History and Role of Proofs in Secondary Mathematics Education: A Pedagogical Perspective Cheryl Clough Dr. King

Statement of Purpose Learn about mathematical proofs: their history, application, and students’ perception of proofs –Explore the different types of proofs –Make proofs more tangible to students –Illustrate why proofs are important Gain a better understanding of what teaching mathematics at the High School level will be Gain experience in an actual classroom setting

Project Components Create a Pedagogical project –E–E–E–Explored the historical aspects of proofs –G–G–G–Gave examples to illustrate the five main types of proof –C–C–C–Created multiple lesson plans –P–P–P–Presented one of the lesson plans to a High School Geometry Class

History Proof became a solid characteristic of mathematics, Thales (6th century B.C.) Proof and geometry were synonymous Contributions by Gauss (1800’s) Secondary school education two- education two- column proof column proof (1900’s) (1900’s) StatementReason 1. Draw a perpendicular from C to AB1. Perpendicular Postulate 2. Geometric Mean Theorem 3. ce = a 2 and cf = b 2 3. Cross product property 4. ce +cf = a 2 + b 2 4. Addition Property of equality 5. c(e + f) = a 2 + b 2 5. Distributive property 6. e + f = c6. Segment Addition Postulate 7. c 2 = a 2 + b 2 7. Substitution property of equality 2.

History of Pythagoras ( B.C.) History of Pythagoras ( B.C.) 6. Born on Island of Samos 28. Studied in Egypt & Babylonia 496. Is considered the first pure mathematician Pythagoras founded a philosophical and religious school in Croton school 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 proof Interested in the abstract idea of proof Pythagorean Theorem Pythagorean Theorem Perfect Number: 28 =

Properties of a Good Proof Flows & Easy to follow Logically guides the reader Each step is clear or clearly justified Reveals the content and the context of the theorem Lead to further discoveries & new theories Gives mathematicians a way to find more revealing ways to look at a given statement

Types of Proof Types of Proofs –Direct Proof –Proof by Contraposition –Proof by Contradiction –Proof by Induction –High School two Column Proof

Lesson Plan Lesson Plan Standards Addressed (Geometry) : –1–1–1–14.0 Students prove the Pythagorean Theorem –1–1–1–15.0 Students use Pythagorean Theorem find missing lengths of sides of right triangle Specific Lesson Objectives: –S–S–S–Students have some knowledge of the history of the Pythagorean Theorem and Pythagoras. –S–S–S–Students will be able to explain and prove the Pythagorean Theorem –S–S–S–Students will be able to compute the missing length of a side of a right triangle

Lesson Plan Cont’d Lesson Sequence Lesson Sequence  History of Pythagoras & Pythagorean Theorem  Geometric proof of Theorem  Algebraic proof of Theorem  Book proof of Theorem (two column style) –Computational Examples -how to use the theorem a 2 +b 2 =c 2 –Practical uses for using the Theorem

Reflection on Experience Lesson went very well Lesson went very well Was organized Was organized Had to overcome my nerves Had to overcome my nerves Classroom teacher compliment Classroom teacher compliment –Nice presentation –students seemed to be engaged and interested Met objectives Met objectives –students should have some knowledge about the history surrounding the Pythagorean Theorem, Pythagoras –Students are capable of explaining, proving and applying the Pythagorean Theorem by one of the methods shown. Engaged students in a thought provoking manner Engaged 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 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 Dr. King Dr. Fogel Garry McGinnin (TO High School Teacher) John Engelstad (web site design and support) For more information visit my web site:

Bibliography Epp, Susanna S, “A Cognitive Approach to Teaching Logic and Proof”, Department of Mathematical Sciences DePaul University Epp, Susanna S, “A Cognitive Approach to Teaching Logic and Proof”, Department of Mathematical Sciences DePaul University Epp, Susanna S. “The role of Logic in Teaching Proof”, The American Mathematical Monthly, December 2003, Volume 110, Issue 10, pg Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Epp, Susanna S. “The role of Logic in Teaching Proof”, The American Mathematical Monthly, December 2003, Volume 110, Issue 10, pg Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Kleiner, Israel “Rigor and Proof in Mathematics: A Historical Perspective”, Mathematics Magazine, December 1991, Volume 64, No. 5, pgs Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Kleiner, Israel “Rigor and Proof in Mathematics: A Historical Perspective”, Mathematics Magazine, December 1991, Volume 64, No. 5, pgs Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Knipping, Christine. Cultural differences and the teaching of proof for all Proceedings of the 3rd International Mathematics Education and Society Conference Copenhagen: Centre for Research in Learning Mathematic. Knipping, Christine. Cultural differences and the teaching of proof for all Proceedings of the 3rd International Mathematics Education and Society Conference Copenhagen: Centre for Research in Learning Mathematic. Markel, William D. “The role of proof in mathematics Education”, School Science and Mathematics; October 1994, Volume 94, Issue 6, pg Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Markel, William D. “The role of proof in mathematics Education”, School Science and Mathematics; October 1994, Volume 94, Issue 6, pg Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Szombathelyi, Anita. “Ideas for developing students’ reasoning; a Hungarian perspective”, The Mathematics Teacher, November 1998, Volume 91, Issue 8, pg Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005 Szombathelyi, Anita. “Ideas for developing students’ reasoning; a Hungarian perspective”, The Mathematics Teacher, November 1998, Volume 91, Issue 8, pg Proquest. California Lutheran University, Thousand Oaks, Ca. January 2005