Chapter 1 Teaching Mathematics in the Era of the NCTM Standards Copyright © Allyn and Bacon 2010 This multimedia product and its contents are protected under copyright law. The following are prohibited by law: any public performance or display, including transmission of any image over a network preparation of any derivative work, including the extraction, in whole or in part, of any images any rental, lease, or lending of the program

Historical Perspectives National Council of Teachers of Mathematics (NCTM) Momentum started in early 1980s 1989 NCTM Curriculum and Evaluation Standards for School Mathematics Principles and Standards for School Mathematics 2000 Copyright © Allyn & Bacon 2010

Copyright © Allyn & Bacon 2010 NCTM The National Council of Teachers of Mathematics (NCTM) is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. Founded in 1920, NCTM is the world’s largest mathematics education organization, with more than 100,000 members and 230 Affiliates throughout the United States and Canada. NCTM holds annual national and regional conferences for American teachers and publishes four print journals and one on-line journal. Its published standards have been highly influential in the direction of mathematics education in the United States and Canada. Copyright © Allyn & Bacon 2010

1989 Curriculum and Evaluation Standards for School Mathematics The controversial 1989 NCTM Standards called for more emphasis on conceptual understanding and problem solving informed by a constructivist understanding of how children learn. The increased emphasis on concepts required decreased emphasis on direct instruction of facts and algorithms. This decrease of traditional rote learning was sometimes understood by both critics and proponents of the standards to mean elimination of basic skills and precise answers, but the NCTM has refuted this interpretation. In reform mathematics, students are exposed to algebraic concepts such as patterns and the commutative property as early as first grade. Standard arithmetic methods are not taught until children have had an opportunity to explore and understand how mathematical principles work, usually by first inventing their own methods for solving problems and sometimes ending with children's guided discovery of traditional methods. The Standards called for a de-emphasis of complex calculation drills. Copyright © Allyn & Bacon 2010

2000 Principles and Standards for School Mathematics The controversy surrounding the 1989 standards paved the way for revised standards which sought more clarity and balance. In 2000, NCTM used a consensus process involving mathematicians, teachers, and educational researchers to revise its standards with the release of the Principles and Standards for School Mathematics (PSSM), which replaced all preceding publications. The new standards were organized around six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) and ten strands which included five content areas (Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability) and five processes (Problem Solving, Reasoning and Proof, Communication, Connections, and Representation). PSSM was not perceived to be as radical as the 1989 standards and did not engender significant criticism. The new standards have been widely used to inform textbook creation, state and local curricula, and current trends in teaching. Copyright © Allyn & Bacon 2010

Copyright © Allyn & Bacon 2010 The Six Principles Equity Curriculum Teaching Learning Assessment Technology Copyright © Allyn & Bacon 2010

Copyright © Allyn & Bacon 2010 The Equity Principle Excellence in mathematics education requires equity—high expectations and strong support for all students. (NCTM 2000, p. 12) The strong message of this principle is high expectations for all students. All students must have the opportunity and support to learn regardless of personal characteristics, backgrounds, or physical challenges. Copyright © Allyn & Bacon 2010

The Curriculum Principle A curriculum is more than a collection of activities: It must be coherent, focused on important mathematics, and well articulated across the grades. (NCTM 2000, p. 14) This principle relates to how important it is to build instruction around big ideas. Students need to see math as an integrated whole and not a collection of bits and pieces. Copyright © Allyn & Bacon 2010

The Teaching Principle Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. (NCTM 2000, p. 16) Teachers must understand deeply the math they are teaching; understand how students learn math, including their individual students; and select instructional tasks and strategies that will enhance learning Copyright © Allyn & Bacon 2010

The Learning Principle Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. (NCTM 2000, p. 20) This principle is based on two fundamental ideas: Learning math with understanding is essential Learning is enhanced where students are required to evaluate their own ideas and those of others, encouraged to make math conjectures, test them, and are helped in developing their reasoning skills Copyright © Allyn & Bacon 2010

The Assessment Principle Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. (NCTM 2000, p. 22) Learning should not be done to students, but for students. Ongoing assessment highlights for students the most important concepts. The assessment should include ongoing observation and interaction. Daily feedback helps students to become independent learners. Teachers can also become better acquainted with each students needs and offer a variety of different assessment techniques. Copyright © Allyn & Bacon 2010

The Technology Principle Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning. (NCTM 2000, p. 24) Calculators, computers, and other technologies should be viewed as essential tools for learning and doing math. By using these tools students are better able to concentrate on the concepts involved in the math they are learning. Copyright © Allyn & Bacon 2010

The Five Content Standards Instead of using different topics for each grade level, it was decided to use a common set of 5 content standards throughout the grades. (Appendix A) Although each standard apply across all grade levels, different emphasis is applied to different grade levels. For example, Number Operations is more heavily emphasized in the earlier grades until high school. Copyright © Allyn & Bacon 2010

The Five Content Standards Numbers and Operations Algebra Geometry Measurement Data Analysis and Probability Copyright © Allyn & Bacon 2010

The Five Process Standards Problem Solving Reasoning and Proof Communication Connections Representation Copyright © Allyn & Bacon 2010

Curriculum Focal Points “Targets” — specify the big ideas — organized grade level and NCTM content strands Designed to be the fundamental core of the curriculum at each grade level Copyright © Allyn & Bacon 2010

Curriculum Focal Points What characteristics qualify a concept or topic to be a curriculum focal point? For inclusion in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, a focal point had to pass three rigorous tests: Is it mathematically important, both for further study in mathematics and for use in applications in and outside of school? Does it “fit” with what is known about learning mathematics? Does it connect logically with the mathematics in earlier and later grade levels? Copyright © Allyn & Bacon 2010

Shifts in Classroom Environment Away From: Classrooms as collections of individuals Teacher as “sole authority” for right answers Mere memorization of procedures Emphasis on mechanistic finding of answers Treating mathematics as a body of isolated concepts and procedures Toward: Classrooms as mathematics communities Logic and mathematical evidence as verification Mathematical reasoning Conjecturing, inventing, and problem solving Connecting mathematics, its ideas, and applications Copyright © Allyn & Bacon 2010

The Teaching Standards Knowledge of Mathematics and General Psychology Knowledge of Student Mathematical Learning Worthwhile Mathematical Tasks Learning Environment Discourse Reflection on Student Learning Reflection on Teaching Practice (These can be found in Mathematics Teaching Today through a 120 day free subscription) Copyright © Allyn & Bacon 2010

Influences and Pressures on Reform The National Assessment of Educational Progress The Third International Mathematics and Science Study Copyright © Allyn & Bacon 2010

Copyright © Allyn & Bacon 2010 State Standards The term “standards” was popularized by NCTM in 1989 Grade-by-grade objectives vary state to state (TN) Testing programs associated with standards put pressure on teachers and students to perform so that test scores are raised at all costs Copyright © Allyn & Bacon 2010

Copyright © Allyn & Bacon 2010 Curriculum Traditional Curricula Standards-Based Curricula Copyright © Allyn & Bacon 2010

An Invitation to Learn and Grow Becoming a Teacher of Mathematics (Things you will need to succeed) Knowledge of mathematics Persistence Positive attitude Readiness for change Reflective disposition Copyright © Allyn & Bacon 2010

Schifter and Fosnot on the Role of Teachers "No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students." From D. Schifter and C. T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993), p. 9. What does this quote mean to you related to teaching mathematics?

Copyright © Allyn & Bacon 2010 Your Responsibilities What does it mean to DO mathematics? How does a person go about LEARNING mathematics? As a future elementary teacher, how should you go about teaching mathematics through problem solving? Copyright © Allyn & Bacon 2010

Online Resources Illuminations: www.illuminations.nctm.org Key Issues in Math: www.mathforum.org/social/index NAEP: National Assessment of Educaitonal Progress (The National Report Card)