1. The Mathematical Processes Targeted Implementation Program Planning Supports, Gr. 7, 8, 9Applied, Ontario Ministry of Education and Training, 2003. TIPS4RM (TIPS for Revised Mathematics), Gr. 7-10, Ontario Ministry of Education and Training, 2006.

2. Mathematical Processes Problem Solving Reasoning and Proving Reflecting Selecting Tools and Computational Strategies Connecting Representing Communicating

3. The Mathematical Processes They are the actions of mathematics. They are the ways of acquiring and using the content, knowledge and skills of mathematics. They are interconnected.

4. Relationship Between Knowledge, Skills and Mathematical Processes The “tree” model on the next slide illustrates the relationship among knowledge, facts and procedures, prior knowledge and understanding, and the seven mathematical processes. Prior knowledge and understanding is activated through learning experiences that provide opportunities to communicate and problem solve. Students mathematically process their prior knowledge and understanding to develop new learning that includes facts, skills and concepts. This new learning now becomes part of a students understanding and prior knowledge that can be used to further process new knowledge and understanding.

5. Relationship Between Knowledge, Skills and Mathematical Processes Reasoning and Proving Reflecting Representing Connecting Selecting Tools and Strategies Problem Solving Communicating CONCEPTS SKILLS CONCEPTS FACTS PRIOR KNOWLEDGE AND UNDERSTANDING

6. Facts and Procedures Facts, such as definitions and formulas, and procedures or algorithms, are one aspect of mathematics.

7. Conceptual Understanding

8. To be a versatile problem solvers in mathematics, students must understand concepts, and see mathematics as a connected whole.

9. Representing Reflecting Reasoning and Proving Connecting Selecting Tools and Computational Strategies Problem Solving Communicating Mathematical Processes

10. The Mathematical Processes are the actions of doing mathematics. They form the primary gear of the metaphor. The interconnected processes of Representing, Reflecting, Reasoning and Proving, Connecting and Selecting Tools and Computational Strategies make up disk of the gear.

11. Mathematical Processes It is essential that our students are capable of communicating their mathematics therefore it is essential that they are given opportunities to practice. Communicating is integral in each of the other processes. Problem Solving is the driver of all mathematics. The solution of rich mathematics questions requires the application of skills and understanding of concepts and incorporates some or all of the mathematical processes.

12. Mathematical Proficiency

13. As the gears mesh, the three components come together to build our model. Interacting with the mathematics through the mathematical processes provides students the opportunity to connect facts and procedures to conceptual understanding. For example, the volume of all right prisms, including cylinders, is found by multiplying the area of the base by the height. Students extend this conceptual understanding of the volume of a right prism to develop the procedure for finding the volume of pyramids by interacting with some of the mathematical processes. The 3 gears are clearly connected and working together to enhance students overall understanding and ability to do mathematics.

14. Connecting Achievement Chart and Mathematical Processes Reasoning and Proving Thinking Problem Solving Knowledge and Understanding Reflecting Communication Application RepresentingCommunicating Selecting Tools and StrategiesConnecting Procedural KnowledgeConceptual Understanding Mathematical Processes

15. Learning mathematics … requires understanding and being able to apply procedures, concepts and processes. In the twenty-first century, all students should be expected to understand and be able to apply mathematics. NCTM, Principles and Standards, 2000.