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Teaching and Learning Mathematics through Problem Solving Facilitator’s Handbook A Guide to Effective Instruction in Mathematics, Kindergarten to Grade (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 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 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 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? 10 Warm Up – About Problems Insert cover vol What are the purposes of problems in terms of learning mathematics? 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 What you think are the key aspects of effective mathematics problems? 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 How 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 Square Units Problem Composite Shape Problem L-shaped Problem 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 Consolidation is the third part of the three-part problem solving-based lesson What does it mean to consolidate learning in a lesson? 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 54 Curriculum Connections – Session A Grade – • 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 – • 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 – • 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 – • 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 – • 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 – • Construct a rectangle, a square, a triangle, and a parallelogram using a variety of tools given the area 56 Curriculum Connections - Session C Grade – • 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 – • 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 Session Before 58 During After (Warm Up) (Working On It) (Reflect and Connect) A Race to Take Up Space The Carpet Problem consolidation problem? B The Size of Things Square Units consolidation Problem problem? C Composite Shape Problem L-shaped Problem consolidation problem? Look Back – Reflect and Connect Solve consolidation problems: one that you wrote and one that a colleague wrote What mathematics you recognize in your solutions and in the solutions of your colleague? What mathematical processes are evident in your solving of the two consolidation problems? 59 Look Back continued 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 How the problems from your resource materials compare to the Criteria for Effective Mathematics Problems? Gather student solutions to a consolidation problem from your resource materials Write problems that better consolidate student learning using the Criteria for Effective Mathematics Problems Gather solutions to your improved consolidation problems from the same 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 you know? 62 Revisiting the Learning Goals continued Describe some key ideas and strategies that you learned about teaching and learning mathematics through problem solving Which ideas and strategies have you implemented in your classroom? Describe your own classroom vignette How have you shared these ideas and strategies with teachers and school leaders at your school? In your region? How did these ideas and strategies impact student learning of mathematics? 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
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