Choices and Challenges: Redefining Mathematics Teaching for the 21st Century

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Choices and Challenges Redefining Mathematics
Teaching for the 21st Century

• Frank Bergman Elementary School In-service
• February 8, 2002
• David S. Allen
• Kansas State University

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Good Morning!!!!!
1. Grapes of Math Activity 2. Introductions 3.
Overview of Scheduled In-service 4. Historical
Discussion on Mathematics Education
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Every weekday, 25 million children study
mathematics in our nations schools. Those at
the younger end, some 15 million of them, will
enter the adult world in the period 1995-2000.
The 40 classroom minutes they spend on
mathematics each day are largely devoted to the
mastery of computational skills which would have
been needed by a shopkeeper in the year 1940
skills needed by virtually no one today. Almost
no time is spent on estimation, probability,
interest, histograms, spreadsheets, or real
problem solvingthings which will be common place
in most of these young peoples later lives.
While the 15 million of them sit there drilling
away on those arithmetic or algebraic exercises,
their future options are bit-by-bit eroded.
(Mathematical Sciences Board, 1986)
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Choices and Challenges

• Today, we in mathematics education face a new
  set of challenges and choices as our changing
  society becomes information based, demanding more
  mathematical literacy from every member.
  (Burrill, 1997)
• Which choices will better enable our students
  to learn mathematics?
• How will the consequences of a given choice
  help students learn algebra and geometry and
  internalize number, measurement, or data?

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Choices and Challenges
Two Major Challenges 1. The need to change
classroom practices. 2. The need to seriously
rethink the definition of basic and what it means
for all students.
In consideration of these statements let us
examine a problem.
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The Rectangle Problem
What percent of the 4 x 10 rectangle is shaded?
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The Rectangle Problem
Possible solutions identified by elementary
teachers.

• Change 6/40 to an equivalent fraction
• (6/40) X (2.5/2.5) 15/100
• Some teachers wondered where the 2.5 came from.

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The Rectangle Problem
Possible solutions identified by elementary
teachers.
2. Each of the ten columns represents 10 percent,
so I took four of the shaded boxes and filled the
first column to make 10 percent and the last two
boxes fill half of the second column to make 5
percent, so the total shaded area is 15 percent.
10 5
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The Rectangle Problem
Possible solutions identified by elementary
teachers.
3. Draw the picture twice. In the first picture,
I can see that I have 6 out of 40. In the second
picture, I have 6 more out of 40 more. If I just
draw half of the picture again, I pick up 3 out
of 20, so that gives a total of 15 our of 100.
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The Rectangle Problem
Understanding percents is basic the
rectangle problem lays the foundation for
computation and visualization as powerful tools.
It provides contextual situation for thinking and
enables students to conceptualize their own
understanding and strategy for solving the
problem. (Burrill, 1997)
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Choices and Challenges
Two Major Challenges 1. The need to change
classroom practices. 2. The need to seriously
rethink the definition of basic and what it means
for all students.
What does it mean to change classroom practices?
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Challenges

• The need to change classroom practices.
• The first challenge is to improve the way we
  teach mathematics.
• Good teaching is not making learning easy! It is
  not making it hard either.
• Good teaching means that students are actively
  engaged in the learning process.
• They are involved with problems, struggle with
  ideas, and they take part in the dialogue.
• Students engaged in the rectangle problem become
  involved in trying to formulate a reasonable
  solution. This is distinctly different than just
  applying a procedure.

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Challenges

• Teaching Students to Reason Mathematically
• 1. Third International Mathematics and Science
  Study (TIMSS)
• Germany, Japan, United States (8th grade
  classrooms)
• Categorized the quality of the mathematical
  reasoning as low, medium, or high.
• Zero percent of lessons in the US emphasized
  high-level mathematical reasoning.
• 2. Students may know the definitions and
  algorithms but real mathematical reasoning comes
  through asking the right questions.
• Try this problem!!!!!

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Acrobats, Grandmas, and Ivan
The problem is to use the information given to
figure out who will win the third round of tug of
war. Round 1 On one side are four acrobats, each
of equal strength. On the other side are five
neighborhood grandmas, each of equal strength.
The result is dead even. Round 2 On one side is
Ivan, a dog. Ivan is pitted against two of the
grandmas and one acrobat. Again its a
draw. Round 3 Ivan and three of the grandmas are
on one side, and the four acrobats are on the
other. Who will win the third round?
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Acrobats, Grandmas, and Ivan
As teachers we must focus the mathematics we
teach on more than definitions and skills. These
are important, but we also need to prepare
students to move to the next level--Make
interpretations, be able to generalize and
recognize generalizations, realize that different
situations need the same mathematics, and use
this mathematics in situations that may be
unfamiliar.
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Challenges

• Communication
• Students do not understand that when they right
  down an answer they should be prepared to explain
  their thinking.
• Find ways to listen to students strategies and
  build upon their thinking to develop their
  understandings.
• Building Connections
• We need to change practices so that students see
  and understand mathematical connections.
• We need to help students see relationships that
  exist between the imbedded mathematics of similar
  problems.

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What is Basic?

• Are addition, subtraction, multiplication, and
  division the only basics?
• To conceptualize, visualize, and predict,
  students must comprehend basics that include
  algebraic reasoning, understanding space and
  shapes, and working with data and measurement.
• Students must be allowed to use skills in
  tackling meaningful problems as a way to learn
  the thinking and reasoning processes with numbers
  that they will need for their future, whatever
  that may be.
• Redefining the basics means teaching mathematics
  that parents may have never seen but will be
  essential for making sense of the world in which
  our students live.

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What is Basic?

• We must make choices from both the old --
  our experience over the years--and the new--what
  we have learned from research-- and the changes
  demanded by new areas of mathematics and a
  different society. (Burrill, 1997)
• But to the degree that discourse about
  purpose in public education concerned itself with
  the public good, it can be understood as a kind
  of trusteeship, and effort to preserve the best
  of the past, to make wise choices in the present,
  and to plan for the future.
• (Tyack and Cuban, 1996)

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What is Basic?

• We must avoid misinterpretations
• everything should be done in cooperative groups
• decrease in emphasis means none at all
• every answer to every problem has to be explained
  in writing
• the teacher is only a guide
• every problem has to involve the real world
• computational algorithms are not allowed
• students should never practice
• manipulatives are the basis for all learning

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What are the Implications?

• How should we redefine mathematics for the 21st
  century?
• Recognize the need to reexamine the concept of
  what is basic.
• Recognize and value teacher experience.
• Recognize research results and subsequent
  curricular changes.
• Recognize the role of reflection in your
  practice.
• Recognize that change does not come easy.
• Recognize that curriculum reform does not have an
  end.
• Recognize the value of communication.
• Recognize that 100 years ago the emphasis was on
  practical and immediate useful education. (They
  needed labors shouldnt we have higher
  expectations for our childrens future.

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Choices and Challenges Redefining Mathematics
Teaching for the 21st Century
Frank Bergman Elementary School
In-service February 8, 2002 David S. Allen Kansas
State University