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Teaching and Learning Mathematics through Problem Solving

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Teaching and Learning Mathematics through Problem Solving Facilitator s Handbook A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – PowerPoint PPT presentation

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Title: Teaching and Learning Mathematics through Problem Solving


1
Teaching and Learning Mathematics through Problem
Solving
Facilitators Handbook A Guide to Effective
Instruction in Mathematics, Kindergarten to
Grade 6 (with reference to Volume Two)
The Literacy and Numeracy Secretariat
Professional Learning Series
2
Aims of Numeracy Professional Learning
  • Promote the belief that all students have learned
    some mathematics through their lived experiences
    in the world and that the mathematics classroom
    is one where students bring that thinking to
    their work
  • Build teachers expertise at setting classroom
    conditions where students can move from their
    informal math understandings to generalizations
    and formal representations of their mathematical
    thinking
  • Assist educators working with teachers of
    students in the junior division to implement
    student-focused instructional methods referenced
    in A Guide to Effective Instruction in
    Mathematics, Kindergarten to Grade 6 to improve
    student achievement

3
Aims continued
  • Have teachers experience mathematical problem
    solving as a model of what effective math
    instruction entails by
  • collectively solving problems relevant to
    students lives that reflect the expectations in
    the Ontario mathematics curriculum
  • viewing and discussing the thinking and
    strategies in the solutions
  • sorting and classifying the responses to a
    problem to provide a visual image of the range of
    experience and understanding of the mathematics
  • analysing the visual continuum of thinking to
    determine starting points and next steps for
    instruction

4
Overall Learning Goals for Problem Solving
  • During this session, participants will
  • become familiar with the notion of learning
    mathematics for teaching as a focus for numeracy
    professional learning
  • experience learning mathematics through problem
    solving
  • solve problems in different ways
  • develop strategies for teaching mathematics
    through problem solving

5
Effective Mathematics Teaching and Learning
  • Mathematics classrooms must be challenging and
    engaging environments for all students, where
    students learn significant mathematics.
  • Students are called to engage in solving rich and
    relevant problems. These problems offer several
    entry points so that all students can achieve,
    given sufficient time and support.
  • Lessons are structured to build on students
    prior knowledge.

Agree, Disagree, Not Sure
6
Effective Mathematics Teaching and Learning
continued
  • Students develop their own varied solutions to
    problems and thus develop a deeper understanding
    of the mathematics involved.
  • Students consolidate their knowledge through
    shared and independent practice.
  • Teachers select and/or organize students
    solutions for sharing to highlight the
    mathematics learning (e.g., bansho, gallery walk,
    math congress).
  • Teachers need specific mathematics knowledge and
    mathematics pedagogy to teach effectively.

Agree, Disagree, Not Sure
7
What Does It Mean to Learn Mathematics for
Teaching?
8
Deborah Loewenberg BallMathematics for Teaching
  • Expert personal knowledge of subject matter is
    often, ironically, inadequate for teaching.
  • It requires the capacity to deconstruct ones own
    knowledge into a less polished final form where
    critical components are accessible and visible.
  • Teachers must be able to do something perverse
    work backward from a mature and compressed
    understanding of the content to unpack its
    constituent elements and make mathematical ideas
    accessible to others.
  • Teachers must be able to work with content for
    students while it is in a growing and unfinished
    state.

9
What Do Teachers Need to Know and Be Able to Do
Mathematically?
  • Understand the sequence and relationship between
    math strands within textbook programs and
    materials within and across grade levels
  • Know the relationship between mathematical ideas,
    conceptual models, terms, and symbols
  • Generate and use strategic examples and different
    mathematical representations using manipulatives
  • Develop students mathematical communication
    description, explanation, and justification
  • Understand and evaluate the mathematical
    significance of students comments and coordinate
    discussion for mathematics learning

10
Why Study Problem Solving?
11
Why Study Problem Solving?
12
Why Study Problem Solving?
13
Why Study Problem Solving?
  • EQAO suggests that
  • a significant number of Grades 3 and 6 students
    exhibited difficulty in understanding the demands
    of open-response problem-solving questions in
    mathematics
  • many Grades 3 and 6 students, when answering
    open-response questions in mathematics, had
    difficulty explaining their thinking in
    mathematical terms

Excerpted from EQAO. (2006). Summary of Results
and Strategies for Teachers Grade 3 and 6
Assessments of Reading, Writing, and Mathematics,
2005 2006
14
An Overview
A Guide to Effective Instruction in Mathematics,
Kindergarten to Grade 6
Volume 5 Teaching Basic Facts and Multidigit
Computations
Volume 1 Foundations of
Mathematics Instruction
Volume 2 Problem Solving and
Communication
Volume 4 Assessment and Home Connections
Volume 3 Classroom Resources and Management
15
In What Ways Does A Guide to Effective
Instruction in Mathematics Describe Problem
Solving?
  • 1. List 2 ideas about problem solving that are
    familiar.
  • 2. List 2 ideas about problem solving that are
    unfamiliar.
  • 3. List 2 ideas about problem solving that are
    puzzling.
  • ltput in graphic of Volume 2 Problem solvinggt

Familiar, Unfamiliar, Puzzling
16
Problem SolvingSession A
  • Activating Prior Knowledge

17
Learning Goals of the Module
  • Experience learning mathematics through
  • problem solving by
  • identifying what problem solving looks, sounds,
    and feels like
  • relating aspects of Polyas problem-solving
    process to problem-solving experiences
  • experiencing ways that questioning and prompts
    provoke our mathematical thinking

18
Curriculum Connections
19
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20
Warm Up Race to Take Up Space
  • Goal To cover the game board with rectangles
  • Players 2 (individual) to 4 (teams of 2)
  • Materials 7x9 square tiles grid game board, 32
    same- colour square tiles per player, 2 dice
  • How to Play
  • Take turns rolling the dice to get 2 numbers.
  • Multiply the 2 dice numbers to calculate the area
    of a rectangle (e.g., 4, 6 ? area 24 square
    units).
  • Construct a rectangle of the area calculated,
    using square tiles of the same colour.
  • Place your rectangle on the game board.
  • Lose a turn if the rectangle you constructed
    cannot be placed on the empty space on the game
    board.
  • The game ends when no more rectangles can be
    placed on the game board. Which player is left
    with the most tiles?

21
Working on It Carpet Problem
Hello Grade 4 students, The carpet you have been
asking for arrives tonight. Please clear a space
in your room today that will fit this new carpet.
The perimeter of the carpet is 12 m. From your
principal
  1. What is the problem to solve?
  2. Why is this problem a problem?
  3. Show two different ways to solve this carpet
    problem.
  4. How do you know we have all the possible
    solutions?

A Guide to Effective Instruction, Vol. 2
Problem Solving, pp. 1825
22
Look Back Reflect and Connect How Were the
Students Solving the Problem?
  • Read one page from the Problem Solving Vignette
    on pp. 1825.
  • Mathematical Processes
  • problem solving
  • reasoning and proving
  • reflecting
  • selecting tools and computational strategies
  • connecting
  • representing
  • communicating

1. What mathematics was evident in the students
development of a solution to the carpet
problem? 2. Describe the mathematical processes
that the students were using to develop a
solution. 3. Provide specific details from the
vignette text to justify your description.
23
Look Back continued
  • Focus on the one or two pages that you read from
    the Problem- Solving Vignette.

Polyas Problem-Solving Process Understand the
Problem Communicate talk to understand the
problem Make a Plan Communicate discuss ideas
with others to clarify strategies Carry Out the
Plan Communicate record your thinking using
manipulatives, pictures, words, numbers, and
symbols Look Back Communicate verify,
summarize/ generalize, validate, and explain
4. What questions does the teacher ask to make
the problem-solving process explicit? 5. What
strategies does the teacher use to engage all the
students in solving this carpet problem?
24
Next Steps in Our Classroom
  1. Describe two strategies from the Problem-
    Solving Vignette that you use and two strategies
    that you will begin to use in your classroom to
    engage students in problem solving.
  2. Keep a written record of the questions that you
    ask to make the problem-solving process explicit.
  3. Practise noticing the breadth of mathematics that
    students use in their solutions.
  4. During class discussions, make explicit comments
    about the mathematics students are showing in
    their solutions.

25
Problem SolvingSession B
Modelling and Representing Area
26
Learning Goals of the Module
  • Solving problems in different ways and developing
    strategies for teaching mathematics through
    problem solving in order to
  • understand the range of students mathematical
    thinking (mathematical constructs) inherent in
    solutions developed by students in a combined
    Grade 4 and 5 class
  • develop strategies for posing questions and
    providing prompts to provoke a range of
    mathematical thinking
  • develop strategies for coordinating students
    mathematical thinking and communication (bansho)

27
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28
(No Transcript)
29
Warm Up The Size of Things
  • How do the areas of the items compare? How do
    you know?

a five-dollar bill
  • 1. Examine the cards in the envelope on your
    table.
  • Order the items on the cards from smallest to
    largest area.
  • How do you know that your order is accurate?
  • 3. Compare the order of your cards with that of
    another group at your table.
  • 4. Discuss any differences you observe.

a credit card
a cheque
an envelope
30
Working on It 4 Square Units Problem
  • Show as many polygons as possible
  • that have an area of 4 square units.
  • a) Create your polygons on a geoboard.
  • b) Record them on square dot paper.
  • c) Label the polygons by the number of sides
    (e.g., triangle, rectangle, quadrilateral,
    octagon).
  • d) Show how you know that each of your polygons
    are 4 square units.

31
Working on It continued
  • What could a polygon look like that is
    a) 1 square unit?
  • b) 2 square units?

32
Working on It continued
  • 3. Show as many polygons as possible that have
    an area of 4 square units.
  • a) Create your polygons on a geoboard.
  • b) Record them on square dot paper.
  • c) Label the polygons by the number of sides
    (e.g., triangle, rectangle, quadrilateral,
    octagon).
  • d) Show how you know that each of your polygons
    is 4 square units.

33
Constructing a Collective Thinkpad Bansho as
Assessment for Learning
Organize student solutions to make explicit the
mathematics inherent in this problem. Solutions
that show similar mathematical thinking are
arranged vertically to look like a concrete bar
graph.
Polygons (composed of other polygons) hexagons,
and so on
Squares
Rectangles
Polygons (composed of rectangles) octagons,
hexagons, and so on
Parallelograms, and so on
Triangles
34
Look Back Reflect and Connect Questioning and
Prompting Students to Share Their Mathematical
Thinking
  • See Volume 2, Problem Solving and Communication
    (pp. 3234).
  • In groups of 2 or 3, share the reading and
  • answer these questions
  • What is the purpose of carefully questioning and
    prompting students during and after problem
    solving?
  • What are a few things that teachers need to keep
    in mind when preparing questions for a
    reflect-and-connect part of a lesson?
  • 3. How should teachers use think-alouds to
    promote learning during a math lesson?

35
Next Steps in Our Classroom
  1. Describe 2 strategies that you use to get
    students to share their mathematical thinking as
    they solve problems.
  2. Describe 2 strategies that you will begin to use
    in your classroom to engage students in
    communicating their mathematical thinking.
  3. Keep a written record of the prompts you use to
    unearth the mathematical ideas during problem
    solving.
  4. Practise noticing the breadth of mathematics that
    students use in their solutions.
  5. During class discussions, make explicit comments
    about the mathematics students are showing in
    their solutions.

36
Problem SolvingSession C
Organizing and Coordinating Student Solutions to
Problems Using Criteria
37
Learning Goals of the Module
  • Develop strategies for teaching mathematics
  • through problem solving by
  • recognizing and understanding the range of
    mathematical thinking (e.g., concepts,
    strategies) in students solutions
  • organizing student solutions purposefully to make
    explicit the mathematics
  • developing strategies for coordinating students
    mathematical thinking and communication (bansho)
  • describing the teachers role in teaching through
    problem solving

38
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39
Curriculum Connections
40
Warm Up Composite Shape Problem
  • What could a composite shape look like that
  • has an area of 4 square units?
  • is composed of 3 rectangles (Grade 5)?
  • is composed of at least one rectangle, one
    triangle, and one parallelogram (Grade 6)?
  • 1. Draw and describe at least one composite shape
    that meets these criteria.
  • 2. Explain the strategies you used to create each
    composite shape.
  • 3. Justify how your composite shape meets the
    criteria listed in the problem.

Think-Aloud
41
Working on It L-shaped Problem
  • 1. What is the area of this shape?
  • a) Show at least 2 different solutions.
  • b) Explain the strategies used.
  • 2. For Grade 5
  • a) Use only rectangles.
  • b) What is the relationship between the
    side lengths and the area of the rectangle?
  • For Grade 6
  • a) Use only triangles.
  • b) What is the relationship between the area of
    a rectangle and the area of a triangle?

42
Constructing a Collective Thinkpad Bansho as
Assessment for Learning
  • What does the teacher need to do to understand
    the range of student responses? (See pp. 4850.)
  • 2. What does the teacher need to know and do to
    coordinate class discussion so it builds on the
    mathematical knowledge from student responses?
    (See pp. 4850.)

Mathematics Chapter/Unit - Gr 5 and 6 Calculating Area of Rectangles and Triangles Mathematics Chapter/Unit - Gr 5 and 6 Calculating Area of Rectangles and Triangles Date
Mathematics Task/Problem What is the area of this figure? Show 2 different solutions. Gr 5 use rectangles. Gr 6 use triangles. Learning Goals (Curriculum Expectations) Possible Solutions 1 2 3 4 5 6
Seating plan to record student responses Seating plan to record student responses Possible Solutions 1 2 3 4 5 6


43
Understanding Range of Gr 5 Responses
Bansho
Will these strategies work for any size L-shaped
figure?
44
Understanding Range of Gr 6 Responses
Whats the relationship between calculating the
area of a rectangle and calculating the area of a
triangle?
Bansho
45
Look Back Reflect and Connect
  • 1. What mathematics is evident in the solutions?
  • Which problem-solving strategies were used to
    develop solutions? (See pp. 3844.)
  • How are the following mathematical processes
    evident in the development of the solutions
  • a) problem solving
  • b) reasoning and proving
  • c) reflecting
  • d) selecting tools and computation strategies
  • e) connecting
  • f) representing
  • g) communicating

46
Look Back continued
  • 4. What are some ways that the teacher can
    support student problem solving? (See pp.
    30-34.)

47
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48
Next Steps in Our Classroom
  • 1. Choose 4 student work samples to analyse and
    describe in terms of
  • a) the mathematics evident in their work
  • b) the problem-solving strategies used to
    develop their solutions.
  • 2. Reflect on and apply 2 of the following
    strategies to support student learning of
    mathematics through problem solving
  • a) bansho
  • b) think-aloud
  • c) any teaching strategy from pp. 3034,

3844, or 4850.
49
Problem SolvingSession D
Selecting and Writing Effective Mathematics
Problems for Learning
50
Learning Goals of the Module
  • Develop strategies for teaching mathematics
  • through problem solving by
  • identifying the purpose of problems for learning
    mathematics
  • analysing the characteristics of effective
    problems
  • analysing problems from resource materials
    according to criteria of effective problems
  • selecting, adapting, and/or writing problems

51
Warm Up About Problems
Insert cover vol 2
  • 1. What are the purposes of problems in terms of
    learning mathematics?
  • 2. How are the ideas about problems, described on
    pp. 67, similar to and different from your ideas?

52
Warm Up continued
Insert cover vol 2
  • 3. What do you think are the key aspects of
    effective mathematics problems?
  • 4. How are the ideas about mathematics problems,
    described on pp. 26-28, similar to and different
    from your ideas?

53
Working on It Analysis of Problems
  • Criteria for Effective
  • Mathematics Problems
  • solution is not immediately obvious
  • provides a learning situation related to a key
    concept as per grade-specific curriculum
    expectations
  • promotes more than one solution and strategy
  • situation requires decision making above and
    beyond choosing a mathematical operation
  • solution time is reasonable
  • encourages collaboration in seeking solutions
  • 1. How do the following problems from sessions
    A, B, and C measure up to the Criteria for
    Effective Mathematics Problems?
  • a. Race to Take Up Space
  • b. Carpet Problem
  • c. The Size of Things
  • d. 4 Square Units Problem
  • e. Composite Shape Problem
  • f. L-shaped Problem
  • 2. What are the relationships among the six
    problems?

54
Working On It continued
  • Criteria for Effective
  • Mathematics Problems
  • solution is not immediately obvious
  • provides a learning situation related to a key
    concept as per grade-specific curriculum
    expectations
  • promotes more than one solution and strategy
  • situation requires decision making above and
    beyond choosing a mathematical operation
  • solution time is reasonable
  • encourages collaboration in seeking solutions
  • 3. Consolidation is the third part of the
    three-part problem solving-based lesson. What
    does it mean to consolidate learning in a lesson?
  • 4. Write consolidation problems for session A,
    B, and C, using the Criteria for Effective
    Mathematics Problems.

55
Curriculum Connections Session A
  • Grade 3
  • Estimate, measure (i.e., using centimeter grid
    paper, arrays), and record area (e.g., if a row
    of 10 connecting cubes is approximately the width
    of a book, skip counting down the cover of the
    book with the row of cubes i.e., counting 10,
    20, 30, ... is one way to determine the area of
    the book cover).
  • Grade 4
  • Determine, through investigation, the
    relationship between the side lengths of a
    rectangle and its perimeter and area (Sample
    problem Create a variety of rectangles on a
    geoboard. Record the length, width, area, and
    perimeter of each rectangle on a chart. Identify
    relationships.)
  • Pose and solve meaningful problems that require
    the ability to distinguish perimeter and area.

56
Curriculum Connections Session B
  • Grade 3
  • Estimate, measure, and record area using standard
    units.
  • Describe through investigation using grid paper,
    the relationship between the size of a unit of
    area and the number of units needed to cover a
    surface.
  • Grade 4
  • Estimate, measure, and record area, using a
    variety of strategies.
  • Determine the relationships among units and
    measurable attributes, including the area of
    rectangles.
  • Pose and solve meaningful problems that require
    the ability to distunuish perimeter and area
  • Grade 5
  • Estimate, measure, and record area using a
    variety of strategies.
  • Estimate and measure the perimeter and area of
    regular and irregular polygons.
  • Create through investigation using a variety of
    tools and strategies, two-dimensional shapes with
    the same area.
  • Grade 6
  • Construct a rectangle, a square, a triangle, and
    a parallelogram using a variety of tools given
    the area.

57
Curriculum Connections - Session C
  • Grade 5
  • Estimate and measure the area of irregular
    polygons using a variety of tools.
  • Determine through investigation using a variety
    of tools and strategies, the relationships
    between the length and width of a rectangle and
    its area and generalize to develop a formula.
  • Grade 6
  • Construct a rectangle, a square, a triangle using
    a variety of tools.
  • Determine through investigation using a variety
    of tools and strategies, the relationship between
    the area of rectangle and the area of triangle by
    decomposing and composing.
  • Solve problems involving the estimation and
    calculation of the area of triangles.

58
Three-Part Lesson Design
Session Before (Warm Up) During (Working On It) After (Reflect and Connect)
A Race to Take Up Space The Carpet Problem consolidation problem?
B The Size of Things 4 Square Units Problem consolidation problem?
C Composite Shape Problem L-shaped Problem consolidation problem?
59
Look Back Reflect and Connect
  • 1. Solve 2 consolidation problems one that you
    wrote and one that a colleague wrote.
  • 2. What mathematics do you recognize in your
    solutions and in the solutions of your colleague?
  • 3. What mathematical processes are evident in
    your solving of the two consolidation problems?

60
Look Back continued
  • 4. What are some ways that the teacher should
    support student problem solving for these
    consolidation problems? (See pp. 3034.)

61
Next Steps in Our Classroom
  • Reflect on your classroom practices in teaching
  • mathematics through problem solving.
  • 1. How do the problems from your resource
    materials compare to the Criteria for Effective
    Mathematics Problems?
  • Gather 4 student solutions to a consolidation
    problem from your resource materials.
  • Write 2 problems that better consolidate student
    learning using the Criteria for Effective
    Mathematics Problems.
  • 4. Gather solutions to your improved
    consolidation problems from the same 4 students.
    Whats the difference in their solutions?

62
Revisiting the Learning Goals
  • During this session, participants will
  • become familiar with the notion of learning
    mathematics for teaching as a focus for numeracy
    professional learning
  • experience learning mathematics through problem
    solving
  • solve problems in different ways to develop
    strategies for teaching mathematics through
    problem solving
  • develop strategies for teaching mathematics
    through problem solving

Which learning goals did you achieve? How do you
know?
63
Revisiting the Learning Goals continued
  • 1. Describe some key ideas and strategies that
    you learned about teaching and learning
    mathematics through problem solving.
  • 2. Which ideas and strategies have you
    implemented in your classroom? Describe your own
    classroom vignette.
  • 3. How have you shared these ideas and strategies
    with teachers and school leaders at your school?
    In your region?
  • 4. How did these ideas and strategies impact
    student learning of mathematics?
  • 5. What are your next steps for continuing to
    learn mathematics for teaching?

64
Revisiting the Learning Goals continued
  • Understanding the sequence and relationship
    between math strands within textbook programs and
    materials within and across grade levels
  • Understanding the relationships among
    mathematical ideas, conceptual models, terms, and
    symbols
  • Generating and using strategic examples and
    different mathematical representations using
    manipulatives
  • Developing students mathematical communication -
    description, explanation, and justification
  • Understanding and evaluating the mathematical
    significance of students comments and
    coordinating discussion for mathematics learning

65
Professional Learning Opportunities
  • Collaborate with other teachers through
  • Co-teaching
  • Coaching
  • Teacher inquiry/study
  • View
  • Coaching Videos on Demand (www.curriculum.org)
  • Deborah Loewenberg Ball webcast
    (www.curriculum.org)
  • E-workshop (www.eworkshop.on.ca)
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