The Standard Program

This program consists of two primary components with an optional third:

Classroom Instruction:
"A Journey Of Discovery Through Mathematics and Music"

This unit is balanced between music and mathematics in content and objectives. In traditional thinking it would be difficult to place it into a particular department within a comprehensive high school; it is truly interdisciplinary. Hence, as presented here it is best suited for self contained classrooms at the elementary level or for use in an after school program. It is designed for a heterogeneous group of students, adaptable for grades 4-8. The topic matrix presented provides an overview of the math and music content, themes and activity modes. A key feature of the unit is the ease with which students naturally develop the fundamental algebraic concepts of variable, function and graphing from the study of musical scales and melody.

Overview of the unit: A thematic question is presented with discussion at the outset of the unit: "Is mathematics an art?" As a route to answering the question students pursue a pathway that will lead to creating musical compositions with mathematics. First, polyrhythm activities (using tambourines and drums) that explore multiples and ratio are conducted and graphed by the students. Singing activities follow with students graphing melodic lines qualitatively and quantitatively. Students then give "names" to melodies, abbreviate the names, and perform basic arithmetic operations on the melodies (notes are represented as numbers). The results are graphed and characteristics of the graphs created by various operations (multiplication and addition) are observed to make general conjectures. The process is then applied to simpler melodies, the simplest being a major scale. Using an abbreviated name for the major scale (x) students find that graphs and algebraic expressions for linear functions emerge. Concepts of algebraic function, transformation and graphing are engaged by students naturally, initially in a musical language, later evolving as pure mathematical expressions. Students also learn fundamental concepts of melodic structure, notation, beats, rhythm and music composition. Student musicians perform their own mathematically generated melodies and discover that the mathematical process can solve problems for musical composers. Reflection, discussion and journal writing on the theme question, "is mathematics an art?" closes the unit. With guitar in hand, Mr. Beall's teaching style is entertaining and interactive.

To maximize student learning gains it is important for teachers to weave some elements of the music and math instruction into subsequent lessons. Teachers are provided with the support, activities and strategies for follow-up instruction in the inservice training component of this program.

Topic Matrix

Unit Theme Question: Is Mathematics An Art?

day	theme	math topics	music topics	activity mode
1	Introduction, theme question, music and math definitions, history, student experience journal entries, "The Multiples of Drummers" The Mathematics of Polyrhythms	counting, multiples, least common multiples, factors, ratio, patterns, problem solving	beat, rhythm, polyrhythms, phrasing, tempo, accents, performance	interactive, listening, ensemble performance calculating, graphing, journal writing,
2	"Multiples of Drummers" (continued) How do musicians use LCMs? How can mathematics help us understand other cultures?	observing patterns, making conjectures, creating a rule for LCM	phrase length, music of diverse cultures, elements of style	paper and pencil calculations, ensemble performance, listening, discussion
3	Melody Graphs: "Name That Graph!" Writing and Singing Melodies as Graphs What do melodies look like?	graphs on 2 axes, qualitative story graphs, quantitative graphs, interpreting graphs	scales, sight singing, melodic contour, melodic and harmonic intervals, pitch	interactive, singing, listening, graphing
4	"Functional Composer"(variation) A Mathematical Solution to Writer's Block Can mathematics be used to compose music?	functions, in/out tables, calculating and graphing transformations of graphs, symbolic representations (variable),	musical notation, music composition and melodic structure, ear training, transposition, modes, instrumental performance	interactive, listening, identifying and calculating, graphing, journal writing, instrumental performance
5	"Functional Composer" (continued) Class performances of math melodies Revisit unit question: Is math an art? What is creativity? Journal writes and discussion	linear equations and their graphs, transformations, problem solving	the major scale, music composition, instrumental performance	listening, calculating, instrumental performance, reflection--journal write and discussion

Student Learning Outcomes

Learning outcomes for the unit are categorized according to knowledge, skill, habits of mind (reasoning, reflection, interpretation, etc.), affect and product. Elementary students are often able to achieve outcomes traditionally associated with grades 9 and 10 as a result of processing musical material in an algebraic language. The level of their mastery of these topics will vary in depth, applicabililty and retention, however a layer of understanding at this point of development fascilitates algebraic understanding in later grades. Return to this page in the future for correlation of the outcomes to the NCTM Principles and Standards for School Mathematics.

	mathematics	music	interdisciplinary
knowledge	Students will know and have an understanding of: -Cartesian coordinate systems: ordered pairs, in/out tables, point plotting -a horizontal axis as time -a variable in algebra -algebraic functions as the pairing of elements in sets -geometric transformations: stretching,shrinking, translation, reflection -algebraic expressions for linear functions -least common multiples: rules to determine LCM, relationship to and meaning of factors -negative integers and number lines -the inductive process applied to mathematics	Students will know and have an understanding of: -scales as the building blocks of melody and harmony -the difference between intervals, chords and melodies -vibrational nature of pitch -musical convention of numbering notes of the scale -melodic vs. harmonic intervals -how to harmonize a melody -simple compositional forms	Students will know and have an understanding that: -the "subjects" they study in school are ultimately interconnected -studying the connections between "subjects" can lead to deeper understanding of the real world.
skill	Students will be able to: - plot points on a Cartesian coordinate system -use and interpret in/out tables -determine an algebraic representation for a line on a graph -find the least common multiple between a set of numbers -experience improved computation skills with basic arithmetic operations on integers	Students will be able to: -perform polyrhythmic beat patterns -sight sing melodies from graphic representations -create melodic variations from an original melody for use in an original composition -aurally identify families of melodic alterations: modal shift, stretching and reflections
habits of mind	Students will: -improve their ability to make conjectures from the observation of patterns -learn to reflect on their learning process -learn persistence in the problem solving process	Students will: - cultivate their artistic judgment with respect to musical decisions -increase their awareness as to the subjective nature of artistic judgment -listen with greater clarity and sensitivity	Students will: -begin to seek connections between other disciplines they study -generate a habit of inquiry to seek depth of understanding
affect	Students will: -gain confidence in their mathematical abilities -become inspired to study mathematics -gain an deeper appreciation for the ways that mathematics informs our lives -see that mathematics is more than a tool, but a way to perceive beauty	Students will: -gain an appreciation of diverse cultures through musical analysis -gain confidence in their musical abilities -gain a fresh perspective and open attitude to their musical art	Students will: -discover new insights into the nature of creativity and the relationship between the work of scientists, mathematicians and artists.
product	Students will create: -polyrhythm graphs -multiples tables for inductive analysis -journal logs -melodic contour graphs -in/out tables	Students will create: -original melodies and/or music compositions -vocal and instrumental performances

Program Customizing

The Standard Program can be customized for the specific needs of teachers, schools and/or districts to specify:

Teacher Inservice Training

A set of teacher inservice workshops accompany the teaching of the music/math unit. Depending on resources and logistics at any particular school site, these workshops will average approximately 1 for every 2 student lessons that are taught. The workshops serve to,

1. demonstrate and discuss strategies for follow-up instruction in the classroom to maximize the learning gains from the unit.
   
   
2. provide the support and materials necessary for teachers to teach the music/math unit independently.
   
   
3. provide professional development for the teaching of mathematics beyond the mathematics and music unit. The big ideas of the teaching strategies and circular design of the unit are connected to general applications of mathematical teaching in classrooms. These ideas include:

• Teaching for understanding; creating an inquiry based classroom
• The art of questioning in the mathematics classroom
• Inductive activities and the constructivist theory of learning
• Building foundations for algebra in elementary grades
• The use of interdisciplinary curricula to motivate, provide relevance, and varied contexts for enhanced depth of understanding
• Spiral curriculum designs
• Group work vs. individual work and the effect of status relationships and the social construction of intelligence on student learning

Community Outreach/Parent Involvement

Parents can be invited to attend the staff workshops, or specialized workshops can be held for them. The topic of music and mathematics is especially interesting to people of all ages, and can provide an effective link to engage parents in the studies of their children. Adult education programs that mirror student programs serve to bring communities and schools together.

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