Teaching and Learning Mathematics through Problem Solving Facilitators

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 2

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 3

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 4

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 5

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 6

What Does It Mean to Learn Mathematics for Teaching? 7

Deborah Loewenberg Ball Mathematics 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. 8

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 evaluate the mathematical significance of students’ comments and coordinate discussion for mathematics learning 9

Why Study Problem Solving? 10

Why Study Problem Solving? 11

Why Study Problem Solving? 12

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 openresponse 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 13

An Overview 14 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 3: Classroom Resources and Management Volume 4: Assessment and Home Connections

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. Familiar, Unfamiliar, Puzzling <put in graphic of Volume 2 – Problem solving> 15

Problem Solving Session A Activating Prior Knowledge 16

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 17

Curriculum Connections 18

Curriculum Connections 19

Warm Up – Race to Take Up Space Goal: To cover the game board with rectangles Players: 2 (individual) to 4 (teams of 2) Materials: 7 x 9 square tiles grid game board, 32 samecolour square tiles per player, 2 dice How 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? 20

Working on It – Carpet Problem Hello Grade 4 students, 1. What is the problem to solve? 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. 2. Why is this problem a problem? From your principal 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 21

Look Back – Reflect and Connect How Were the Students Solving the Problem? Read one page from the “Problem Solving Vignette” on pp. 18– 25. 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. Mathematical Processes a) problem solving b) reasoning and proving c) reflecting d) selecting tools and computational strategies e) connecting f) representing g) communicating 22

Look Back continued 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? 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 23

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. 24

Problem Solving Session B Modelling and Representing Area 25

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) 26

Curriculum Connections 27 Specific Expectations Gr 3 Specific Expectations Gr 5 Specific Expectations Gr 4

Curriculum Connections 28 Specific Expectations

Warm Up – The Size of Things 29 How do the areas of the items compare? How do you know? 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. a five-dollar bill 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 you know that each of your polygons are 4 square units. 30

Working on It continued 31 1. What could a polygon look like that is a) 1 square unit? b) 2 square units? a) one square unit 2. a) What is the area of this polygon? b) How can you reason about half-square units? b) two 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 you know that each of your polygons is 4 square units. 32

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. Squares Rectangles Polygons Parallelograms, (composed of and so on rectangles): octagons, hexagons, and so Triangles Polygons (composed of other polygons): hexagons, and so on 33

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-andconnect part of a lesson? 3. How should teachers use think-alouds to promote learning during a math lesson? 34

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. 35

Problem Solving Session C Organizing and Coordinating Student Solutions to Problems Using Criteria 36

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 37

Curriculum Connections 38

Curriculum Connections 39

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 40

Working on It – L-shaped Problem 4 cm 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? 41 8 cm 3. For Grade 6: 6 cm a) Use only triangles. b) What is the relationship between the area of a rectangle and the area of a triangle? 4 cm

Constructing a Collective Thinkpad Bansho as Assessment for Learning 1. 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. Possible Solutions Learning Goals (Curriculum Expectations) Seating plan to record student responses 1 2 3 4 5 42

Understanding Range of Gr 5 Responses 8 cm 4 cm 8 cm 4 cm 6 cm 4 cm 6 cm 8 cm 4 cm 8 cm 4 cm 6 cm 4 cm 6 cm Will these strategies work for any size L-shaped figure? 4 cm 8 cm 4 cm 6 cm 43 Bansho

Understanding Range of Gr 6 Responses 4 cm 8 cm 4 cm 6 cm 4 cm 8 cm 4 cm 6 cm 4 cm 8 cm 4 cm 6 cm 8 cm 44 4 cm 6 cm What’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? 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 solving b) reasoning and proving c) reflecting d) selecting tools and computation strategies e) connecting f) representing g) communicating 45

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

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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. 30– 34, 38– 44, or 48– 50. 48

Problem Solving Session D Selecting and Writing Effective Mathematics Problems for Learning 49

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 50

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. 6– 7, similar to and different from your ideas? 51

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? 52

Working 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. b. c. d. e. f. Race to Take Up Space Carpet Problem The Size of Things 4 Square Units Problem Composite Shape Problem 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 53

Working 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 • 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 54

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. 55

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, twodimensional 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. 56

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. 57

Three-Part Lesson Design 58 Sessio Before (Warm Up) n During After (Working On It) (Reflect and Connect) A Race to Take Up Space The Carpet Problem consolidation problem? B The Size of Things 4 Square Units consolidation Problem 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? 59

Look Back continued 4. What are some ways that the teacher should support student problem solving for these consolidation problems? (See pp. 30– 34. ) 60

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? 61

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? 62

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? 63

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 64

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) 65