Teaching and Learning Mathematics through Problem Solving

Teaching and Learning Mathematics through Problem Solving

Teaching and Learning Mathematics through Problem Solving

The Literacy and Numeracy Secretariat Professional Learning Series

Facilitator’s HandbookA Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6(with reference to Volume Two)

2

Aims of Numeracy Professional LearningAims 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

• 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

Aims continuedAims continued

4

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

Overall Learning Goals for Problem SolvingOverall Learning Goals for Problem Solving

5

Effective Mathematics Teaching and LearningEffective 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 continuedEffective 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?What Does It Mean to Learn Mathematics for Teaching?

8

Deborah Loewenberg BallMathematics for TeachingDeborah 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.

9

What Do Teachers Need to Know and Be Able to Do Mathematically? 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?Why Study Problem Solving?

11


12


13Why Study Problem Solving?Why Study Problem Solving?

Excerpted from EQAO. (2006). Summary of Results and Strategies for Teachers: Grade 3 and 6 Assessments of Reading, Writing, and Mathematics, 2005 –2006

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

14

A Guide to Effective Instruction inMathematics, Kindergarten to Grade 6

Volume 3: Classroom

Resources and Management

Volume 4: Assessment and

Home Connections

Volume 5: Teaching Basic

Facts and Multidigit Computations

Volume 2: Problem Solving and

Communication

Volume 1: Foundations of

Mathematics Instruction

An OverviewAn Overview

15In What Ways Does A Guide to Effective Instruction in Mathematics Describe Problem Solving?

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.

<put in graphic of Volume 2 – Problem solving>

Familiar, Unfamiliar, Puzzling

16Problem SolvingSession AProblem SolvingSession A

Activating Prior Knowledge

17Learning Goals of the ModuleLearning 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

18

Curriculum ConnectionsCurriculum Connections

19


20Warm Up – Race to Take Up SpaceWarm Up – Race to Take Up SpaceGoal: 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 diceHow to Play:1. Take turns rolling the dice to get 2 numbers.2. Multiply the 2 dice numbers to calculate the area of a

rectangle (e.g., 4, 6 area = 24 square units).3. Construct a rectangle of the area calculated, using

square tiles of the same colour. 4. Place your rectangle on the game board.5. Lose a turn if the rectangle you constructed cannot

be placed on the empty space on the game board.6. The game ends when no more rectangles can be placed on the

game board. Which player is left with the most tiles?

21Working on It – Carpet ProblemWorking 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. 18–25

22Look Back – Reflect and Connect How Were the Students Solving the Problem?Look Back – Reflect and Connect How Were the Students Solving the Problem?

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

Mathematical Processesa) problem solvingb) reasoning and

provingc) reflectingd) selecting tools and

computational strategies

e) connectingf) representingg) 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.

23Look Back continued Look Back continued

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

Polya’s Problem-Solving Process

Understand the ProblemCommunicate – talk to understand the problem

Make a PlanCommunicate – discuss ideas with others to clarify strategies

Carry Out the PlanCommunicate – 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?

24Next Steps in Our ClassroomNext 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.

25Problem SolvingSession BProblem SolvingSession B

Modelling and Representing Area


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


Specific Expectations Gr 3



28


Specific Expectations

29Warm Up – The Size of Things Warm Up – The Size of Things

How do the areas of the items compare? How do you know?

a five-dollar bill

a cheque

a credit card

an envelope

1. Examine the cards in the envelope on your table.

2. Order the items on the cards from smallest to largest area.

3. 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.

30

Working on It – 4 Square Units ProblemWorking 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.

31Working on It continuedWorking on It continued

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

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

1. What could a polygon look like that is a) 1 square unit?

b) 2 square units?

a) one square unit b) two square units

32Working on It continuedWorking on It continued3. 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.

33Constructing a Collective Thinkpad Bansho as Assessment for LearningConstructing 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.

Squares Rectangles Polygons (composed of rectangles): octagons, hexagons, and so on

Parallelograms, and so on

Triangles Polygons (composed of other polygons): hexagons, and so on

34

Look Back – Reflect and Connect Questioning and Prompting Students to Share Their Mathematical Thinking

Look Back – Reflect and Connect Questioning and Prompting Students to Share Their Mathematical Thinking

See Volume 2, Problem Solving and Communication (pp. 32–34). In groups of 2 or 3, share the reading and answer these questions:1. What is the purpose of carefully questioning and

prompting students during and after problem solving?

2. 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?

35Next Steps in Our ClassroomNext Steps in Our Classroom1. 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.

36Problem SolvingSession CProblem SolvingSession C

Organizing and Coordinating Student Solutions to Problems Using Criteria


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

38


39


40Warm Up – Composite Shape ProblemWarm 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

41Working on It – L-shaped ProblemWorking 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?

3. For Grade 6:

a) Use only triangles.

b) What is the relationship between the area of a rectangle and the area of a triangle?

4 cm

8 cm

4 cm

6 cm

42Constructing a Collective Thinkpad Bansho as Assessment for LearningConstructing a Collective Thinkpad Bansho as Assessment for Learning1. 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.)

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 Solutions1

2

3

4

5

6

Seating plan to record student responses

43Understanding Range of Gr 5 Responses Understanding Range of Gr 5 Responses

Bansho

4 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

4 cm

8 cm

4 cm

6 cm

4 cm 4 cm 4 cm

4 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

4 cm4 cm 4 cm

8 cm 4 cm

6 cm

4 cm8 cm

4 cm6 cm

Will these strategies work for any size L-shaped figure?

44Understanding Range of Gr 6 Responses Understanding Range of Gr 6 Responses 4 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

8 cm

4 cm

6 cm

4 cm

8 cm

4 cm

6 cm

4 cm 4 cm

8 cm 4 cm

6 cm

4 cm

8 cm

4 cm

6 cm

What’s the relationship between calculating the area of a rectangle and calculating the area of a triangle?

4 cm 4 cm 4 cm

Bansho

45Look Back – Reflect and ConnectLook Back – Reflect and Connect1. What mathematics is evident in the

solutions?2. Which problem-solving strategies

were used to develop solutions? (See pp. 38–44.)

3. How are the following mathematical processes evident in the development of the solutions: a) problem solvingb) reasoning and provingc) reflectingd) selecting tools and computation

strategies e) connectingf) representingg) communicating

46Look Back continuedLook Back continued4. What are some ways that the teacher can support student

problem solving? (See pp. 30-34.)

47

48Next Steps in Our ClassroomNext Steps in Our Classroom1. 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.

49Problem SolvingSession DProblem SolvingSession D

Selecting and Writing Effective Mathematics Problems for Learning


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

51Warm Up – About ProblemsWarm Up – About Problems1. 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?

Insert cover vol 2

52

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?

Warm Up continuedWarm Up continuedInsert cover vol 2

53Working on It – Analysis of ProblemsWorking on It – Analysis of Problems

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 Spaceb. Carpet Problem c. The Size of Thingsd. 4 Square Units Probleme. Composite Shape Problemf. L-shaped Problem

2. What are the relationships among the six problems?

Criteria for EffectiveMathematics 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

54Working On It continuedWorking On It continued

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.

Criteria for EffectiveMathematics 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

55Curriculum Connections – Session ACurriculum Connections – Session AGrade 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.

56Curriculum Connections – Session BCurriculum Connections – Session BGrade 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 areaGrade 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.

57Curriculum Connections - Session CCurriculum Connections - Session CGrade 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.

58Three-Part Lesson DesignThree-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


C Composite Shape Problem

L-shaped Problem


59Look Back – Reflect and ConnectLook 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?

60Look Back continuedLook Back continued4. What are some ways that the teacher should support student

problem solving for these consolidation problems? (See pp. 30–34.)

61Next Steps in Our Classroom 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?

2. Gather 4 student solutions to a consolidation problem from your resource materials.

3. 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?

62Revisiting the Learning GoalsRevisiting 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?

63Revisiting the Learning Goals continuedRevisiting 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?

64Revisiting the Learning Goals continuedRevisiting 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

65Professional Learning OpportunitiesProfessional Learning OpportunitiesCollaborate 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)

Documents

Teaching and Learning Mathematics through Problem Solving