SCALE Middle School Math Forum

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SCALE Middle School Math Forum
December 11-12, 2005 Madison, Wisconsin
Denver Public Schools Los Angeles Unified School
District Madison Metropolitan School
District Providence (RI) Public
Schools California State University, Dominguez
Hills California State University,
Northridge University of Pittsburgh University of
Wisconsin-Madison
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Improving Middle School Student Math Proficiency
What will make a difference?
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Tonight we address middle school math teacher
in-service learning
Teacher In-service learning
Student
What are possible factors?
For middle school math teachers
What you know you should understand
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The views discussed tonight are informed
primarily by
Our understanding of the relevant research
literature
Our understanding of mathematics
20 years of math PD work at CSUDH
Five years of quilting/symmetry work in Madison
The MMSD Math Masters courses
The SCALE/QED Math Institutes
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Math Explanation Structures Math Immersion
Resources
Math Immersion Concept paper
Jackie Barab Matt Felton Eunice Krinsky Terry
Millar
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Now let us return to the
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Student
Adding It Up Helping Children Learn Mathematics
Mathematical Proficiency

• conceptual understanding
• procedural fluency
• strategic competence
• adaptive reasoning
• productive disposition

National Research Council 2001
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Adaptive Reasoning
Adaptive reasoning refers to the capacity to
think logically about the relationships among
concepts and situations. Such reasoning is
correct and valid, stems from careful
consideration of alternatives, and includes
knowledge of how to justify the conclusions. In
mathematics, adaptive reasoning is the glue that
holds everything together, the lodestar that
guides learning. Adding It Up
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Teacher?
Knowledge Understanding
xxx
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Hung-Hsi Wu The most difficult step in
becoming a good teacher is to achieve a firm
mastery of the mathematical content knowledge.
Without such a mastery, good pedagogy is
impossible.
Liping Ma As I read this research, I kept
thinking about the issue of teachers
mathematical knowledge. Could it be that the
learning gap was not limited to students? If
so, there would be another explanation for U.S.
students mathematical performance.
Tom Carpenter Teachers need flexible knowledge
that they can adapt to their students and the
demands of situations that arise in their
classes. This kind of knowledge cannot be
embedded in curriculum materials or scripted into
instructional routines.
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Flavors of Knowledge
Content Knowledge
Pedagogical Knowledge
Pedagogical Content Knowledge Shulman
Knowledge of and about Mathematics Ball
Knowledge Packages Ma
Common and Specialized Content Knowledge Ball
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Knowledge vs Understanding
Many people know that p is approximately 3.14
Fewer people understand that p is approximately
3.14
Typically neither middle school students nor
teachers need to understand that p is
approximately 3.14
But there are many things about p that they can
understand
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p is the ratio of the length of the circumference
of any circle to the length of its diameter
This is a definition and can be known and
understood
p
p
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Middle school students can understand p is less
than 4

lt
p
4



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Middle school students can understand p is
greater than 3
gt


p

gt


3
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Punch Line
For middle school mathematics teachers
What you know you should understand.
And this raises a question
What do you have when you have an understanding?
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Perkins What do you have when you have an
understanding?
An Explanation Structure
It is a rich network of explanatory
relationships that are encoded mentally in any of
the many ways the mind has available through
words, images, cases in point, anecdotes, formal
principles, and so on. This explanation structure
is more than a memorized explanation It is
extensible and revisable Inside Understanding
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Math Explanation Structures should help inform
middle school math teacher (pre-service and
in-service) learning through the use of networks
of connected problems that we call Math
Immersion Resources
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Math Explanation Structures should have
connective threads that mirror the cognitive
contours of mathematics
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Connective threads are themes that run through
explanation structures connecting diverse
elements.
These connective threads can be developed through
properly designed professional learning.
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Example of connective threads in mathematics
Similarity Sameness Equality Equivalence Congruenc
e Isomorphism
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Explanation structures and thus professional
learning should have
Extended Explanation Structure
Explanation Structure
both big ideas
and connective threads.
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Goal of examples this evening
Motivate a different long-term approach to
middle school math teacher professional
understanding that uses some math immersion
resource-based professional development
Tonight we will model this with immersion-like
games from a mathematical perspective
involving adaptive reasoning and
connective threads and
building on variations of sameness
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Cautionary Note!

• These examples and the props are not themselves
  intended for middle school teacher mathematics
  professional development nor the classroom.

• Rather, they are intended to provide a common
  conceptual space that is both unfamiliar and
  accessible to most of tonights diverse audience.
  

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2-Player Game Sum of 15
Equipment
For each integer, 1 through 9 inclusive, one chip
with that number
1
2
3
4
5
6
7
9
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Play of the Game
Players alternate selecting chips from the above
collection
A win
A player whose set of chips has a subset of three
chips whose numbers add to exactly 15 has a win.
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2-Player Game Sum of 15
Example
1
2
3
4
5
6
7
9
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Player 1
Player 2
Play
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2-Player Game Sum of 150
Equipment
For the multiples of ten, 10 through 90
inclusive, one chip with that number
10
20
30
40
50
60
70
90
80
Play of the Game
Players alternate selecting chips from the above
collection
A win
A player whose set of chips has a subset of three
chips whose numbers add to exactly 150 has a win.
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Is the sum of 150 game the same
as the sum of 15 game?
The elements of 150
The correspondence
The elements of 15
Math Talk An isomorphism is a correspondence
between the elements of two structures that is
structure preserving
20 50 80
10(2 5 8)
150
1015
15
2 5 8
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The rest of this evening will be spent
on Tic-Tac-Toe and variations
How many different first moves are there?
We will investigate other variations of
sameness within a Tic-Tac-Toe framework.
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Variations on a Theme How interesting can
Tic-Tac-Toe be??
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Tic-Tac-Dull vertical and horizontal wins
Tic-Tac-Toe include diagonals
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Tic-Tac-Cylinder
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Tic-Tac-Donut
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Tic-Tac-Mobius
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Math Immersion-type investigations
Using tic-tac-X games

• Investigations
• Beginning
• Intermediate
• Advanced
• Really advanced!

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Math Immersion-type investigations
For Tic-Tac-X, X Dud-V, Dud-H, Dull, Toe,
Cylinder, Donut, Mobius
Beginning Investigations

1. How many different wins are possible?
2. For each element (square) of the game, how many
   wins can that element be in?
3. Can the players cooperate so that each has an
   edge-through-edge win (for example, a horizontal
   win)?
4. Can the players cooperate so that each has a
   corner-through-corner win (for example, a
   diagonal win)?

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Math Immersion-type investigations
For Tic-Tac-X, X Dud-V, Dud-H, Dull, Toe,
Cylinder, Donut, Mobius
Intermediate Investigations
5. Does either player have a winning strategy?
6. If the players cooperate, is a draw always
possible?
7. Is it possible for two different elements
(squares) to be in more than one win?
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Math Immersion-type investigations
For Tic-Tac-X, X Dud-V, Dud-H, Dull, Toe,
Cylinder, Donut, Mobius
Advanced Investigations
8. How many final positions are there?
9. Which pairs of games are isomorphic?
10. If game A is isomorphic to game B, and game B
is isomorphic to game C, then is it always the
case that game A is isomorphic to game C?
11. Is the sum of 15 game isomorphic to
tic-tac-toe?
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Math Immersion-type investigations
Really Advanced investigations
12. How many different Tic-Tac-X games are there
satisfying

• Can be played on a 3x3 tick-tac-toe surface
• The collection of wins is closed under the
  symmetries
• of the square
• c. There is a topological space in which the 3x3
  surface
• can be embedded so that each win is a geodesic

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Stereolithography Apparatus at the Milwaukee
School of Engineering
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Stereolithography 3D Systems, Inc., Valencia, CA

• 3D graphing calculator!
• Entire operation is driven by systems of
  (parametric) equations
• Input mathematics
• Output solid shape.

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Stereolithography
1- laser 2- mirror 3- positioning mechanism 4-
liquid polymer with photoinitiator 5- part
Math in
Reality out
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2-Player Game Sum of 15
Analysis
All possible wins
How many wins include 1?
How many wins include 2?
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the structure is the collection of possible wins
Tic-Tac-Toe Elements are the spaces for marks
the structure is the collection of possible wins
Sum of 15 Elements are the chips
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Sum of 15 elements
Tic-Tac-Toe elements
1
2
3
4
Isomorphism
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6
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The correspondence is structure preserving Under
this correspondence, every win in sum of 15
corresponds to a win in tic-tac-toe, and visa
versa
8
9
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Teachers
(Extensible and Revisable)
Principles of Learning
Mathematics
Pedagogy
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Conclusion?
Develop teacher math explanation structures by
Developing math immersion resource-based PD by
Developing new kinds of partnerships
Explanation structure
Immersion resource
New partnerships