Coaching Mathematical Thinking

1. Coaching Mathematical Thinking

2. Agenda Part I: What is problem solving? Part II: Word Problems vs. Problem Solving

3. What is problem solving?

4. Why problem solve? • Why is it important to subject our students to frustrating tasks? • What is our role in the development of our students’ problem-solving abilities? • How can teamwork help in the problem-solving process?

5. Teachers play an important role in the development of students’ problem-solving dispositions by creating and maintaining classroom environments, from pre-kindergarten on, in which students are encouraged to explore, take risks, share failures and successes, and question one another. In such supportive environments, students develop confidence in their abilities and a willingness to engage in and explore problems, and they will be more likely to pose problems and to persist with challenging problems. —NCTM, 2001

6. Opportunities for Problem Solving Directions: As a team, identify the multiple opportunities for problem solving in your current math lessons. Record notes and examples from your discussion in the space provided.

7. Discuss and Defend Word problems do not necessarily involve problem solving.

8. Number Maze Activity Directions: • Find a way through this maze, starting with a score of 0 on the left and ending with a score of 10 on the right. • Each time you cross an arithmetic sign and a number, you must do the math, keeping track of your changing score as you go. • Doubling back on your path is not allowed.

9. Number Maze Activity—Reflection • What did you have to know to solve this problem? • What did you have to think about to solve this problem? • How did you have to feel to solve this problem? • What did you have to do to solve this problem?

10. Elements of Mathematical Problem Solving • Concepts and skills • Cognitive processes • Attitude • Metacognition

11. Elements of Mathematical Problem Solving: Concepts and Skills Concepts • Number concepts • Geometry concepts • Algebra concepts • Statistics concepts • Measurement concepts Skills • Estimation and approximation • Mental calculation • Communication • Use of mathematical tools • Arithmetic manipulation • Algebraic manipulation • Handling of data

12. Elements of Mathematical Problem Solving: Cognitive Processes • Translation • Deductive reasoning • Inductive reasoning • Planning • Execution

13. Elements of Mathematical Problem Solving: Cognitive Processes Translation: A sentence-by-sentence conversion of the words and numbers in a problem into a mental representation. Number Maze Experience: Translation of numbers and operation symbols throughout the maze into an ongoing calculation in one’s head or on paper

14. Elements of Mathematical Problem Solving: Cognitive Processes Deductive, or logical, reasoning: A method of reasoning in which new information is deduced from existing information and a conclusion is drawn. Number Maze Experience: I know that I have to end up with 10, and my last operation before reaching 10 is to add 2; therefore, my number at that point in the maze has to be 8.

15. Elements of Mathematical Problem Solving: Cognitive Processes Inductive reasoning: A method of reasoning in which a conclusion is reached based on recognition of patterns or structures in mathematics. 2 x 1 = 2, 2 x 2 = 4, 2 x 3 = 6, 2 x 4 = 8, 2 x 5 = 10 When multiplying sequential numbers by 2, one observes that the product is an even number. Therefore, one can conclude that the product of any number and 2 is an even number.

16. Elements of Mathematical Problem Solving: Cognitive Processes Planning: The steps, strategies, and techniques one will use to organize and begin to solve the problem. • Problem-solving steps • Problem-solving strategies

17. Elements of Mathematical Problem Solving: Cognitive Processes Planning • Problem-solving steps • Understand the problem. • Read and retell. • Find the parts. • Make a plan with your team. • Choose a strategy. • Choose an operation. • Solve. • Check your answer. • Talk about the solution with your team. • Use Random Reporter to randomly select a student to share the answer and explain his or her thinking.

18. Elements of Mathematical Problem Solving: Cognitive Processes Planning • Problem-solving strategies • Sort and list. • Model the problem. • Find a pattern. • Guess and check. • Solve a simpler problem. • Work backwards. “I am going to work backwards starting with the total of 10 and ask myself what calculation would give me the number at each place of the maze.”

19. Elements of Mathematical Problem Solving: Cognitive Processes Execution: The process of carrying out the steps, strategies, and operations to solve the problem. What × 5 will give me 10? [2] What – 1 gives me 2? [3] What ÷ by 3 gives me 3? [9]

20. Elements of Mathematical Problem Solving: Attitude • Enjoying the process of mathematics • Using mathematics confidently • Persisting with the recursive nature of problem solving • Trying different approaches when the first approach does not work

21. Elements of Mathematical Problem Solving: Metacognition • Constant monitoring of strategies used to carry out tasks • Seeking alternative ways of performing a task • Checking the appropriateness and reasonableness of answers

22. Student Outcomes of Mathematical Problem Solving • Conceptual understanding • Use of strategies and reasoning • Communication • Computation and execution • Mathematical insight

23. Putting It All into Practice:Strategic Questioning • Read the problem assigned to your team. • Think of the questions/prompts you would use to guide students through the process of solving the problem. • Record the questions/prompts on the problem-solving template provided for your assigned problem. • Record your final version on the transparency provided, and choose someone from your team to share your questions/prompts with the whole group.

24. Next Steps