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LEARNING
PLACEtoVALUE
Transitioning
from Elementary
Middle
Learning
Place
Value
Through
a
THROUGH
A
MEASUREMENT
School Cultivating Problem Solvers
Measurement
Context
CONTEXT
Linda Venenciano, Fay Zenigami, and Seanyelle Yagi
Curriculum Research & Development Group
Linda Venenciano, Hannah Slovin, and Fay Zenigami
University of Hawai‘i at Mānoa
University of Hawai‘i at Mānoa
2016 Math Summit
Best Practices for Math Transitions
November 18, 2016
A familiar mathematics
problem
Find the area of a rectangle that is
16 inches long and 9 inches wide.
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How problem solvers think
What processes help good problem solvers reason mathematically, communicate
their thinking, and link their learning to prior mathematics?
Based on the work of Russian psychologist, Krutetskii (1976), we believe
successful problem solvers use three cognitive processes––
•  Reversibility
•  Flexibility
•  Generalization
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Reversibility
The ability to restructure the direction of a mental process from a direct to a
reverse train of thought.
Example:
Find the dimensions of a rectangle whose area is 144 square inches.
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Flexibility
The ability to solve problems in multiple ways or use information from one
problem to solve another problem.
Example:
a.  Find the area of a rectangle that is 16 inches long and 9 inches wide.
b.  Find the area of a rectangle that is 32 inches long and 9 inches wide.
c.  Find the area of a rectangle that is 16 inches long and 18 inches wide.
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Generalization
The ability to deduce from particular cases to form a concept or to use a general
concept to solve specific cases.
Examples:
How is the area of a rectangle changed when one dimension is doubled?
Find all possible rectangles that have an area of 144 square inches.
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What processes do you
recognize?
Write an addition/subtraction fact team for each situation:
a) The sum of two positive numbers is positive.
b) The sum of a positive number and a negative number is negative.
c) The sum of a positive number and a negative number is negative.
d) The sum of two negative numbers is a negative.
Algebra I: A Process Approach PS 2-4
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A good problem solving task
•  Is challenging yet accessible
•  Promotes communication
•  Connects to other topics
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Tasks that are challenging yet
accessible lead students to––
• 
Engage in the mathematics via multiple entry points
• 
Employ different strategies as they work on the problem
• 
Draw on what what they know as they work through the task
• 
Look for and use a solution pathway that is not previously known or
apparent
• 
Explore and understand the nature of mathematical relationships,
processes, and concepts
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Tasks that promote
communication and class
discourse lead students to––
• 
Explain beyond procedural steps and checking answers
• 
Pose questions, make conjectures, justify, and extend the problem situation
• 
Problem solve productively and make the processes visible to all
• 
Participate
• 
Engage
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Tasks that connect to multiple
topics lead students to––
•  Know mathematics as a progression of interconnected topics
•  Review or practice previously learned concepts and skills, and preview
upcoming ones
•  Be resourceful and creative when solving a problem
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Adapt this task
The following is a task that would typically appear in a 6th
grade textbook. Adapt the task to address one or more of the
processes –– reversibility, flexibility, generalization
Write the following fraction in simplest form,
.
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Mahalo
[email protected]
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Possible adaptations for
“Write 6/9 in simplest form.”
●  Tia simplified her fraction to ⅔. What was Tia’s fraction?
●  Use a visual representation to prove that ⅔ is equivalent
to Tia’s fraction.
●  Kala said his fraction is equivalent to ⅔ and his numerator
is m. What is Kala’s denominator?
●  Kala said his fraction is equivalent to ⅔ and his
denominator is m. What is Kala’s numerator?
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