History and Role of Proofs History and Role of Proofs in Secondary Mathematics in Secondary Mathematics Education: A Pedagogical Education: A Pedagogical

PerspectivePerspective

Cheryl CloughCheryl Clough

Dr. KingDr. King

Statement of PurposeStatement of Purpose

Learn about mathematical proofs: Learn about mathematical proofs: their history, application, and their history, application, and students’ perception of proofsstudents’ perception of proofs– Explore the different types of proofsExplore the different types of proofs– Make proofs more tangible to studentsMake proofs more tangible to students– Illustrate why proofs are importantIllustrate why proofs are important

Gain a better understanding of what Gain a better understanding of what teaching mathematics at the High teaching mathematics at the High School level will be School level will be Gain experience in an actual Gain experience in an actual classroom settingclassroom setting

Project ComponentsProject Components

Create a Pedagogical projectCreate a Pedagogical project– Explored the historical aspects of Explored the historical aspects of

proofsproofs– Gave examples to illustrate the five Gave examples to illustrate the five

main types of proof main types of proof – Created multiple lesson plans Created multiple lesson plans – Presented one of the lesson plans to Presented one of the lesson plans to

a High School Geometry Classa High School Geometry Class

HistoryHistory

ProofProof became a solid became a solid characteristic of mathematics, characteristic of mathematics, Thales (6th century B.C.) Thales (6th century B.C.)

Proof and geometry were Proof and geometry were synonymoussynonymous

Contributions by Gauss Contributions by Gauss (1800’s)(1800’s)

Secondary school Secondary school education two-education two- column proof column proof (1900’s)(1900’s)

Statement Reason

1. Draw a perpendicular from C to AB 1. Perpendicular Postulate

2. Geometric Mean Theorem

3. ce = a2 and cf = b2 3. Cross product property

4. ce +cf = a2 + b2 4. Addition Property of equality

5. c(e + f) = a2 + b2 5. Distributive property

6. e + f = c 6. Segment Addition Postulate

7. c2 = a2 + b2 7. Substitution property of equality

f

b

b

cande

a

a

c2.

History of PythagorasHistory of Pythagoras(572- 501 B.C.)(572- 501 B.C.)

6.6. Born on Island of SamosBorn on Island of Samos28.28. Studied in Egypt & BabyloniaStudied in Egypt & Babylonia496.496. Is considered the first pure mathematician Is considered the first pure mathematician 8128.8128. Pythagoras founded a philosophical and religious Pythagoras founded a philosophical and religious school in Crotonschool in Croton Main areas of study: Arithmetic, Music, Geometry and Main areas of study: Arithmetic, Music, Geometry and

Astronomy Astronomy

Studied properties of numbers such as: even and odd Studied properties of numbers such as: even and odd numbers, triangular numbers, perfect numbers, etc. numbers, triangular numbers, perfect numbers, etc. Believed that “each number had its own personality - Believed that “each number had its own personality -

masculine or feminine, beautiful or ugly” masculine or feminine, beautiful or ugly” Interested in the abstract idea of proofInterested in the abstract idea of proof

Pythagorean TheoremPythagorean Theorem

8589869056.

137438691328.

2305843008139952128.

Perfect Number:

28 = 1+2+4+7+14

33550336.

Properties of a Good ProofProperties of a Good Proof

Flows & Easy to followFlows & Easy to follow

Logically guides the reader Logically guides the reader

Each step is clear or clearly justifiedEach step is clear or clearly justified

Reveals the content and the context Reveals the content and the context of the of the theorem theorem

Lead to further discoveries & new Lead to further discoveries & new theories theories

Gives mathematicians a way to find Gives mathematicians a way to find more more revealing ways to look at a revealing ways to look at a given given statementstatement

Types of ProofTypes of Proof

Types of ProofsTypes of Proofs– Direct ProofDirect Proof– Proof by ContrapositionProof by Contraposition– Proof by ContradictionProof by Contradiction– Proof by InductionProof by Induction– High School two Column ProofHigh School two Column Proof

Lesson PlanLesson Plan Standards Addressed Standards Addressed (Geometry) :(Geometry) :

– 14.0 Students prove the Pythagorean Theorem14.0 Students prove the Pythagorean Theorem– 15.0 Students use Pythagorean Theorem find 15.0 Students use Pythagorean Theorem find

missing lengths of sides of right trianglemissing lengths of sides of right triangle

Specific Lesson ObjectivesSpecific Lesson Objectives: : – Students have some knowledge of the history Students have some knowledge of the history

of the Pythagorean Theorem and Pythagoras. of the Pythagorean Theorem and Pythagoras. – Students will be able to explain and prove the Students will be able to explain and prove the

Pythagorean TheoremPythagorean Theorem– Students will be able to compute the missing Students will be able to compute the missing

length of a side of a right trianglelength of a side of a right triangle

Lesson Plan Cont’dLesson Plan Cont’d

Lesson SequenceLesson Sequence History of Pythagoras & Pythagorean TheoremHistory of Pythagoras & Pythagorean Theorem Geometric proof of TheoremGeometric proof of Theorem Algebraic proof of TheoremAlgebraic proof of Theorem Book proof of Theorem (two column style)Book proof of Theorem (two column style)

– Computational ExamplesComputational Examples -how to use the theorem -how to use the theorem aa22+b+b22=c=c22

– Practical usesPractical uses for using the Theorem for using the Theorem

Reflection on ExperienceReflection on Experience Lesson went very wellLesson went very well Was organized Was organized Had to overcome my nervesHad to overcome my nerves Classroom teacher compliment Classroom teacher compliment

– Nice presentation Nice presentation – students seemed to be engaged and interested students seemed to be engaged and interested

Met objectives Met objectives – students should have some knowledge about the history students should have some knowledge about the history

surrounding the Pythagorean Theorem, Pythagorassurrounding the Pythagorean Theorem, Pythagoras– Students are capable of explaining, proving and applying Students are capable of explaining, proving and applying

the Pythagorean Theorem by one of the methods shown. the Pythagorean Theorem by one of the methods shown. Engaged students in a thought provoking mannerEngaged students in a thought provoking manner Lesson was shorter than anticipated Lesson was shorter than anticipated Reaffirmed my desire to teach. Reaffirmed my desire to teach.

Survey of Lesson TaughtSurvey of Lesson Taught

Answer on a scale 1 to 5 with one being no/bad/waste of time/don’t understand and 5 being yes/good/wow! I learned something/understand

You can now find the missing side of a right triangle ____4.6______

The lesson was interesting _____4.3_________You can prove the Pythagorean theorem _____3.9______The lesson was clear easy to understand _______4.5_______The overhead pictures used were helpful______4.5______The instructor was prepared and organized_____4.7______Would you listen to another lesson by this

teacher_____4.7_______

Special Thanks Special Thanks

Dr. KingDr. King

Dr. FogelDr. Fogel

Garry McGinnin (TO High School Teacher)Garry McGinnin (TO High School Teacher)

John Engelstad (web site design and John Engelstad (web site design and support)support)

For more information visit my web site:For more information visit my web site:

http://public.clunet.edu/~clclough/Capstone/http://public.clunet.edu/~clclough/Capstone/

Bibliography Bibliography Epp, Susanna S, Epp, Susanna S, “A Cognitive Approach to Teaching Logic and Proof”“A Cognitive Approach to Teaching Logic and Proof”, ,

Department of Mathematical Sciences DePaul UniversityDepartment of Mathematical Sciences DePaul University http://www.cs.cornell.edu/Info/People/gries/symposium/sepp.htmhttp://www.cs.cornell.edu/Info/People/gries/symposium/sepp.htm

Epp, Susanna S. “Epp, Susanna S. “The role of Logic in Teaching Proof”The role of Logic in Teaching Proof”, The American , The American Mathematical Monthly, December 2003, Volume 110, Issue 10, Mathematical Monthly, December 2003, Volume 110, Issue 10,

pg. pg. 886. Proquest. California Lutheran University, Thousand Oaks, 886. Proquest. California Lutheran University, Thousand Oaks, Ca. Ca. January 2005January 2005

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Knipping, Christine. Cultural Knipping, Christine. Cultural differences and the teaching of proof differences and the teaching of proof for allfor all Proceedings of the 3rd International Mathematics Education Proceedings of the 3rd International Mathematics Education and and Society Conference. 2002 Copenhagen: Centre for Research in Society Conference. 2002 Copenhagen: Centre for Research in

Learning MathematicLearning Mathematic.. http://www.congress-consult.com/mes3/Symposia/KnippingReid.doc http://www.congress-consult.com/mes3/Symposia/KnippingReid.doc Markel, William D. “Markel, William D. “The role of proof in mathematics Education”The role of proof in mathematics Education”, ,

School School Science and Mathematics; October 1994, Volume 94, Issue 6, Science and Mathematics; October 1994, Volume 94, Issue 6, pg. 291. pg. 291. Proquest. California Lutheran University, Thousand Proquest. California Lutheran University, Thousand Oaks, Ca. Oaks, Ca. January 2005January 2005

Szombathelyi, Anita. “Szombathelyi, Anita. “Ideas for developing students’ reasoning; a Ideas for developing students’ reasoning; a Hungarian perspective”Hungarian perspective”, The Mathematics Teacher, November , The Mathematics Teacher, November

1998, 1998, Volume 91, Issue 8, pg. 677. Proquest. California Lutheran Volume 91, Issue 8, pg. 677. Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005University, Thousand Oaks, Ca. January 2005