Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL An Over view What is The Open-Ended Approach Akihiko Takahashi, Ph.D. DePaul University, Chicago IL

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Traditional Instruction Based on the belief that teachers can make the unknown known by imparting teachers’ knowledge to their students (Gattegno, 1970). Students are viewed as passive recipients of knowledge in traditional instruction (Brown, 1994).

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL As a result, teachers always tell students, “You should know it because I told you”. However, there is no way for teachers to actually know whether they can pass their knowledge to their students successfully. Therefore, teachers proceed to give the students exercises, to make sure that the knowledge gets securely into their students. Teachers also give reviews to let students hold the knowledge and test whether students still hold the knowledge. This cycle of reviewing and testing has gone on for years because teachers know that many of their students do not retain the knowledge they are presented with (Gattegno, 1970).

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Students’ Beliefs from Traditional Mathematics Instruction: The processes of formal mathematics have little or nothing to do with discovery or invention. Students who understand the subject matter can solve assigned mathematics problems in five minutes or less. Only geniuses are capable of discovering, creating, or really understanding mathematics. One succeeds in school by performing the tasks, to the letter, as described by the teacher. ………………. Schoenfeld (1988)

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Reform mathematics students can learn mathematics by constructing their own concepts of mathematics (National Research Council, 1989) students are viewed as active constructors, rather than passive recipients (Brown, 1994) one of the more important concepts of teachers’ roles is to stimulate students to learn mathematics and support their development (Gttegno,1970)

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Instruction as Interaction Adding it up, (National Research Council, 2001)

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Traditional problems used in mathematics teaching in both elementary and secondary schools classroom have a common feature: that one and only one correct answer is predetermined. The problems are so well formulated that answers are either correct or incorrect and the correct one is unique. Closed Problems Open-ended problems Problems that are formulated to have multiple correct answers. The Open-Ended Approach Shimada et.al.,1977, Becker & Shimada, 1997

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Open-ended Approach 1.An open-ended problem is presented first 2.The lesson proceeds by using many correct answers to the given problem to provide experience in finding something new in the process. This can be done through combining students; own knowledge, skills, or ways of thinking that have previously been learned. The Open-Ended Approach Shimada et.al.,1977, Becker & Shimada, 1997

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL The Open-Ended Approach Shimada et.al.,1977, Becker & Shimada, 1997

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Typical flow of a mathematics class 1)Demonstrates a procedure 2)Assigned similar problems to students as exercises 3)Homework assignment 1)Presents a problem to the students without first demonstrating how to solve the problem 2)Individual or group problem solving 3)Compare and discuss multiple solution methods 4)Summary, exercises and homework assignment

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Three Major Types of the Open-Ended Approach 1)A lesson using problems with multiple solutions. Week1 2)A lesson using problems with multiple solution methods. Week2 3)A lesson using an activity called ‘problem to problem’ Week3

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL The Water-Flask Problem A transparent flask in the shape of a right rectangular prism is partially filled with water. When the flask is placed on a table and tilted, with one edge of its base being fixed, several geometric shapes of various sizes are formed by the cuboid’s face and surface of the water. The shapes and sizes may vary according to the degree of tilt or inclination. Try to discover as many invariant relations (rules) concerning these shapes and sizes as possible. Write down all your findings.

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Solutions

Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005 by Akihiko Takahashi, DePaul University, Chicago IL Advantages of the open-ended approach 1)Students participate more actively in the lesson and express their ideas more frequently. 2)Students have more opportunities to make comprehensive used of their mathematical knowledge and skills. 3)Even low-achieving students can respond to the problem in some significant ways of their own. 4)Students are intrinsically motivated to give proofs. 5)Students have rich experiences in the pleasure of discovery and receive the approval of fellow students.