Introduction to Proof Techniques in a Geometry Course MAA MathFest 2019 Cincinnati, OH

1. Introduction to Proof Techniques in a Geometry CourseMAA MathFest2019Cincinnati, OH Carol J. Bell Northern Michigan University cbell@nmu.edu

2. Background of Course • Originally intended for future secondary math teachers; future elementary math teachers took a separate geometry course • Fall 2013 credit hours increased from 3 to 4 • Fall 2016 became course for future K-12 teachers • Fall 2018 course name changed (Geometry I to Geometry for Educators) • Course content focuses on Euclidean Geometry with one other geometry introduced, such as Taxi Cab geometry • Another geometry course added for math majors in department

3. Previous Course Ideas and Assessments • Straightedge and Compass Constructions • “Make Geometric Constructions,” MI Math Standards (2010) https://www.michigan.gov/documents/mde/K-12_MI_Math_Standards_REV_470033_7_550413_7.pdf • In-class Explorations with Geometer’s Sketchpad (explore, conjecture, prove) • Online class discussions (Proofs without Words) • For additional ideas on using PWW see: Bell, C. J. (2011). Proofs without words: A visual application of reasoning and proof. Mathematics Teacher, 104(9), 690 – 695. • Projects & Presentations • Sketchpad project example • Class project on tessellations showcased at NMU’s Celebration of Student Scholarship • Presentation of homework problems – graded on written work (15% of grade) and presentation of problems (5% of grade)

4. Need for Change • Weakness in students’ knowledge of geometry • Weakness in students’ ability to do proofs • Expansion of course to include future elementary math teachers • Different set of needs than future secondary math teachers

5. Geometry Knowledge of Teachers (examples) • Early Childhood (Pre-K to grade 2) Teachers should possess strong understandings of geometry, including spatial relationships. Be able to analyze the components and properties of shapes. Quantify 2D and 3D spaces. • Upper Elementary Teachers should possess strong understandings of geometry and measurement. Provide frameworks for connecting linear measurement with measures of area and volume. Use measurement tools and construct figures. • Middle Level Teachers should be able to connect geometry to measurement and algebra. They have strategies for connecting geometry to ratios, proportions, and algebraic thinking when they explore scale drawings and transformations. Make connections among length, area, and volume concepts and formulas. Explain why Pythagorean Theorem is true. • High School Teachers should understand the role of diagrams and definitions in geometry. Understand geometry from the perspective of transformations. Use dynamic geometry software to investigate and understand variance and invariance of geometric objects. Reference: Association of Mathematics Teacher Educators. (2017).  Standards for Preparing Teachers of Mathematics. Available online at amte.net/standards.  

6. Focus on Proof • The concept of proof or justification encompasses convincing, explaining, and understanding. (Huang, 2005) • Proof is an important component of mathematics teaching for every age group. (NCTM, 2000) • Justification of mathematical statements in the early grades may constitute nothing more than mathematical verification. (Jones, 1997) • At the middle school level, students make conjectures, make and test generalizations, and learn the difference between mathematical explanation and experimental evidence. (Jones, 1997) • In high school, students are expected to extend their mathematical reasoning into understanding and use more rigorous arguments, leading to notions of proof. (Jones, 1997) • By the time students complete the eighth grade they should be able to explain a proof of the Pythagorean Theorem and its converse. (CCSSI, 2010)

7. Fall 2016 Project • Proof and Justification Questionnaire given at beginning of semester. • Research project: explored pre-service mathematics teachers’ understanding of the proof process at the elementary, middle, and high school levels. Examined what constitutes proof and justification at the K-12 level. • Geometry students developed methods to help elementary, middle, or high school students further their understanding and justification of the Pythagorean Theorem and its converse. For the results of this project see: Bell, C. J. (2017). Pre-service teachers understanding of proof and justification at the elementary, middle, and high school levels. Proceedings of the Arts, Humanities, Social Sciences and Education Conference, USA.

8. Fall 2017 Introduced Proof Portfolio • Idea came from reading: Spencer, G. (2017, April/May). A revise-and-resubmit proof portfolio. MAA Focus, 10 – 13. • Discovered website on using ProofBlocks (flow chart) • Dirksen, J., Dirksen, N., Hwang, J., & Cheng, I. (2008). ProofBlocks: A Visual Approach to Proof. Retrieved from http://www.proofblocks.com/ • Dirksen, J., Dirksen, N., & Cheng, I. (2010). ProofBlocks: A visual approach to proof. Mathematics Teacher, 103(8), 571 – 576. • Had a collection of proof processes that students should know • Proof portfolio replaced the final exam

9. Proof Techniques in Geometry • Straightedge and compass constructions with step-by-step explanation • Indirect Proof • ProofBlocks (flow chart proofs) • Two-column proof • Paragraph proof • Coordinate geometry proofs • Math induction • Proof without words • Use Sketchpad to construct, explore, write a conjecture, and prove statement • Prove a biconditional statement – used PWW ideas from: Nirode, W. (2017). Proofs without words in geometry. Mathematics Teacher, 110(8), 580 – 586.

10. Work in Progress • Do more in-class activities to help in understanding the different proof techniques • Introduce more hands-on activities, including paper-folding constructions • Develop more activities relevant to future K-6 teachers

11. References • Association of Mathematics Teacher Educators. (2017).  Standards for Preparing Teachers of Mathematics. Available online at amte.net/standards. • Bell, C. J. (2017). Pre-service teachers understanding of proof and justification at the elementary, middle, and high school levels. Proceedings of the Arts, Humanities, Social Sciences and Education Conference, USA. • Bell, C. J. (2011). Proofs without words: A visual application of reasoning and proof. Mathematics Teacher, 104(9), 690 – 695. • Common Core State Standards Initiative. (2012). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf • Dirksen, J., Dirksen, N., & Cheng, I. (2010). ProofBlocks: A visual approach to proof. Mathematics Teacher, 103(8), 571 – 576. • Dirksen, J., Dirksen, N., Hwang, J., & Cheng, I. (2008). ProofBlocks: A Visual Approach to Proof. Retrieved from http://www.proofblocks.com/ • Huang, R. (2005). Verification or proof: Justification of Pythagoras’ Theorem in Chinese mathematics classrooms. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 161–168. Melbourne: PME. • Jones, K. (1997). Student-teachers' conceptions of mathematical proof. Mathematics Education Review, 9, 21–32. • National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. • Nirode, W. (2017). Proofs without words in geometry. Mathematics Teacher, 110(8), 580 – 586. • Spencer, G. (2017, April/May). A revise-and-resubmit proof portfolio. MAA Focus, 10 – 13.