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


1
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
2
The Problem Situation
  • NCLBtestachievementnewmatholdmathalgebrafractionde
    cimallimited
  • TimeunlimitedpressureNCLBthinkingvalueaddedatandab
    ovedatadrivenNCTMteacherleadershiptimeNCLB
  • timeNCLBachievementproblems

3
Limited Teacher Preparation
  • Mathematics
  • Formative Evaluation

4
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
5
A Comprehensive ResponseTeacher Development
  • Three graduate courses in mathematics
  • One course in Formative Evaluation
  • Teaching coaching and collaboration
  • Scaffolds for student problem-solving

6
ADDING VALUE THROUGH TEACHER DEVELOPMENT math
strategies scaffolds for problem solving
formative evaluation teacher collaboration
School Progress
7
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

8
  • math thinking
  • relationships proportion shape and size
    estimation
  • which operation to use sequence strategies
    value

9
Program Neutral
10
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.

11
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13
Teacher CompetenceTeacher Commitment
?Student Progress
14
Fifth Grade ITBS Math Gains 2005 to 2006
ComparisonsTreatment and Limited Treatment
Groups (YEAR 2)
15
Eighth Grade ITBS Math Gains 2005 to 2006 (YEAR
2)Teacher Commitment 1 vs. 3 vs. 4
16
Sixth Grade ITBS Math Gains 2005 to 2006 (YEAR
2)Teacher Competence Gain 0 vs. 1 vs. 2
17
Ask the Learners How do you learn? What is
difficult? What helps you learn?
18
Summary of Student Attitudes and Beliefs, pre- to
post-treatment changes in frequency and character
count
Content Analysis Year 1
19
Summary of Motivation for Learning Math, pre- to
post-treatment changes in frequency and character
count
Content Analysis Year 1
20
Summary of Student Perception of Teacher
Techniques, pre- to post-treatment changes in
frequency
Content Analysis Year 1
21
Co-Presenters HOW James Lynn, Project
Manager New Leaders for New Schools

22
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.

23
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.
24
Other Representations
25
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
26
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.

27
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
28
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

29
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)
30
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.
31
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.

32
Developing Algebraic Thinking
  • Formal

Pre-formal
Informal
33
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.
34
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.
35
Literacy in Mathematics Class
36
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.
37
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.

38
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
39
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

40
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

41
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

42
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

43
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

44
HOW Molly Reed, Teacher Leader Gray
Elementary School
45
The Teacher Connection
46
At the classroom level Algebra
Projects Writing in Math Problem of the Week
(POW)
47
Algebra Projects
48
Algebra Projects
49
Writing in Math
50
Writing in Math
51
Problem of the Week
52
Problem of the Week
53
Problem of the Week
54
Problem of the Week
55
POW Gains1st Semester 2006-2007
56
Multiplying the Solution Across the
Grade-level Math Club Multi-age Grouping
57
Sharing ideas across the grade level
58
Sharing ideas across the grade level
59
Math Club
60
Multi-age Grouping
61
Multi-age Grouping
62
Multi-age Grouping
63
Multi-age Grouping
64
Multi-age Grouping
65
Making Math a Priority Math Night (grades
3-5) School-wide Inservices Curriculum
Backmapping
66
NEXT STEPS Mirna Diaz Ortiz, Principal
Nobel School
67
Next Steps for Your School
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