NCTM's Curriculum Research Articles

Asymptotes and Other Limiting Functions

Mathematics and the way we teach it are undergoing dramatic changes. The NCTM's Curriculum and Evaluation Standards (1989) has determined a direction and focus for the future of mathematics education. The study of mathematics should be an active experience for students—one in which they explore the world of mathematics, make conjectures based on patterns discovered, and induce the mathematical concept. The idea of a guided-discovery approach for students in mathematics is hardly new. Socrates taught this way. However, he had the luxury of a lighter student load and no state or federal mandates on expected student outcomes. In short, Socrates had the time.

Connecting Research to Teaching: Proportional Reasoning

The attainment of proportional reasoning is considered a milestone in students' cognitive development. According to the NCTM's Curriculum and Evaluation Standards (1989), this ability is “of such great importance that it metits whatever time and effort must be expended to assure its careful development” (p. 82). As teachers and researchers know, students' understanding of proportionality develops slowly over a number of years. This article reports research findings regarding the learning and teaching of proportional reasoning that have potential for making contributions to classroom practice.

Young's Vision

The boxed paragraph's objective can also be considered one of the fundamental concepts found in NCTM's Curriculum and Evaluation Standards for School Mathematics, published in 1989. Making the underlying assumption that a metamorphosis was drastically needed in our nation's mathematics curriculum, the standards document thoroughly examined and analyzed all aspects of the current methods by which mathematics is presented at all grade levels (K–12), as well as what is considered to be essential material that must be presented to students. (For an overview of the Curriculum and Evaluation Standards, see Thompson and Rathmell [1988].) However, the succinct statement below is not the product of the NCTM's standards; rather it is the wisdom of Jacob William Albe1t Young (1865–1948), a tum-of-thecentury pioneer in the field of educational pedagogy. (J. W. A. Young is not to be confused with John Wesley Young, 1879–1932, a professor of mathematics at Dartmouth College during the same era.)

Activities: Pythagorean Dissection Puzzles

The Pythagorean theorem is one of the most familiar mathematical formulations, quoted in movies, television, and cartoons. Yet, as familiar as it may be to many people, it is often not completely understood. In the NCTM's Curriculum and Evaluation Standards (1989, 113–14), Standard 12: Geometry for grades 5–8 states, “One of the most important properties in geometry, the Pythagorean theorem, is introduced in the middle grades. Students can a Fig 1 Pythagorean theorem discover this relationship through explorations …” (see fig. 1).

Connecting Research to Teaching: Probability and Statistics

This issue introduces a new department to the Mathematics Teacher, “Connecting Research to Teaching.” Articles will focus on mathematical and pedagogical ideas related to the NCTM's Curriculum and Evaluation Standards (1989) and the Professional Standards for Teaching Mathematics (1991). Authors will strive to present information to help teachers (1) understand students' conceptions or misconceptions of important ideas, (2) consider various approaches to teaching, and (3) offer activities that probe students' understanding. Although research offers no one correct answer to the many perplexing problems surrounding teaching and learning mathematics, the suggestions and perspectives may help teachers pursue their work with new insights. It is hoped that the department will also stimulate researchers to reflect on connecting research to the classroom. Communication and collaboration between teachers and researchers will benefit both groups and help each grow in appreciation of the other's tasks.

One Point Of View: It's Time to Use Our “OOB” Detectors!

The NCTM's Curriculum and Evaluation Standards for School Mathematics was thoughtfully conceived and continues to influence the school mathematics curriculum. A strength oft he document eems to be that the standards trike a rea onable balance between what ought to be and what can be. Because of this, the book has stimulated our thinking and facilitated realistic, po itive changes in many schools.

Multiple Strategies: Product of Reasoning and Communication

The NCTM's Curriculum and Evaluation Standards (1989) supports the idea that problems can be solved in more than one correct way. This multiple-strategy approach contains the seeds of motivation, success, and mind stretching. The curriculum standards that focus on reasoning and communication skills are integral to delivering mathematics education that generates the cre ative, problem-solving, divergent thinker that the business community would like to employ.

One Point Of View: Mathematics Integration: Common-Sense Approach to Curriculum Development

Eight years ago I interviewed Mike, a second grader, and asked him to define “mathematics.” His response was that “math is something you do in the momjng.” I have referred to his answer in my workshops and writing as evidence of the isolation of the different subject areas (Jacobs 1989). Children viewed mathematics as a time of day, a textbook, or a change in teacher's attitude, given the predisposition of the teacher toward mathematics. But, eight years ago the NCTM curriculum and evaluation standards had not hit America's schools (NCTM 1989). Because of the declarations made in that document, today's students have a better chance of using mathematics through the course of their day. More students today view mathematics as a common, everyday activity. Through thoughtful integration of the curriculum, students use mathematics

Star Numbers and Constellations

The actual number of digits in a number's representation can sometimes fall into the same category as the number itself. Note, for example, that the number of digits in 5378 is an even integer, as is the number 5378; or that the number of digits in 108 is an exact square, as is the number 108 itself. Such comparisons lend a certain appeal to easily overlooked relationships and open the door to varied forms of generalization and learning by discovery, all of which conform perfectly to NCTM's curriculum standards. Our area of exploration is accordingly that of the integers and their digital representations. Brought into focus are fascinating connections, as well as an assortment of whole-number classifications. Let us begin with a challenging example and the more rigorous treatment afforded by a precise definition.

Bulgarian Solitaire

Bulgarian solitaire is a topic in discrete mathematics that is easily understood by both the most and least able students and challenges each at a different level. It is one of many mathematical recreations made popular by Martin Gardner (1983). It offers an opportunity for classes to find patterns, solve problems, and find further problems to study. NCTM's Curriculum and Eualuation Standards (1989) emphasizes the value of finding patterns as an aid to problem solving and problem posing. Bulgarian solitaire is openended, and lessons can be tailored to nearly any level, from elementary school to senior secondary school.

Solution Revolution

In Consortium, the newsletter of the Consortium for Mathematics and its Applications, Froelich (1988) explored graphical strategies for solving the equation 2x = x10. He described his use of a function plotter as he sought a graphical solution for that equation. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has proposed the increased use of function plotters and other technology to investigate and solve problems. Equations such as 2x = x10 can be used to illustrate the potential of that technology and to consider its implications for the teaching and learning of mathematics.

A Quadrilateral Hierarchy to Facilitate Learning in Geometry

The NCTM's Curriculum and Evaluation Standards (1989) challenges mathematics educators to rethink both content and instructional style. Specifically, it calls for “a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures … to one that emphasizes conceptual understandings, multiple representations, and connections …”(p. 125). We would like to illustrate such an approach for the study of quadrilaterals in a year-long high school geometry course. This strand is part of a complete course described in more detail in Craine (1985).

Integrating Transformation Geometry into Traditional High School Geometry

The grades 9-12 section of NCTM's Curriculum and Evaluation Standards for School Mathematics defines transformation geometry as “the geometric counterpart of functions” (1989, 161). Further, the Standards document recognizes the importance of this topic to the high school mathematics curriculum by listing it among the “topics to receive increased attention” (p. 126). Also included on this list is the integration of geometry “across topics.”

A Generalized Area Formula

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) states that mathematics instruction should reflect a constructvist's view of the learning process, that is, that learners are not passive absorbers of knowledge but rather active participants in the construction of knowledge. Such instruction would approach mathematics as interrelated concepts and skills to be understood rather than as independent concepts and skills to be memorized. It would also encourage learners to examine concepts and their relationships in such a way as to encourage pattern recognition, generalizing, hypothesizing, proving, and other higher-order thinking skills.

A Joint Vision for Classroom Assessment

The NCTM's Curriculum and Evaluation Standards for School Mathematics proposes a vision for assessment in the mathematics classroom that would “help teachers better understand what students know and make meaningful instructional decisions” (NCTM 1989, 189). For assessment to be truly aligned with the mathematics curriculum, teachers would give more emphasis to taking a holistic view of mathematics—using multiple assessment methods, including written, oral, and performance formats, and incorporating calculators, computers, and manipulatives as part of assessment. Thus, mathematics teachers would be empowered to trust their own abilities and judgments in the area of mathematics assessment (Clarke, Clarke, and Lovitt 1990; Cooney and Badger 1990; Schoen 1989; Webb and Briars 1990).

Teaching Mathematics with Technology: Exploring Geometry with The Geometer's Sketchpad

Have you ever thought of exploring geometric relationships on the computer? The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) challenges us to pay increased attention to “developing an understanding of geometric objects and relationships” and “using geometry in solving problems” (p.70). The document also advises placing decreased emphasis on memorizing geometric vocabulary, facts, and relationships. This article describes several geometric investigations that were developed in the spirit of the curriculum standards and are appropriate for use at advanced elementary and middle school levels, as well as with older students. They encourage exploration, creativity, and discovery.

Further Thoughts on the Standards: A Reaction to Apple

As always, I enjoy and learn from the thoughtful analyses and well-argued comments about educational practices by my colleague Michael Apple. In reading his often flattering, yet critical, analysis of NCTM's Curriculum and Evaluation Standards/or School Mathematics (1989) and Professional Standards for Teaching Mathematics (1991), I must admit he is right on several counts. The lens through which he views the world is not one familiar to most mathematics educators. Relating any message about schooling to the patterns of differential economic, political, and cultural power is not common in our field. Furthermore. the five issues he raises—the financial crisis in education, the nature of inequality in schools, the role of mathematical knowledge as a filter that maintains those inequalities. the possibilities and limitations of a curriculum grounded in problem situations, and the complex realities of teachers' lives—are critical issues facing all educators interested in changing the current system. Clearly. the Standards documents by themselves are not well situated with respect to these broader ideological and political issues. In fact, none of the five issues, although considered, were the driving forces behind the NCTM's reform documents.

Do the Standards Go Far Enough? Power, Policy, and Practice in Mathematics Education

Although NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and Professional Standards for Teaching Mathematics (1991) are generating considerable interest, there has been little discussion of their ideological and social grounding and effects. By placing the Standards within the growing conservative movement in education, this paper raises a number of crucial issues about the documents, including the depth of the financial crisis in education and its economic and ideological genesis and results; the nature of inequality in schools; the role of mathematical knowledge in our economy in maintaining these inequalities; the possibilities and limitations of a mathematics curriculum that is more grounded in students' experiences; and the complicated realities of teachers' lives. Without a deeper understanding of these issues, the Standards will be used in ways that largely lend support only to the conservative agenda for educational reform.

Grade Assignment Based on Progressive Improvement

The NCTM's Curriculum and Evaluation Standards for School Mathematics states, “Evaluation is a tool for implementing the Standards and effecting change systematically” (1989, 189). Tests are one facet of evaluation, and we maintain that mathematics classes are strongly affected by the way in which test scores are used to generate final course grades. In the traditional secondary school mathematics class, current grading practices tend to drive instruction by putting constraints on specific course content and its organization. In turn, content and its organization affect testing and therefore grading. The interaction of these factors is an aspect of assessment that is not specifically discussed by the NCTM's evaluation standards. The purpose of this article is to examine the impact of grading on mathematics instruction and on the implementation of the curriculum and evaluations standards.

Supermarket Challenge

Every so often teacher present a lesson that far exceeds their original expectations for students' learning. This delight occurred with a lesson I developed for my s ixth-grade mathematics class in an effort to implement the NCTM's curriculum standards. My objectives were to integrate mathematics with other curriculum areas and draw on a real-life situation to motivate and interest my students.