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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

(with reference to Volume Two)

The Literacy and Numeracy Secretariat Professional Learning Series

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

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

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

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

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

What Does It Mean to Learn Mathematics for Teaching?

Deborah Loewenberg BallMathematics for Teaching

- Expert personal knowledge of subject matter is often, ironically, inadequate for teaching.
- It requires the capacity to deconstruct one’s 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.

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

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

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

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.

<put in graphic of Volume 2 – Problem solving>

In What Ways Does A Guide to Effective Instruction in Mathematics Describe Problem Solving?Familiar, Unfamiliar, Puzzling

Problem SolvingSession A

Activating Prior Knowledge

Learning Goals of the Module

Experience learning mathematics through

problem solving by:

- identifying what problem solving looks, sounds, and feels like
- relating aspects of Polya’s problem-solving process to problem-solving experiences
- experiencing ways that questioning and prompts provoke our mathematical thinking

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?

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

- What is the problem to solve?
- Why is this problem a problem?
- Show two different ways to solve this carpet problem.
- How do you know we have all the possible solutions?

A Guide to Effective Instruction, Vol. 2 – Problem Solving, pp. 18–25

Read one page from the “Problem Solving Vignette” on pp. 18–25.

Mathematical Processes

problem solving

reasoning and proving

reflecting

selecting tools and computational strategies

connecting

representing

communicating

Look Back – Reflect and Connect How Were the Students Solving the Problem?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.

Look Back continued

Polya’s 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

Focus on the one or two pages that you read from the “Problem- Solving Vignette.”

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?

Next Steps in Our Classroom

- 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.
- Keep a written record of the questions that you ask to make the problem-solving process explicit.
- Practise noticing the breadth of mathematics that students use in their solutions.
- During class discussions, make explicit comments about the mathematics students are showing in their solutions.

Problem SolvingSession B

Modelling and Representing Area

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)

Specific Expectations Gr 3

Specific Expectations Gr 4

Specific Expectations Gr 5

Specific Expectations

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

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.

b) two square units

2. a) What is the area of this polygon?

b) How can you reason about half-square units?

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

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.

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

See Volume 2, Problem Solving and Communication (pp. 32–34).

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?

Look Back – Reflect and Connect Questioning and Prompting Students to Share Their Mathematical ThinkingNext Steps in Our Classroom

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

Problem SolvingSession C

Organizing and Coordinating Student Solutions to Problems Using Criteria

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 teacher’s role in teaching through problem solving

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

8 cm

4 cm

6 cm

Working on It – L-shaped Problem1. 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?

Constructing a Collective Thinkpad Banshoas Assessment for Learning

What does the teacher need to do to understand the range of student responses? (See pp. 48–50.)

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. 48–50.)

4 cm

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4 cm

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6 cm

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8 cm

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Understanding Range of Gr 5 ResponsesWill these strategies work for any size L-shaped figure?

Bansho

4 cm

4 cm

4 cm

4 cm

8 cm

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8 cm

4 cm

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Understanding Range of Gr 6 ResponsesWhat’s the relationship between calculating the area of a rectangle and calculating the area of a triangle?

Bansho

Look Back – Reflect and Connect

1. What mathematics is evident in the solutions?

- Which problem-solving strategies were used to develop solutions? (See pp. 38–44.)
- 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

Look Back continued

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

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. 30–34,

38–44, or 48–50.

Problem SolvingSession D

Selecting and Writing Effective Mathematics Problems for Learning

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

Warm Up – About Problems

1. What are the purposes of problems in terms of learning mathematics?

2. How are the ideas about problems, described on pp. 6–7, similar to and different from your ideas?

Warm Up continued

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?

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?

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

Working on It – Analysis of Problems3. 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.

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

Working On It continuedCurriculum Connections – Session A solving-based lesson. What does it mean to consolidate learning in a lesson?

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.

Curriculum Connections – Session B solving-based lesson. What does it mean to consolidate learning in a lesson?

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.

Curriculum Connections - Session C solving-based lesson. What does it mean to consolidate learning in a lesson?

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.

Three-Part Lesson Design solving-based lesson. What does it mean to consolidate learning in a lesson?

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?

Look Back – Reflect and ConnectLook Back continued one that a colleague wrote.

4. What are some ways that the teacher should support student problem solving for these consolidation problems? (See pp. 30–34.)

Next Steps in Our Classroom one that a colleague wrote.

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. What’s the difference in their solutions?

Revisiting the Learning Goals one that a colleague wrote.

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?

Revisiting the Learning Goals continued one that a colleague wrote.

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?

Revisiting the Learning Goals continued one that a colleague wrote.

- 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

Professional Learning Opportunities one that a colleague wrote.

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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