Solving the Math Problem: Think-Pair-Share Professional Leadership

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Solving the Math Problem Think-Pair-Share
Professional Leadership

Algebra Connections Teacher Education in Clear
Instruction and Responsive Assessment of Student
Learning of Algebra Patterns and Problem Solving
This project is a research grant funded by the
Institute of Education Sciences, U.S. Department
of Education, Award Number R305M040127 DePaul
University Center for Urban Education http//teach
er.depaul.edu/AlgebraConnections.html
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The Problem Situation

• NCLBtestachievementnewmatholdmathalgebrafractionde
  cimallimited
• TimeunlimitedpressureNCLBthinkingvalueaddedatandab
  ovedatadrivenNCTMteacherleadershiptimeNCLB
• timeNCLBachievementproblems

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Limited Teacher Preparation

• Mathematics
• Formative Evaluation

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Limited Teacher Preparation Limited
Student Development
Source National Assessment of Educational
Progress, 2005 National Assessment Results
http//nces.ed.gov/nationsreportcard/nrc/reading_
math_2005/s0027.asp?tab_idtab2subtab_idTab_1pr
intverchart
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A Comprehensive ResponseTeacher Development

• Three graduate courses in mathematics
• One course in Formative Evaluation
• Teaching coaching and collaboration
• Scaffolds for student problem-solving

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ADDING VALUE THROUGH TEACHER DEVELOPMENT math
strategies scaffolds for problem solving
formative evaluation teacher collaboration
School Progress
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Co-Presenters

• WHAT Barbara Radner, Ph.D., Director DePaul
  University Center for Urban Education
• HOW James Lynn, Project Manager New Leaders
  for New Schools
• Molly Reed, Teacher Leader Gray Elementary
  School
• NEXT STEPS Mirna M. Diaz Ortiz,
  Principal Nobel School

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• math thinking
• relationships proportion shape and size
  estimation
• which operation to use sequence strategies
  value

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Program Neutral
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A Scaffold for Learners and Teachers

• Researchers have shown that self-explanations
  during learning or problem solving are positively
  correlated with learning and problem-solving
  measures (p. 197). Neuman and Schwarz suggest
  three broad categories of self-explanation
  clarification, inference, and justification.
  Clarification entails explaining the problem
  space. Justification entails giving reasons
  that a particular solution step was taken.
  Inference entails generating new knowledge
  having the general form of Ifthen.
• Neuman, Y Leibowitz, L., Schwarz, B. (2000).
  Patterns of verbal mediation during problem
  solving A sequential analysis of explanation.
  The Journal of Experimental Educational, 68(3),
  197-213.

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Teacher CompetenceTeacher Commitment
?Student Progress
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Fifth Grade ITBS Math Gains 2005 to 2006
ComparisonsTreatment and Limited Treatment
Groups (YEAR 2)
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Eighth Grade ITBS Math Gains 2005 to 2006 (YEAR
2)Teacher Commitment 1 vs. 3 vs. 4
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Sixth Grade ITBS Math Gains 2005 to 2006 (YEAR
2)Teacher Competence Gain 0 vs. 1 vs. 2
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Ask the Learners How do you learn? What is
difficult? What helps you learn?
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Summary of Student Attitudes and Beliefs, pre- to
post-treatment changes in frequency and character
count
Content Analysis Year 1
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Summary of Motivation for Learning Math, pre- to
post-treatment changes in frequency and character
count
Content Analysis Year 1
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Summary of Student Perception of Teacher
Techniques, pre- to post-treatment changes in
frequency
Content Analysis Year 1
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Co-Presenters HOW James Lynn, Project
Manager New Leaders for New Schools

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The Pond Border Problem

• A company specializes in installing fish ponds
  for residential landscaping. A square pond, 10
  feet on each side, is to be surrounded by a
  ceramic-tile border. The border will be one tile
  wide all around and each tile is 1 foot by 1 foot.

10 ft 10 ft 10 ft 10 ft
10 ft 10 ft 10 ft 10 ft

10 ft 10 ft 10 ft
10 ft 10 ft 10 ft
10 ft 10 ft 10 ft

• Your challenge is to figure out how many tiles
  will be needed without counting the tiles
  individually.
• Write down as many ways as you can for doing
  this, giving the specific arithmetic involved in
  detail.
• For each method that you find, draw a diagram
  that indicates how the method works.

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One Representation
This student had counted ten tiles along each edge of the pond and then added two tiles to each edge since the border extended one tile in each direction past the edge of the pond next the student multiplied by four since there were four edges finally the student subtracted 4 because four of the tiles had each been counted twice. This students arithmetic looked like that shown at the right.
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Other Representations
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Representing the Situations using Arithmetic
12 10 x 2 x 2 24 20 24 20 44 12 11 11 10 44 10 x 4 40 4 44 12 10 x 12 x 10 144 100 144 100 44 11 x 4 44
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Generalizing the Pond Border Problem

• The company decides that it would like to have a
  general formula for the case of the square pond
  that gives the number of tiles needed as a
  function of the size of the pond.
• Using s for the length of a side of the square
  pond, find a general formula for the number of
  tiles needed.

s

• Find a formula for each of the different ways of
  seeing the problem.
• Make your formula match the arithmetic as closely
  as possible.

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One Generalization
Example For the case where the student had counted ten tiles along the edge, extended by one tile in each direction, and then subtracted four for the double counting, the matching formula would be
4 (s 2) 4
ARITHMETIC ? ALGEBRA
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Building Bridges from Arithmetic to Algebra
12 10 x 2 x 2 24 20 24 20 44 12 11 11 10 44 10 x 4 40 4 44 12 10 x 12 x 10 144 100 144 100 44 11 x 4 44

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Building Bridges from Arithmetic to Algebra
12 10 x 2 x 2 24 20 24 20 44 12 11 11 10 44 10 x 4 40 4 44 12 10 x 12 x 10 144 100 144 100 44 11 x 4 44

2(s2)2s
(s2)2(s1)s
4s4
(s2)2 - s2
4(s1)
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The Pond Border Problem Extension 1

• Suppose the pond is not a square. For example,
  what if the pond were 8 feet by 6 feet, as shown
  in the diagram below?
  

Explore examples like this and then develop an
expression for the number of tiles needed for the
border of a pond that is m feet by n feet.
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The Pond Border Problem Extension 2

• Consider the problem of creating a border 2 feet
  wide. For example, for a pond 10 feet by 10
  feet, the border would look like the diagram
  below. How many tiles would be needed?

• In general, how many tiles would be need for a a
  border like this for a pond that is s feet by s
  feet?
• And what about for a rectangular pond that is m
  feet by n feet?
• Generalize the problem further by considering a
  border that is r feet wide all around.

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Developing Algebraic Thinking

• Formal

Pre-formal
Informal
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Algebraic Habits of Mind
Doing-undoing
Building rules to represent functions
Abstracting from computation
Driscoll, M. (1999). Fostering algebraic
thinking A guide for teachers grades 6-10.
Portsmouth, NH Heinemann.
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Fundamental Components of Algebraic Thinking
Understanding Patterns, Relations, and Functions
Analyzing Change in Various Contexts
Exploring Linear Relationships
Using Algebraic Symbols
Burke, M., et al. (2001). Navigating through
algebra in grades 9-12. Reston, VA National
Council of Teachers of Mathematics.
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Literacy in Mathematics Class
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Powerful Practices in Mathematics Class
Modeling
Justifying
Generalizing
Carpenter, T. P., Lehrer, R. (1999). Teaching
and learning mathematics with understanding. In
E. Fennema T. A. Romberg (Eds.), Classrooms
that promote mathematical understanding (pp.
1932). Mahwah, NJ Erlbaum.
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Toward a More Expansive View of Algebra

• Implications of our research findings include
  the need to broaden teachers conceptions of what
  it means for students to think algebraically so
  that their focus shifts away from particular
  representations (e.g., symbol use is inherently
  algebraic) and towards the student thinking
  behind these representations. Teachers who
  understand these links will be better equipped to
  facilitate student connections between
  representations.
• Asquith, P., Stephens, A., Grandau, L., Knuth, E.
  Alibali, M.W. (2005). Investigating
  middle-school teachers perceptions of algebraic
  thinking. Paper presented at the American
  Educational Research Association Annual Meeting,
  Montreal, Canada.

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Algebra Curriculum Foundational Concepts
Graphing in the x-y plane
Multi-step problem solving
ALGEBRA CONTENT
Understanding variables and patterns
Signed number operations
Exponents
Fractions, percents, and proportional reasoning
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Algebra in Grades K 2

• Students begin their study of algebra in early
  elementary grades by learning about the use of
  pictures and symbols to represent variables. They
  look at patterns and describe those patterns.
  They begin to look for unknown numbers in
  connection with addition and subtraction number
  sentences. They model the relationships found in
  real-world situations by writing number sentences
  that describe those situations. At these grade
  levels, the study of algebra is very much
  integrated with the other content standards.
  Children should be encouraged to play with
  concrete materials, describing the patterns they
  find in a variety of ways.
• New Jersey Mathematics Curriculum Framework

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Algebra in Grades 3 4

• Although the formality increases in grades 3 and
  4, it is important not to lose the sense of play
  and the connection to the real world that were
  present in earlier grades. As much as possible,
  real experiences should generate situations and
  data which students attempt to generalize and
  communicate using ordinary language. Students
  should explain and justify their reasoning orally
  to the class and in writing on assessments using
  ordinary language. When introducing a more formal
  method of communicating, such as the language of
  algebra, it is helpful to revisit some of the
  situations used in previous grades.
• New Jersey Mathematics Curriculum Framework

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Algebra in Grades 5 6

• It is important that students continue to have
  informal algebraic experiences in grades 5 and 6.
  Students have previously had the opportunity to
  generalize patterns, work informally with open
  sentences, and represent numerical situations
  using pictures, symbols, and letters as
  variables, expressions, equations, and
  inequalities. At these grade levels, they will
  continue to build on this foundation.
• Algebraic topics at this level should be
  integrated with the development of other
  mathematical content to enable students to
  recognize that algebra is not a separate branch
  of mathematics. Students must understand that
  algebra is an expansion of the arithmetic and
  geometry they have already experienced and a tool
  to help them describe situations and solve
  problems.
• New Jersey Mathematics Curriculum Framework

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Algebra in Grades 7 8

• Students in grades 7 and 8 continue to explore
  algebraic concepts in an informal way. By using
  physical models, data, graphs, and other
  mathematical representations, students learn to
  generalize number patterns to model, represent,
  or describe observed physical patterns,
  regularities, and problems. These informal
  explorations help students gain confidence in
  their ability to abstract relationships from
  contextual information and use a variety of
  representations to describe those relationships.
  Manipulatives such as algebra tiles provide
  opportunities for students with different
  learning styles to understand algebraic concepts
  and manipulations. Graphing calculators and
  computers enable students to see the behaviors of
  functions and study concepts such as slope.
• New Jersey Mathematics Curriculum Framework

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Algebra in Grades 7 8

• Students need to continue to see algebra as a
  tool which is useful in describing mathematics
  and solving problems. The algebraic experiences
  should develop from modeling situations where
  students gather data to solve problems or explain
  phenomena. It is important that all concepts are
  presented within a context, preferably one
  meaningful to students, rather than through
  traditional symbolic exercises. Once a concept is
  well-understood, the students can use traditional
  problems to reinforce the algebraic manipulations
  associated with the concept.
• New Jersey Mathematics Curriculum Framework

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HOW Molly Reed, Teacher Leader Gray
Elementary School
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The Teacher Connection
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At the classroom level Algebra
Projects Writing in Math Problem of the Week
(POW)
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Algebra Projects
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Algebra Projects
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Writing in Math
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Writing in Math
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Problem of the Week
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Problem of the Week
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Problem of the Week
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Problem of the Week
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POW Gains1st Semester 2006-2007
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Multiplying the Solution Across the
Grade-level Math Club Multi-age Grouping
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Sharing ideas across the grade level
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Sharing ideas across the grade level
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Math Club
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Multi-age Grouping
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Multi-age Grouping
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Multi-age Grouping
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Multi-age Grouping
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Multi-age Grouping
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Making Math a Priority Math Night (grades
3-5) School-wide Inservices Curriculum
Backmapping
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NEXT STEPS Mirna Diaz Ortiz, Principal
Nobel School
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Next Steps for Your School
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