Two Proofs Same Theorem 1 Traditional approach 2

Two Proofs, Same Theorem 1. Traditional approach 2. Teaching for Understanding approach via Problem-Based Instructional Task

Theorem to Prove 1. Alternate interior angles formed by two parallel lines cut by a transversal are congruent. 2. `Twas brillig, and the slithy toves Did gyre and gimble in the wabe: All mimsy were the borogoves, And the mome raths outgrabe. (Will #1 sound any more meaningful to students than #2? Can we help make it meaningful for them? )

(from Through the Looking-Glass and What Alice Found There, 1872)

Problem-Based Instructional Tasks: • Help students develop a deep understanding of important mathematics • Emphasize connections, especially to the real world • Are accessible yet challenging to all students • Can be solved in several ways • Encourage student engagement and communication • Encourage the use of connected multiple representations • Encourage appropriate use of intellectual, physical, and technological tools

Teaching for Understanding means: • Developing deep conceptual and procedural knowledge of mathematics • Posing problem-based instructional tasks • Engaging students in tasks and providing guidance and support as they develop their own representations and solution strategies • Promoting discourse among students to share their solution strategies and justify their reasoning • Summarizing the mathematics and highlighting effective representations and strategies • Extending students’ thinking by challenging them to apply their knowledge in new situations, especially in real-world situations • Listening to students and basing instructional decisions on their understanding

Traditional Proof Lesson • Engage in sample lesson from traditional text. • Then reflect and analyze (see next slide).

Traditional Proof Lesson • Problem-based instructional task? Discuss and explain. (re: attributes previous slide) • Focus on teaching for understanding? Discuss and explain. (re: attributes previous slide) • Launch-Explore-Summarize? Discuss and explain. • Inductive reasoning Deductive proof? [Find a pattern, make a conjecture, by examining data or examples. Then use if-then reasoning to prove. ] Discuss and explain.

Video of CMIC Lesson As you watch the video, focus on: • Use of questioning • Problem-Based Instructional Task attributes • Teaching for Understanding attributes

CMIC Proof Lesson • Problem-based instructional task? Discuss and explain. (re: attributes previous slide) • Focus on teaching for understanding? Discuss and explain. (re: attributes previous slide) • Launch-Explore-Summarize? Discuss and explain. • Inductive reasoning Deductive proof? [Find a pattern, make a conjecture, by examining data or examples. Then use if-then reasoning to prove. ] Discuss and explain.

Source for video: STREAM video series COMAP Contemporary Mathematics in Context “Shapes and Geometric Reasoning” www. comap. com/highschool/projects/stream/ Videos available for all NSF high school projects

Two Proofs, Same Theorem 1. Traditional approach 2. Teaching for Understanding approach via Problem-Based Instructional Task