Teaching Mathematics through Problem Solving Sonja Karsh & Joyce Tonner August 22, 2006

1. Teaching Mathematics through Problem Solving Sonja Karsh & Joyce Tonner August 22, 2006

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

3. 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

4. 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 data management • study, practise, and reflect on the critical aspects of content-focused coaching Instructional Strategies: Identify learning goals

5. 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. The “before” … from Coaching binder 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 that students on which students are working Instructional Strategies: Conferencing

8. The “before” cont’d … from Coaching binder • The coach and teacher determine what the evidence of student achievement will be • 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

9. The “during” … from Coaching binder 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

10. The “during” cont’d … from Coaching binder • 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)

11. The “after” … from Coaching binder 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 Instructional Strategies: Conferencing

12. The problem Is there a way you can ensure more consistent winning at the game of Rock~Paper~Scissors? Coach’s Corner - Pre-lesson conference Content-focused coaching: The teacher explains the goals of the lesson and how she/he plans to teach it. Concept: Theoretical probability Approach: Teaching through Problem Solving What questions would you ask? Instructional Strategies: Problem Solving and Metacognitive Reflections

13. How do you play the game? • When trying to make a decision arbitrarily, or at random, two players face each other and count to three. • After counting to three, each player places her/his hand in either a rock, a paper, or scissors position. • Rock wins over scissors. • Scissors wins over paper. • Paper wins over rocks. • If both players’ choice is the same, the game is tied, and they must play another time to determine a win.

14. Make your predictions • Are all outcomes equally likely to occur? Why or why not? • If you think they are not equally likely, which one is more likely? Why? • What is the probability that each one will occur? Instructional Strategies: Make Predictions

15. Play “Rock~Paper~Scissors” Is there a way you can ensure more consistent winning at the game of Rock~Paper~Scissors? Instructional Strategies: Use of Games

16. Your results • Did the results in your respective dyads support the predictions you had made? Talk about the way that they did or did not support your predictions. • Would the results have changed if we had played only two games? If we had played 50 games? How would they have changed? • Is there anything your partner did that influenced your next move? • What strategies did you use during the game? (i.e., how did you decide your next move) • Do you see a trend on your recording sheet in terms of how the “winners” came out? (e.g., There were two Rocks in a row followed by two Papers, then two Scissors.) Instructional Strategies: Analysing Trend in Data to Confirm or Refute Predictions and Reveal the Math

17. the number of favourable outcomes the total number of possible outcomes Theoretical Probability Theoretical probability is • the mathematical calculation of the chances that an event will happen in theory • if all outcomes are equally likely, it is calculated as What is the theoretical probability of each outcome occurring in the game of “Rock~Paper~Scissors”? sample space Instructional Strategies: Accountable Talk

18. Paper Paper Rock Scissors Paper Rock Rock Scissors Paper Scissors Rock Scissors Sample Space (Possible Outcomes) • A list of all the possible outcomes of an event Instructional Strategies: Use of Graphic Organizer

19. Paper 1 Paper Rock 9 Scissors Paper Rock Rock Scissors Paper Scissors Rock Scissors Determining Theoretical Probability The theoretical probability of each outcome occurring is: 1 2 3 4 5 6 7 8 9 1 out of 9 1:9 Instructional Strategies: Representation

20. Going Deeper – Exploring Relationships – More Players • What would happen if you played with three people or four people? • List the sample space (possible outcomes). • Choose a way to represent the sample space. Instructional Strategies: Going Deeper – Extending Problems

21. Going Deeper – Exploring Relationships – More Elements What would happen if you added more elements to the game? • Play the game for one case. • Record your outcomes. Case 1 – Rock, Paper, Scissors Case 2 – Rock, Paper, Scissors, ___ Case 3 – Rock, Paper, Scissors, ___, ___ Instructional Strategies: Doing math for teaching

22. The “Reckoning” Analyse how the sample space changes with 2, 3, 4, or 5 players. • Look at your data • Organize your data • Discuss trends, similarities, and differences in each case and with the varying number of players.

23. 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

24. 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

25. The “Reckoning” Generalize your findings. • Talk about the relationships between the number of elements and the number of outcomes. Compare the results with the original Rock~Paper~Scissors game results. • Tell how you could predict the number of outcomes for any number of players and any number of elements. 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 from Coaching binder What questions would you ask? What observations would you make? What suggestions would you add to the lesson?

26. 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 Activating Prior Knowledge anticipation chart

27. The post-lesson conference 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. 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. Instructional Strategies: Pose Questions and/or Problems

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