Transcript The Open-Ended Approach

An Overview
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:
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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
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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
The Open-Ended Approach
Shimada et.al.,1977, Becker & Shimada, 1997
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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
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Problems that are formulated to have multiple correct
answers.
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
Open-ended Approach
1.
2.
An open-ended problem is presented first
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.
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)
2)
3)
Demonstrates a
procedure
Assigned similar
problems to students as
exercises
Homework assignment
1)
2)
3)
4)
Presents a problem to the
students without first
demonstrating how to
solve the problem
Individual or group
problem solving
Compare and discuss
multiple solution methods
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.
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