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

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 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.

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?

Why Study Problem Solving?

Why Study Problem Solving?

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

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

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 Polyas problem-solving

process to problem-solving experiences - experiencing ways that questioning and prompts

provoke our mathematical thinking

Curriculum Connections

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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. 1825

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.

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?

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)

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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.

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

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?

Next 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 teachers role in teaching through

problem solving

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Curriculum Connections

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

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?

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

Understanding Range of Gr 5 Responses

Bansho

Will these strategies work for any size L-shaped

figure?

Understanding Range of Gr 6 Responses

Whats 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. 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

Look Back continued

- 4. What are some ways that the teacher can

support student problem solving? (See pp.

30-34.)

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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.

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

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?

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?

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?

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.

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.

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.

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.

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?

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?

Look Back continued

- 4. What are some ways that the teacher should

support student problem solving for these

consolidation problems? (See pp. 3034.)

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?

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?

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?

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

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)