Making beautiful music in Making beautiful music in the mathematics the mathematics

classroom classroom

Matt DeLongMatt DeLongINMAA OnlineINMAA Online

Based on an Alder Award PresentationBased on an Alder Award Presentationat Mathfest 2005at Mathfest 2005

A Surprising EmailA Surprising Email

Identity and IntegrityIdentity and Integrity

Here is a secret hidden in plain sight: Here is a secret hidden in plain sight: good teaching cannot be reduced to good teaching cannot be reduced to technique; good teaching comes technique; good teaching comes from the identity and integrity of the from the identity and integrity of the teacherteacher..– Parker Palmer, Parker Palmer, The Heart of a TeacherThe Heart of a Teacher

My IdentityMy Identity

• I’m a mathematician: TU Professor I’m a mathematician: TU Professor and UM Ph.D.and UM Ph.D.

• My father was a high school math My father was a high school math teacherteacher

• I’m a musician: Singer, choir director, I’m a musician: Singer, choir director, faculty quartetfaculty quartet

• I come from a family of amateur I come from a family of amateur musicians (I have a brother with a musicians (I have a brother with a graduate degree in music)graduate degree in music)

Les ChoristesLes Choristes

Stand and DeliverStand and Deliver

My IntegrityMy Integrity

• I teach at a Christian college where I teach at a Christian college where foundational discussions are encouraged foundational discussions are encouraged and beauty is accepted as a guide to truth and beauty is accepted as a guide to truth and as a motivation to study.and as a motivation to study.

• Finally, brothers, whatever is true, whatever is noble, whatever is right, whatever is pure, whatever is lovely, whatever is admirable—if anything is excellent or praiseworthy—think about such things.—Philippians 4:8Philippians 4:8

An Image ProblemAn Image Problem

Last time, I asked: "What does Last time, I asked: "What does mathematics mean to you?" And some mathematics mean to you?" And some people answered: "The manipulation of people answered: "The manipulation of numbers, the manipulation of numbers, the manipulation of structures." And if I had asked what structures." And if I had asked what music means to you, would you have music means to you, would you have answered: "The manipulation of answered: "The manipulation of notes?“—Serge Lang, notes?“—Serge Lang, The Beauty of The Beauty of Doing MathematicsDoing Mathematics

One Possible SolutionOne Possible Solution

If we view beauty as central to If we view beauty as central to mathematics we would be introducing mathematics we would be introducing ideas that, while they may have ideas that, while they may have application or may build some tools, application or may build some tools, would certainly let students see a would certainly let students see a much bigger picture of mathematics much bigger picture of mathematics at a much earlier stage in their at a much earlier stage in their mathematical development.—Sam mathematical development.—Sam Stueckle, Stueckle, Beauty in Mathematics Beauty in Mathematics

PassionPassion

We became teachers for reasons of We became teachers for reasons of the heart, animated by a passion for the heart, animated by a passion for some subject and for helping people some subject and for helping people learn. learn. – Parker Palmer, Parker Palmer, The Heart of a TeacherThe Heart of a Teacher

Technique and PassionTechnique and Passion

Teaching mathematics and making Teaching mathematics and making music are similar in that they require music are similar in that they require both both goodgood techniquetechnique and and passionpassion to be most effective. to be most effective.

TechniqueTechnique

• LectureLecture

• Cooperative learningCooperative learning

• Appropriate use of technologyAppropriate use of technology

• Answering questionsAnswering questions

• Etc.Etc.

MALAMALA

The MALA framework is a tool for focusing discussions about classroom management.– Motivation– Activity– Learning outcomes– AttributionDeLong, Winter, YackelPRIMUS, June 2003

FAST-SLOFAST-SLO

Highly encapsulated knowledge - much is tacit and implicit.

External schematic representation

of what the expert knows, the connections that exist, applications, how this new

material is connected to what learners already know,

etc.

1. Performance 2. Conditions

3. Level

1. Performance 2. Conditions

3. Level

Subject matter expert/instructor

De-encapsulation FAST Codification SLOs

DeLong, Winter, Yackel PRIMUS, September 2005

Beauty in MathematicsBeauty in MathematicsMathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show—Bertrand Russell, The Study of Mathematics

Beauty in Mathematics IIBeauty in Mathematics II

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way.—G. H. Hardy, A Mathematician's Apology

Beauty in Mathematics IIIBeauty in Mathematics III

It is more important to have beauty in one’s equations than to have them fit experiment.—Paul Dirac, Scientific American

Beauty in Math IVBeauty in Math IV

You get these feelings of You get these feelings of beautybeauty … …

it’s more about the whole story that was it’s more about the whole story that was just told about a certain area of truth. You just told about a certain area of truth. You get this feeling of enlightenment…. get this feeling of enlightenment…. Something that was sort of unclear in your Something that was sort of unclear in your mind before suddenly falls into place. You mind before suddenly falls into place. You feel like you’ve learned a truth. You’ve feel like you’ve learned a truth. You’ve gotten closer to understanding the world.gotten closer to understanding the world.—Manjul Bhargava, —Manjul Bhargava, Morning EditionMorning Edition

DefinitionsDefinitions

• Beauty is that quality or combination Beauty is that quality or combination of qualities which affords keen of qualities which affords keen pleasure to the senses, especially pleasure to the senses, especially that of sight, or which charms the that of sight, or which charms the intellectual or moral faculties.—intellectual or moral faculties.—ShorterShorter Oxford English DictionaryOxford English Dictionary

• Beauty is that which pleases in mere Beauty is that which pleases in mere contemplation.—Thomas Aquinas contemplation.—Thomas Aquinas

Attributes of BeautyAttributes of Beauty

Surprise at the unexpected Surprise at the unexpected

• Mathematics:Mathematics:

• Music: Music: McClure’s McClure’s KyrieKyrie 6

1 2

1 2

n n

4222

642

4

111

!7!5!31

xx

xxx

Perception of relationshipsPerception of relationships

• Mathematics: Mathematics:

• Music: Music: Hampton’s Hampton’s Praise His Holy Praise His Holy NameName

01ie

xixeix sincos

Organized complexityOrganized complexity

• Mathematics: Mandelbrot SetMathematics: Mandelbrot Set

• Music: Music: He, Watching Over IsraelHe, Watching Over Israel, , MendelssohnMendelssohn

Brevity/SimplicityBrevity/Simplicity

• Mathematics: Bhaskara’s ProofMathematics: Bhaskara’s Proof

• Music: Music: Schubert’s Schubert’s SanctusSanctus

EleganceElegance

• Mathematics: Euclid’s proof of the Mathematics: Euclid’s proof of the infinitude of primesinfinitude of primes

• Music: Mozart’s Music: Mozart’s Ave Verum CorpusAve Verum Corpus

121 npppN

Awe at the presence of the Awe at the presence of the infiniteinfinite

• Mathematics: Mathematics: Cantor’s proof of the Cantor’s proof of the non-denumerability of the continuumnon-denumerability of the continuum

• Music: Handel’sMusic: Handel’s Messiah Messiah

4321 bbb.bb

nnn ab ,9,0

But Can It Be Taught?But Can It Be Taught?

It’s like asking why Beethoven’s Ninth Symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.—Paul Erdösaul Erdös

But Can It Be Taught? IIBut Can It Be Taught? II

The mathematical esthetic sense is The mathematical esthetic sense is not distributed uniformly among the not distributed uniformly among the population, and there is a large population, and there is a large number of people who find number of people who find mathematics not only not beautiful, mathematics not only not beautiful, but repulsive.... There is nothing to but repulsive.... There is nothing to be done about those who cannot be done about those who cannot appreciate mathematics.—Underwood appreciate mathematics.—Underwood Dudley, Dudley, Indiana Mathematics TeacherIndiana Mathematics Teacher

But Can It Be Taught? IIIBut Can It Be Taught? III

Music is passionate and very Music is passionate and very sensual. Mathematicians may sensual. Mathematicians may see math that way, but try to see math that way, but try to explain that to anyone else and explain that to anyone else and you sound like an idiot.—Dan you sound like an idiot.—Dan Naiman, Naiman, Johns Hopkins MagazineJohns Hopkins Magazine

Investigations in Investigations in MathematicsMathematics

• General education General education mathematics course mathematics course for liberal arts for liberal arts majorsmajors

• Course objective: Course objective: To help each studentTo help each student– uncover the beauty of uncover the beauty of

mathematics and mathematics and become fascinated by become fascinated by it,it,

Investigations FeedbackInvestigations Feedback

• My attitude towards mathematics My attitude towards mathematics has improved as a result of this class.has improved as a result of this class.– 26 % Strongly Agree, 61 % Agree26 % Strongly Agree, 61 % Agree

• I enjoyed this course.I enjoyed this course.– 26 % Strongly Agree, 61 % Agree26 % Strongly Agree, 61 % Agree

Problem SolvingProblem Solving

• Required freshman Required freshman course for all course for all mathematics majorsmathematics majors

• Course objective: Course objective: To help each studentTo help each student– uncover the beauty of uncover the beauty of

mathematics and mathematics and become fascinated by become fascinated by it,it,

Problem Solving FeedbackProblem Solving Feedback

• I have a better appreciation for the I have a better appreciation for the beauty of mathematics than I did beauty of mathematics than I did before taking this course.before taking this course.– 18 % Strongly Agree, 73 % Agree18 % Strongly Agree, 73 % Agree

Lessons From ConductingLessons From Conducting

The successful conductor … assumes The successful conductor … assumes responsibilities as an interpreter of responsibilities as an interpreter of music, a teacher of skills, an music, a teacher of skills, an organizer of experiences, a self-organizer of experiences, a self-motivated learner, and a sensitive motivated learner, and a sensitive human being.—Decker and Kirk, human being.—Decker and Kirk, Choral Conducting: Focus Choral Conducting: Focus

on Communicationon Communication

InterpreterInterpreter

• Knows the material extremely wellKnows the material extremely well

• Communication is unencumbered Communication is unencumbered and clear (not confusing)and clear (not confusing)

• Is thoroughly preparedIs thoroughly prepared

• Gives sections importance Gives sections importance

• Makes choice of "repertory"Makes choice of "repertory"

Teacher of skillsTeacher of skills

• Skillful themselvesSkillful themselves

• Understands student developmentUnderstands student development

• Ably guide skill developmentAbly guide skill development

• Puts skills into contextPuts skills into context

Organizer of experiencesOrganizer of experiences

• Encourages growth and enjoymentEncourages growth and enjoyment

• Guides love of the disciplineGuides love of the discipline

• Encourages appreciation of the Encourages appreciation of the discipline as artdiscipline as art

• Guides ability to read and interpretGuides ability to read and interpret

• Encourages working togetherEncourages working together

Self-motivated learnerSelf-motivated learner

• Engages in exploration, research, Engages in exploration, research, learninglearning

• Has a fascination with discoveryHas a fascination with discovery

Sensitive human beingSensitive human being

• Understands the diverse interests of Understands the diverse interests of the studentsthe students

• Responds to the interests of the Responds to the interests of the studentsstudents

• Accepts responsibility for broadening Accepts responsibility for broadening the students’ horizonsthe students’ horizons

• Guides students in an ever-increasing Guides students in an ever-increasing awareness of their responsibility in awareness of their responsibility in learninglearning

Making Beautiful MusicMaking Beautiful Music

… … mathematics should be taught as mathematics should be taught as music is taught. Students should make music is taught. Students should make mathematics mathematics together together … not alone. … not alone. Creative ideas should be stressed over Creative ideas should be stressed over the “right way to do it.” … And finally, the “right way to do it.” … And finally, students should perform mathematics; students should perform mathematics; they should they should singsing mathematics and mathematics and dancedance mathematics.—Jim Henle, mathematics.—Jim Henle, American American Mathematical MonthlyMathematical Monthly