1. An Instructional Approach to Foster Understanding Week 1

2. Why is Understanding Mathematics so Important? One’s knowledge are interconnected and one can reconstruct them when needed. One can flexibly apply those ideas to new situations. One can flexibly apply those ideas to new situations. One feels great! One feels great!

3. Data From Classroom Observations Fall 2009

4. How Can Classrooms be Designed to Foster Understanding? Week 1

5. Why Use a Problem-based Approach? Students must experience the intellectual need for the concept to be learned “For students to learn what we intend to teach them, they must have a need for it, where by ‘need’ is meant intellectual need, not social or economic need.” (Harel, 2007)

6. Why Use a Problem-based Approach? “The process of knowing is developmental in the sense that it proceeds through a continual tension between accommodation and assimilation.” (Harel, 2007, p. 265) Students must experience the intellectual need for the concept to be learned Understanding involves grappling and resolving the inconsistency between one’s existing conception and a problem situation

7. Why Use a Problem-based Approach? Students must experience the intellectual need for the concept to be learned Understanding involves grappling and resolving the inconsistency between one’s existing conception and a problem situation Problems are the means to experience the limitation of their existing knowledge and the need for a new piece of knowledge

8. Why Use a Problem-based Approach? Students must experience the intellectual need for the concept to be learned Understanding involves grappling and resolving the inconsistency between one’s existing conception and a problem situation Problems are the means to experience the limitation of their existing knowledge and the need for a new piece of knowledge Problem situations are opportunities for students to develop mathematical habits of mind

9. How Can Classrooms be Designed to Foster Understanding? 1.Allowing mathematics to be problematic for students Pose challenging problems within students reach Challenge students to use what they know for a simpler problem (e.g. ¾ + ½) to solve a more complex problem (e.g. 5¾ + 1½) “Refrain from stepping in and doing too much of the mathematical work too quickly” (Hiebert & Wearne, 2003)

10. How Can Classrooms be Designed to Foster Understanding? 1.Allowing mathematics to be problematic for students (Hiebert & Wearne, 2003) In comparing 8 th grade lessons in the TIMSS study among U.S., Germany, and Japan, “U.S. teachers almost always stepped in to show students how to solve the problems; the mathematics they left for students to think about and do was rather trivial. Teachers in the other two countries allowed students more opportunities to wrestle with the challenging aspects of the problems.” (p. 6-7)

11. How Can Classrooms be Designed to Foster Understanding? 1.Allowing mathematics to be problematic for students (Hiebert & Wearne, 2003) 2.Examining increasingly better solution methods Allow students to use their own methods but commit them to search for better ones (more efficient, flexible, understandable) Provide opportunities for students to share their methods, to hear others’ methods, and to examine the strengths and weaknesses of various methods “Examining methods encourage students to construct mathematical relationships, and constructing relationships is at the heart of understanding” (p. 9)

12. How Can Classrooms be Designed to Foster Understanding? 1.Allowing mathematics to be problematic for students (Hiebert & Wearne, 2003) 2.Examining increasingly better solution methods What information to be shared? Conventions such as order of operations, exponents, function notation Alternative solutions not presented by students Ideas embedded in students’ solution methods When to share information? As needed 3.Providing appropriate information at the right time

13. How Can Classrooms be Designed to Foster Understanding? 1.Allowing mathematics to be problematic for students (Hiebert & Wearne, 2003) 2.Examining increasingly better solution methods Hiebert, J., & Wearne, D. (2003). Developing understanding through problem solving. In H. L. Schoen (Ed.), Teaching mathematics through problem solving: Grade 6-12 (pp. 3-13). Reston, VA: National Council of Teachers of Mathematics. 3.Providing appropriate information at the right time

14. Why “Lesson Study” Project? Apply what you have learned from the last three series of workshops into your own classrooms Improve your pedagogical content knowledge Experience the process of designing, planning, and teaching lessons Build a professional learning community

15. What does a “Lesson Study” Project entail? Step 2: Formula learning goals Step 3:Design activities and plan a classroom lesson Present your lesson and get inputs from the other groups Step 1: Identify a topic and its related key ideas Try out your activities during a workshop session

16. What does a “Lesson Study” Project entail? Step 4:Implement/observe the lesson Collect student work Step 5:Analyze student work Step 6: Reflect and report the results (relate to goals) Refine the lesson for future use Write up a brief report Step 1: Identify a topic and its related key ideas Step 2: Formula learning goals Step 3:Design activities and plan a classroom lesson

17. Tentative Schedule Groups A & B Groups C & D Project 1Project 2Project 1Project 2 1.Sep 30S1, S2S1, S2 2.Oct 14 S3S2, S3 3.Nov 4 S5, S6 S3 4.Nov 18S1, S2S5, S6 5.Dec 2S3S1, S2 Teach & Observe Lesson #1 Teach & Observe Lesson #2

18. Preparation for Workshop 3 (Sep 30) Search for activities/problems for your topic that are likely to help you achieve your agreed learning objectives Think about the lesson for your topic Share resources/notes/reflections and post your questions/comments/ideas at http://thinkmaththink.wikispaces.com/ (Note: there is a discussion tab for each page/group) http://thinkmaththink.wikispaces.com/