Principles and Standards for School Mathematics

1. Principles and Standards for School Mathematics National Council of Teachers of Mathematics

2. Principles and Standards supplies guidance and vision while leaving specific curriculum decisions to the local level. This document is intended to— • set forth a comprehensive and coherent set of goals for mathematics for all students from prekindergarten through grade 12 that will orient curricular, teaching, and assessment efforts during the next decades; • serve as a resource for teachers, education leaders, and policymakers to use in examining and improving the quality of mathematics instructional programs;

3. guide the development of curriculum frameworks, assessments, and instructional materials; • stimulate ideas and ongoing conversations at the national, provincial or state, and local levels about how best to help students gain a deep understanding of important mathematics.

4. Chapter 1A Vision for School Mathematics “…those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping the future. Mathematical competence opens doors to productive futures. A lack of mathematical competence keeps those doors closed.”

5. Chapter 1 cont’d “…everyone needs to understand mathematics. All students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding. There is no conflict between equity and excellence.”

6. Chapter 2Principles for School Mathematics “They (the six principles) describe crucial issues that, although not unique to school mathematics, are deeply intertwined with school mathematics programs.”

7. The Six Principles Can Influence: • the development of curriculum frameworks • the selection of curriculum materials • the planning of instructional units or lessons • the design of assessments • the assignment of teachers and students to classes • instructional decisions in the classroom • the establishment of supportive professional development programs for teachers.

8. The Equity Principle • “Expectations must be raised—mathematics can and must be learned by all students.” • “All students should have access to an excellent and equitable mathematics program…”

9. The Curriculum Principle “A school mathematics curriculum is a strong determinant of what students have an opportunity to learn and what they do learn.”

10. The Teaching Principle “Students' understanding of mathematics, their ability to use it to solve problems, and their confidence in, and disposition toward, mathematics are all shaped by the teaching they encounter in school.”

11. The Learning Principle • “In the twenty-first century, all students should be expected to understand and be able to apply mathematics.” • “…conceptual understanding is an important component of proficiency, along with factual knowledge and procedural facility.”

12. The Assessment Principle Assessment should be: • not merely done to students, but for students, to guide and enhance their learning. • an integral part of instruction that informs and guides teachers as they make instructional decisions.

13. The Technology Principle Technology • enhances mathematics learning. • supports effective mathematics teaching. • influences what mathematics is taught.

14. Chapter 3 Overview of Standards for Mathematics Education in Prekindergarten through Grade 12

15. Content and Process Standards Four Grade Bands Prekindergarten-grade 2 Grades 3-5 Grades 6-8 Grades 9-12 Chapters 4-7

16. Chapter 8 Steps Needed to Move Toward the Vision in Principles and Standards

18. Process Standards

19. Communication StandardInstructional 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

20. Communication Task Grades 3 - 5 Pretend you are a jeweler. Sometimes people come in to get rings resized. When you cut down a ring to make it smaller, you keep the small portion of gold in exchange for the work you have done. Recently you have collected these amounts: Now you have a repair job to do for which you need some gold. You are wondering if you have enough. Work together with your group to figure out how much gold you have collected. Be prepared to show the class your solution.

21. Communication Task Grades 6-8A certain rectangle has length and width that are whole numbers of inches, and the ratio of its length to its width is 4 to 3. Its area is 300 square inches. What are its length and width?

22. Communication Task Grades 9 - 12Imagine you are talking to a student in your class on the telephone and want the student to draw some figures. The other student cannot see the figures. Write a set of directions so that the other student can draw the figures exactly as shown in the figure below.

23. Connections StandardInstructional 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 of mathematics

24. Connections Task Grades 3 - 5Estimate the cost of 12 notebooks.

25. Problem Solving StandardInstructional 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

26. Representation StandardInstructional 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

27. Reasoning and Proof StandardInstructional programs from prekindergarten through grade 12 should enable all students to— • •Recognize reasoning and proof as fundamental aspects of mathematics • Make and investigate mathematical conjectures • Develop and evaluate mathematical arguments and proofs • Select and use various types of reasoning and methods of proof