One of the defining events in the history of mathematics education was the launching of Sputnik 1 by the Soviet Union in 1957. This marked the start of the space age and the space race between the United States and the Soviet Union. Concern that the United States was falling behind in the areas of math and science triggered major national reforms in these areas. These reforms brought about the “New Math” of the 1960s and 1970s. The emphasis of this New Math was on set language and properties, proof, and abstraction. However, the New Math curriculum failed to meet the challenge of increasing the nation’s mathematical prowess as a whole. Some would even say that the New Math created more math confusion than it eliminated, which brought about the trend of Back to Basics in the late 1970s and early 1980s. Back to Basics emphasized arithmetic computation and rote memorization of algorithms and basic arithmetic facts.

### The 1989 NCTM Standards

In the later 1980s the focus shifted to critical thinking. In 1989 the National Council of Teachers of Mathematics (NCTM) released a groundbreaking document, Curriculum and Evaluation Standards for School Mathematics. This publication, sometimes referred to as the “NCTM Standards,” stresses problem solving, communication, connections, and reasoning. The key assumptions underlying the 1989 NCTM curriculum standards for Grades K–4, listed below, are addressed throughout this textbook.

The 1989 NCTM Standards include 13 curriculum standards addressing both content and emphasis. One theme common to the NCTM Standards and to the recent changes in mathematics education is that “the study of mathematics should emphasize reasoning so that students can believe that mathematics makes sense” (NCTM, 1989, p. 29). Although not discussed here, this document also includes similar sets of assumptions and standards for Grades 5 through 8 and for Grades 9 through 12.

The 1989 NCTM Standards list five goals for students. Although these are goals stated for elementary students, it is especially important that teachers of elementary students have attained them.

### Professional and Assessment Standards for Teaching Mathematics

A second groundbreaking document released by the National Council of Teachers of Mathematics was Professional Standards for Teaching Mathematics. This set of standards “present[s] a vision of what teaching should entail to support the changes in curriculum set out in the Curriculum and Evaluation Standards. This document spells out what teachers need to know to teach toward new goals for mathematics education and how teaching should be evaluated for the purpose of improvement” (NCTM, 1991, p. vii). NCTM followed with the 1995 release of Assessment Standards for Teaching Mathematics. NCTM produced this important document because “new assessment strategies and practices need to be developed that will enable teachers and others to assess students’ performance in a manner that reflects the NCTM’s reform vision for school mathematics” (NCTM, 1995, p. 1).

In the 1990s the major focus of reform in mathematics education was directed toward teaching pedagogy. Numerous studies and articles promoted the use of manipulatives and technology in the classroom (Burns, 1996; Hatfield, 1994; National Association for the Education of Young Children [NAEYC], 1996; Roth, 1992). Key ideas of this era included the use of developmentally appropriate activities and the constructivist approach to teaching. The NCTM Standards continued to gain support and popularity among mathematics educators, and many states developed grade-level scope and sequences and competency-based model programs that reflected these standards. Proficiency testing became more widespread, with some states requiring a certain level of competency in subject areas such as mathematics for grade promotion.

### Standards Update: The 2000 Principles and Standards for School Mathematics

In April 2000, the National Council of Teachers of Mathematics released its Principles and Standards for School Mathematics. This document updates the 1989 Curriculum and Evaluation Standards and includes some components of both Professional Standards for Teaching Mathematics and Assessment Standards for Teaching Mathematics as well.

With such far-reaching significant goals (see the standards box below), the 2000 Principals and Standards will certainly serve as a major influence in changes and trends in mathematics education and reform in the years to come. However, the focus of this document remains on curriculum, and so Professional Standards for Teaching Mathematics and Assessment Standards for Teaching Mathematics will both also continue to play major roles in math education and reform.

The 2000 Principles and Standards identifies six principles of high-quality mathematics education.

#### Content Standards

The 2000 Principles and Standards document describes in detail standards and expectations for grade levels Pre-K–2, 3–5, 6–8, and 9–12 for each of the five content strands:

- Number and Operations
- Algebra
- Geometry
- Measurement
- Data Analysis and Probability

Each of the content strands will be covered within the material of this textbook. Number and Operations is especially important in the early childhood curriculum. The Algebra strand includes patterns, relations, functions, and mathematical models. You may be surprised to see the various types of algebra that are included in today’s early childhood curriculum.

#### Process Standards

Process standards differ from content standards in that the process standards are not subject matter that can be learned but are the methods by which content knowledge can be acquired. The 2000 Principles and Standards document describes in detail the five process standards:

- Problem Solving
- Reasoning and Proof
- Communication
- Connections
- Representation

Problem solving is the heart of any solid mathematics curriculum. Reasoning is also important. Students who are exposed to the logic behind mathematical procedures are more likely to be able to learn and correctly apply those procedures than students who attempt to apply rules without regard to their reasonableness (Carpenter, Franke, Jacobs, Fenemma, & Empson, 1998; Hiebert & Wearne, 1996; NCTM, 2003).

Communication is especially important for assessment. Students must learn to explain, write, draw, or otherwise show what they have learned. A variety of nonverbal forms of communication must be used in the early childhood classroom. Teachers must often devise alternate means of assessment and communication when dealing with students with disabilities or students who have limited-English abilities.

Connections refers to connections among mathematics topics as well as connections to other subject areas and to real-life situations. By stressing connections, one can show the relevance and importance of mathematics. Students must also be able to make connections among mathematical representations (Coxford, 1995).

There are often a variety of representations for a single mathematical concept. For example, to represent the amount twenty-five cents, one can use a word phrase such as “twenty-five cents” or “a quarter,” use actual coins (or representations of coins) to show the amount in a variety of ways (one quarter, two dimes and a nickel, etc.), draw a pictorial representation, or use a symbolic expression such as 25¢ or $0.25. By learning several representations for a single concept, teachers can adapt their teaching methods to the needs and abilities of their students. Students should learn a variety of representations to best express and use their mathematical knowledge.

### The Influence of Recent Legislation

The need for reform in mathematics education has not gone unnoticed by U.S. legislation. In 1994 Congress enacted into law the Goals 2000 Educate America Act. On January 8, 2002, President Bush signed into law the No Child Left Behind Act of 2001. These laws mandate that all states implement accountability systems and that teachers and schools are held accountable for the education of all students. The Goals 2000 Educate America Act includes the following terms and definitions:

(1) The terms “all students” and “all children” mean students or children from a broad range of backgrounds and circumstances, including disadvantaged students and children, students or children with diverse racial, ethnic, and cultural backgrounds, American Indians, Alaska Natives, Native Hawaiians, students or children with disabilities, students or children with limited-English proficiency, school-aged students or children who have dropped out of school, migratory students or children, and academically talented students and children;

(4) The term “content standards” means broad descriptions of the knowledge and skills students should acquire in a particular subject area;

(9) The term “performance standards” means concrete examples and explicit definitions of what students have to know and be able to do to demonstrate that such students are proficient in the skills and knowledge framed by content standards;

(11) The term “State assessment” means measures of student performance which include at least one instrument of evaluation, and may include other measures of student performance, for a specific purpose and use which are intended to evaluate the progress of all students in the State toward learning the material in State content standards in one or more subject areas

Goals 2000 Educate America Act, 1994

In compliance with these laws, nearly every state in the United States has developed its own set of content standards, performance standards, and assessment measures. At the time of this printing, the Eisenhower National Clearinghouse Web site provides links to the state frameworks for mathematics and science education for 49 of the 50 states. Many of these frameworks include content standards that reflect the NCTM goals and standards. Some representative examples of objectives from state models or frameworks are provided within this text. You may wish to download or purchase the appropriate document for your state.

### State and Local Standards

Many school districts have developed their own sets of standards and assessment measures. Such documents are generally based on the appropriate state model or framework or on the national standards set by the NCTM. Local district standards may break one national or state objective into several smaller objectives or specific tasks and may specify which objectives or tasks are to be covered during each grading period of the school year. Compare the Number and Operations NCTM standard versus the standard developed by West Point Elementary.

The tasks described by the West Point Elementary Scope and Sequence Document address the national standard expectation components of mental computation, estimation, and calculators. Other components of this expectation are addressed within other objectives (not reprinted here) from the West Point Scope and Sequence Document. Notice the progression of the difficulty level of the tasks throughout one year and into the next. This is typical of the teaching methods of today. A topic is not introduced, taught once, and then forgotten until the following year. Important instructional objectives are revisited and reinforced throughout the school year as necessary. Research indicates that an incremental approach, where concepts and skills build on prior concepts, skills, and knowledge, is highly effective in student retention of material (Klingele & Reed, 1984).

© ______ 2005, Allyn & Bacon, an imprint of Pearson Education Inc. Used by permission. All rights reserved. The reproduction, duplication, or distribution of this material by any means including but not limited to email and blogs is strictly prohibited without the explicit permission of the publisher.

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