The Process Standards from the National Council of Teachers of Mathematics, expanded with examples, analogies and frameworks.

1. NCTM Process Standards What are you doing?

2. PSSM K-12 Process Standards (page 52) Introduction to the Process Standards The 5 mathematical processes are: Problem Solving – finding your own way towards a new solution or understanding Reasoning and Proof –seeing and establishing relationships among ideas and facts Communication –sharing or recording your understanding Connection –relating math ideas to each other and to phenomena outside mathematics Representation – making or understanding the modeof communication

3. Process Standards PSSM K-12 Process Standards (page 52-70) Introduction to the The processes are not created equal – problem solving is central. The other fourconnectin powerful ways to problem solving. These areconcurrent, not isolated. Ina rich math activity there will be several processes going on.

4. PSSM K-12 Process Standards (page 60) Introduction to the Communication Standard Instructional programs from prekindergarten through grade 12 should enable all students to Organize and consolidate their mathematical thinking through communication; Communicate their mathematical thinking coherently and clearly to peers, teachers, and others; Analyze and evaluate the mathematical thinking and strategies of others; Use the language of mathematics to express mathematical ideas precisely.

5. Communication Framework Clear Coherent Complete Consolidated C C C C

6. Communication Standard Analogy I’m going to Detroit next week, can you tell me about your trip there? Unclear. What does that have to do with my question? I’m stuck in Grand Rapids – going in circles.

7. Communication Standard Analogy Clear (trip to Detroit) but incoherent (doesn’t make sense) Isn’t this Chicago? But I was headed for Detroit.

8. Communication Standard Analogy Clear and coherent but incomplete Youonly made it as far as Lansing.

9. Communication Standard Analogy Clear, coherent, and complete but not always consolidated Did you learn anything from your trip? Would you go to that casino again? mrfrumm@flickr

10. Communication Standard Analogy Clear, coherent, complete and consolidated. What a great trip – here’s what I learned... jbcurio@flickr

11. Problem Solving Standard PSSM K-12 Process Standards (page 52) Introduction to the Instructional programs from prekindergarten through grade 12 should enable all students to build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving.

12. Problem Solving Georg Polya first to describe it. It’s really a never ending cycle!

13. Representation Standard PSSM K-12 Process Standards (page 67) Introduction to the Instructional programs from prekindergarten through grade 12 should enable all students to Create and use representations to organize, record, and communicate mathematical ideas; Select, apply, and translate among mathematical representations to solve problems; Use representations to model and interpret physical, social, and mathematical phenomena.

14. Representation Make Interpret Bruno Postle @ Flickr Diagram for panoramic image photos Translate Model

15. PSSM K-12 Process Standards (page 64) Introduction to the Connection Standard Instructional programs from prekindergarten through grade 12 should enable all students to— Recognize and use connections among mathematical ideas; Understand how mathematical ideas interconnect and build on one another to produce a coherent whole; Recognize and apply mathematics in contexts outside mathematics.

16. Reading Strategy – Schema Readers activate schema before, during, and after reading. add and make changes to their schema based on new information. use schema to relate text to their world knowledge, other reading, and personal experience.

17. Reading Strategy – Schema Readers use schema to retain text information better and longer. use their schema for specific authors and styles to better understand. recognize their own inadequate back-ground information & can create it. srqpix@flickr

18. Schema in Math Mathematicians use current understandings as first steps in problem-solving. consider what they know about the general topic consider if they have seen similar problems connect to a simpler problem After problem-solving, connect to harder problems. Use schema to develop their own problems.

19. Schema in Research Researchers frequently choose topics they know and care about. use their prior knowledge and experience to launch investigations and ask questions. consider what they already know to decide what they need to find out self-evaluate according to experience of what constitutes high quality products/presentations.

21. To a Similar Problem

22. To a Simpler Problem

23. To a Generalized ResultProblem

25. Make and investigate mathematical conjectures;

26. Develop and evaluate mathematical arguments and proofs;

28. Reasoning and Proof (57) “Doing mathematics involves discovery. Conjecture – that is, informed guessing – is a major pathway to discovery.”

29. Reasoning and Proof (58) “Along with making and investigating conjectures, students should learn to answer the question, Why does this work?”

30. Reasoning Make Sense Make Conjectures Make Arguments Advice to Sink in Slowly