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

2. 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?)

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

4. 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

5. 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

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

7. 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.

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

9. 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.

10. 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

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