Teaching Mathematics through Problem Solving

1. Teaching Mathematics through Problem Solving Sonja Karsh and Joyce Tonner Student Achievement Officers, Literacy and Numeracy Secretariat

2. Teaching Through Problem Solving “ It’s the questions that drive mathematics. Solving problems and making up new ones is the essence of mathematical life.” Fosnot and Dolk, 2001 Instructional Strategies: Quotations and Think-Pair-Share

3. Learning Goals During this session, numeracy coaches will: • develop an understanding of the difference between learning through problem solving andlearning about problem solving • solve problems to deepen their understanding of the big ideas of measurement • study, practise, and reflect on the critical aspects of content-focused coaching Instructional Strategies: Identify learning goals

4. Job-embedded structures we will use today: • Co-teaching • Teacher inquiry / study • Coaching

6. The meaning of “3” Discuss with your elbow partner instances in your professional life where “3” has significance. Instructional Strategies: Accessing Prior Knowledge through Connections

7. Building the Who’s Whoof Teaching through Problem Solving In your group, reflect on the characteristics we can list on our Who’s Who of “Teaching Through Problem Solving.” Instructional Strategies: Creating an anticipation chart

8. 1: The “before” … from front matter of Coaching binder (Coaching Institute 2006), www.curriculum.org The pre-lesson conference • The teacher clarifies lesson goals and objectives • The coach becomes familiar with the teacher’s thinking, beliefs, and knowledge • The coach and the teacher agree to be accountable for effective student learning • The coach and teacher collaboratively design the lesson • The coach and teacher develop a shared view of understandings, strategies, concepts and skills on which students are working Instructional Strategies: Conferencing

9. The “before” continued • The coach and teacher determine what will be the evidence of student achievement • The coach and teacher identify resources, materials and procedures to be used in a lesson • The coach and teacher anticipate students’ responses • The coach and teacher describe the lesson’s relationship to the curriculum

10. 2: The “during” … from front matter of Coaching binder (Coaching Institute 2006), www.curriculum.org Lesson • The coach’s role is collaborative • The coach is partner with the teacher in working towards shared goals, not a critic of the teacher’s practice • The teacher and coach co-create conditions to make the lesson one in which students learn Instructional Strategies: Co-Teaching

11. The “during” continued • The teacher and coach negotiate how they collaborate; (e.g., the teacher and coach collaboratively respond to emergent and dynamic student thinking and adjust instruction as they gather evidence of learning)

12. 3: The “after” … from front matter of Coaching binder (Coaching Institute 2006), www.curriculum.org The post-lesson conference • The teacher and coach talk about how the lesson plan was implemented • The teacher and coach talk about the degree of success of the lesson • The teacher and coach discuss problems that arose and whether or not the students learned what they were supposed to learn • The teacher and coach analyse the students’ work and use the evidence as feedback for planning the next lesson

13. Measurement Strand: Attributes, Units, and Measurement Sense Grade 1 Identify and describe common two dimensional shapes Grade 2 Estimate, measure, and record the distance around the objects, using non standard units Grade 3 Estimate, measure, and record the perimeter of two-dimensional shapes, through investigation using standard units Grade 5 Estimate and the perimeter and area of regular and irregular polygons, using a variety of tools Grade 6 Select and justify the appropriate metric unit to measure length or distance in a given real-life situation Grade 4 Estimate, measure, using a variety of tools and strategies, and record the perimeter and area of polygons Big Idea: The closer a shape approaches a square, the smaller the perimeter Connections from other strands: Number Sense and Numeration - Ratio

14. The Three Part Lesson: 1 2 3 The “Warm Up” A candy box holds 4 chocolates. What are the dimensions of the box that creates the longest perimeter? Instructional Strategies: Activating Prior Knowledge

15. The Three Part Lesson: 1 2 3 The “Problem” Ms. T. promised her class that she would give them one minute of play time for every unit of a perimeter they could create.  If they create a shape 60 cm long, they will receive an extra hour to play any game of their choice!  What is the greatest perimeter the students can create inside a 10 cm by 10 cm grid? Coach’s Corner - Pre-lesson conference What questions would you ask? Content-focused coaching: The teacher explains the goals of the lesson and how she/he plans to teach it. Concept:Measuring perimeter Approach: Teaching through Problem Solving Instructional Strategies: Problem Solving and Metacognitive Reflections

16. Extensions If you received one minute for each segment of .75cm, how much time would you have now? Build the net of the polyhedron that has the greatest perimeter.

17. Math Congress Please post your work so we can share the thinking. Coach’s Corner - The lesson: Co-teaching • Joyce and Sonja confer to select the work to be shared. The purpose is to show a range of approaches and strategies, and highlight the big idea(s) of the lesson. • The relationship is collaborative and focused on student learning, not on the teacher’s performance. • The discussion focused on the representations that support their use. Co-teachers set the criteria. In this case, help in making generalizations. Instructional Strategies: Math congress

18. Math Congress Other criteria for selecting student work to be shared. • comparing solutions of others – same / different • giving feedback that can be used to edit solution presentation • different strategies • different representations • conceptual models • emerging math concepts and/or properties • error analysis Instructional Strategies: Math congress

19. Your results The Three Part Lesson: 1 23 • What strategies did you use to increase the perimeter of your shape? • How do you know you have found the greatest perimeter possible? • What is the relationship between the perimeter and the number of sides of a shape? • What is the relationship between the perimeter and the number of corners in a shape? Instructional Strategies: Analysing Data to Reveal the Math

20. Possible student solutions • Able to count units • Able to recognize we are counting small lengths, not dots or vertices • Able to use squares and rectangles • Able to recognize counting corners results in two units • Able to write their thinking in a number sentence • Able to connect the ratio of the lengths to the amount of time they win

21. Coach’s Corner - Lesson • The coach’s role is collaborative • The coach is a partner with the teacher in working towards shared goals, not a critic of the teacher’s practice • The teacher and coach co-create conditions to make the lesson one in which students learn • What questions would you ask? • What observations would you make? • What suggestions would you add to the lesson?

22. Adding to the Who’s Who In your group, reflect on what we have done to date. List any additional characteristics we could add to our Who’s Who of “Teaching Through Problem Solving.” Each group will be asked to report on one characteristic. Instructional Strategies: Re-visiting anticipation chart

23. The post-lesson conference • The teacher and the coach talk about how the lesson plan was implemented, and with what level of success. • They discuss what problems arose and whether or not the students learned what they were supposed to learn • This process involves looking at students’ work • The feedback gathered here often contributes to data used for planning the next lesson. Instructional Strategies: Pose Questions and/or Problems

24. Coach’s Corner - Post-lesson conference With a partner, role play the post-lesson conference. • What observations would you make? • What questions would you ask? • How would you determine next steps with the teacher? Remember to keep the discussion focused on students’ learning.

25. Content-focused Coaching • Pre-lesson conference • The lesson • Post-lesson conference Instructional Strategies: Reviewing key terms and structures

26. Core Issues in Math Lesson Design • Lesson goals • Lesson plan and design • Students’ relevant prior knowledge • Relationship between the nature of the task and the activity on one hand and the lesson goals on the other hand • Strategies for students to make public their thinking and understanding • Evidence of students’ understanding and learning • Students’ difficulties, confusions, and misconceptions • Ways to encourage collaboration in an atmosphere of mutual respect • Strategies to foster relevant student discussions

27. Thank you for sharing your thoughts, ideas, and expertise today. Enjoy the rest of the LNS Coaching Institute.