible rhythmic combinations.
Objectives
 To make connections between various types of improvisation and uses for improvisation.
 To observe that there are myriad combinations of rhythms to choose from when improvising jazz and blues music, and recognize that while the variations seem infinite, they are in fact finite.
 To estimate the number of possible variations given a number of rhythms to choose from to fill one 4 beat measure.
 To experiment with creating various combinations in an attempt to verify estimates (trial and error).

To derive a mathematical relationship that will allow students to calculate the actual number of possible musical permutations given the limited set of options to choose from.
 To compare the number of actual permutations with previous estimates and account for discrepancies.
 To gain understanding of scales and chords so that students can estimate the possible number of notes in a given measure of a 12 bar blues progression (any given measure uses one chord only).
 To calculate the actual number of permutations of notes per measure.
 To combine calculations for rhythms with calculations for notes to find the overall permutations possible in one measure of a 12 bar blues progression.
 To notate a 12 bar blues progression using a different combination of notes and rhythms for each of the 12 bars, and then perform it on a keyboard or virtual piano online.
Materials Needed
 VCR, TV, and PBS Ken Burns JAZZ documentary, Episode One "Gumbo." Verbal cue is "…a form called the Blues. And it's a useful form that is elastic because it is simple… " (24:48  25:17).

Examples of improvisations on tape, CD, or from the PBS JAZZ Web site.
Suggestions: For an example of early stages of improvisation, see "Afro" on Erykah Badu's Baduizm album; for more polished examples, try Ella Fitzgerald's Ella Live in Berlin "How High the Moon" which has a long and impressive scat section.
Other suggested pieces include: Ella Fitzgerald's The Intimate Ella "Black Coffee", or The Best of Ella Fitzgerald First Lady of Song "Can't We Be Friends?" and "I Won't Dance" (both with Louis Armstrong).

Fraction Notation Chart and Rhythms Worksheet

White board and dry erase markers AND/OR overhead projector, transparencies and marker

For the last activity before assessment: access to a classroom with headphones so students can listen individually to a piece of jazz music and try their hand at improvising various rhythms along with the music percussively (snapping, tapping, drumming on table).
Procedures and Activities
Introduction
Ask students to describe what improvisation means. Discuss improvisation in drama and explain that to get a feel for improvising, students will play a drama game. The game is entitled "You shouldn't have!" and it involves two people who will be improvising dialogue and pantomime. The first person mimes passing a gift to the other person (the gift needs to have a definite shape and size), who receives it, opens it and declares, "You shouldn't have!" followed by an improvised line telling what the gift is. The actors need to play off of each other because the person receiving the gift needs to make up a reasonable thing for the gift to be, considering how the giver pantomimed the shape, size, weight, etc. of the gift.
Next link the theme of improvisation to students' lives by eliciting examples of improvisation in everyday life (e.g., conversations with other people, dancing, etc.). Discuss what the prerequisites for good improvisation are. For example, to converse well, one needs to know the language fluently—think about how hard it is to improvise a conversation in a language that you are learning in school. To dance well one needs to have learned some dance steps and moves—think of how nerveracking it can be to get on the dance floor when you don't know how to do a certain type of dance.
Make the link to improvisation in music by listening to a piece of music that contains improvising, such as Ella Fitzgerald's How High The Moon. Ask for student impressions about the improvisation. How difficult do they think it would be to do that? Have they ever had to make up a story on the spot? Discuss how important and difficult it is to make up good details in both a story and a scat song.
Then watch the PBS JAZZ video clip from Episode One about the blues being simple and elastic, which allows for an infinite number of variations. After watching, discuss two key points:
 How many different variations can students imagine for Ella's scat, their dramatic improvisations, or for how the teacher conducts class?
 What does the concept of infinite really mean? How many infinite things can they think of? How do you know if something is infinite or just really extensive?
Doing The Math (Small Groups, Experimentation, Teacher Guidance)
Let's just consider the number of different rhythms that the jazz musician has to decide between when improvising. Look over the rhythmic notation and fraction note chart from the lesson entitled "Rhythmic Innovations." For practice, try the Rhythms Worksheet. Estimate how many different combinations of rhythms could be made to fill one measure.
Have students work in small groups to experiment with writing various combinations of rhythms. Encourage them to look for an efficient and systematic way to find how many possible permutations exist. Suggest that they begin by breaking down the measure into beats and try to figure out how many different ways one can notate a single beat.
They should come up with a fairly comprehensive listing of all the possible ways to notate one beat. If they miss something, supplement their list (a List of Ways To Notate One Beat may be printed and distributed). This list should go up on the board and be copied down by each group. (This list will act as the finite number of rhythmic possibilities that will be considered in making combinations of measures.)
Once there is a class list of the different ways to notate one beat, the groups are to experiment with combinations of four different beats to create unique measures of music.
After the groups have worked for a while, but before they are thoroughly frustrated, ask groups to read off their measures to the teacher or a scribe who will write them on the board. Nudge them along through the problem solving process by beginning to organize types of combinations. For example, when students give measures starting with a quarter note, write those on one side of the board separate from measures that start with 2 eighth notes. Then begin to ask the class, "how many of you have measures that start with quarter notes? Let's put all of those together..." (and so on).
Discuss any observable patterns in the measures. Ask, "How many different ways can a measure be completed if it starts out with a quarter note? What about if it starts with two eighth notes?" (This is rather overwhelming to figure out, so tell them that we can break it down and make it a simpler version of the same problem so that we can figure it out.)
Explicit Problem Solving
Show students how to break the problem down into smaller chunks, and how to create a simpler mathematical model to test the concept. For example, tell them that we will concern ourselves with only 4 different ways to notate a beat and then figure out how many different combinations are possible for a 2 beat measure.
Four simple ways to notate one beat are: a quarter note, two eighth notes, four sixteenth notes, or a dottedeighth sixteenth.
If we label the four different notations for a beat A (quarter note), B (two eighth notes), C (four sixteenth notes) and D (a dottedeighth sixteenth), then we can arrange them into groups of two systematically:
AA  BB  CC  DD 
AB  BA  CA  DA 
AC  BC  CB  DB 
AD  BD  CD  DC 
Essentially, there are 2 beats and one can choose between 4 different notations for each beat; therefore to find the number of possible combinations you multiply 4 x 4= 16. Or 4^{2}.
Now that the problem has been broken down and we have created a mathematical model and tested it out, we can apply our model to the problem at hand.
We created a finite and comprehensive list of all the ways to notate one beat. Instead of just four ways, now we have as many ways as are on our classcreated list (M). Also, we want to make measures with 4 beats instead of 2.
Essentially, there are four beats and one can choose between M different notations for each beat; therefore to find the number of possible combinations, M x M x M x M, or M^{4}.
Now students will apply this method to the problem at hand. Given a certain number of ways to notate one beat, how many different ways can one notate one measure?
Student Improvisation And Reflection (Individual Work)
The final part of the lesson will be for students to try their hand at improvising. This should be done with headphones so each student may listen closely to a selected jazz piece and improvise rhythms by clapping, tapping, humming, etc. (The headphones allow for less distraction.) Students will process this experience in part by writing a reflection piece about how a jazz musician must decide on rhythms in a split second when they are improvising. What does this tell us about the base of knowledge that gifted jazz musicians must have?
Assessment Suggestions
Reapply the method to the problem after adding the condition that any and all types of rests may be used (whole rest, half rest, quarter rest, eighth rest, and sixteenth rest). Students will need to figure out how many ways one beat can be filled with rests and then extend that to all four beats in the manner that they have been taught.
Extensions/Adaptations
For students who benefit from more visual and/or handson activities, the possible ways to notate one beat can be written on notecards. Then the students can rearrange the cards to create different permutations while another student, a scribe, or an aide records the permutations on a sheet of paper. For students with poor motor control, the notecards can be mounted on a thicker medium to make them easier to pick up and rearrange (foam core board or just thin foam rubber are two suggestions).
If students have difficulty organizing their combinations, the teacher can suggest creating a chart and model how to do it, or if need be offer them a premade chart to fill in ("Organizing Permutations" handout attached).
Standards
This lesson correlates to the following math and technology standards established by the Midcontinent Regional Educational Laboratory (McREL) at http://www.mcrel.org/standardsbenchmarks/index.asp:
 Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions.
 Generalizes from a pattern of observations made in particular cases, makes conjectures, and provides supporting arguments for these conjectures (i.e., uses inductive reasoning).
 Uses formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.
 Understands various sources of discrepancy between an estimated and a calculated answer.
 Uses recurrence relations (i.e., formulas expressing each term as a function of one or more of the previous terms, such as the Fibonacci sequence or the compound interest equation) to model and to solve realworld problems (e.g., home mortgages, annuities).
 Understands counting procedures and reasoning (e.g., use of the Addition Counting Principle to find the number of ways of arranging objects in a set, the use of permutations and combinations to solve counting problems).
 Understands that mathematics is the study of any pattern or relationship, but natural science is the study of those patterns that are relevant to the observable world.
 Understands that theories in mathematics are greatly influenced by practical issues; realworld problems sometimes result in new mathematical theories and pure mathematical theories sometimes have highly practical applications.
 Understands that science and mathematics operate under common principles: belief in order, ideals of honesty and openness, the importance of review by colleagues, and the importance of imagination.
 Understands that mathematics provides a precise system to describe objects, events, and relationships and to construct logical arguments.
About the Author
Amy Lein has taught mathematics and the performing arts for the past 5
years—but usually not at the same time! Being an advocate for involving
all of our multiple intelligences in education, she jumped at the chance to
develop lessons linking jazz music to mathematics. Currently she is
teaching math and science to high school students with special needs, and
continues to play classical violin and sing jazz in her free time.
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