Essential Mathematics for the Twenty-first Century: The Position of the National Council ofSupervisors of MathematicsAuthor(s): Iris M. CarlSource: The Mathematics Teacher, Vol. 82, No. 5 (MAY 1989), pp. 388-391Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27966291 .

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Essential Mathematics for

the Twenty-first Century : The Position of the National Council of Supervisors of Mathematics

Since A Nation at Risk was released five years ago, we have experienced a most serious and sustained national effort to improve education. That attention has been welcomed, and it re

quires our continued support. Today, reformers continue their search for ways to effect the curricular changes needed.

Evidence is mounting that principal among their major concerns is mathematics. The NCSM's "Essential Mathematics for the Twenty-first Century" was created for the purpose of contribu

ting to current efforts to reform mathematics education. This NCSM position statement is intended to complement the mathematics curriculum as

developed in the NCTM's Standards and reflect that commitment to the evolving changes. The

components are the basic mathematical competencies that represent those elements critical to the revision and expansion of mathematics programs. For the many critical audiences, math ematics educators, the broader educational community, and other decision makers, it is de

signed to be a concise proposal for mathematics curriculum reform. A successful response from the mathematics education community to the reform effort will

require many levels and forms of support* "Essential Mathematics for the Twenty-first Cen

tury" is one facet developed from the perspective of school-based mathematics educators, whose

specific assignments are curriculum and instruction. We hope that it will assist in the process of

reaching a consensus for excellence in mathematics education that will meet the demands of the

twenty-first century.

Iris M. Carl, President National Council of Supervisors

of Mathematics

This position statement has been endorsed by the National Council of Teachers of Mathematics.

Introduction

Students

who enter kindergarten in 1988 can expect to graduate from high school

in the year 2001. Yet these students who will graduate in the twenty-first century still frequently face a computation-dom inated curriculum more suitable for the nineteenth century. To address this incon

sistency, the National Council of Supervi sors of Mathematics (NCSM) now updates its 1977 basic skills position statement to describe the essential mathematical com

petencies that citizens will need to begin adulthood in the next millenium. This posi tion by NCSM is intended to complement and support positions on mathematics edu cation by the National Council of Teachers

of Mathematics and other professional groups.

Our technological world is changing at an ever-increasing rate, and our respon

sibilities in international affairs continue to increase. As the demands of society change, so do the essential competencies needed by individuals to live productively in that society. All students, including those of all races and both sexes, will need

competence in essential areas of mathemat ics.

What Is Essential?

The NCSM views as "essential" those com

petencies that are necessary for the doors to employment and further education to

388 Mathematics Teacher

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remain open. Essential mathematics, as de scribed in this paper, represents the math ematical competence students will need for

responsible adulthood. The students we educate today can

expect to change jobs many times during their lifetimes. The jobs they hold will de

velop and change around them. Often, spe cific job skills will not transfer from one

position to another. To prepare for mobil

ity, students must develop a thorough un

derstanding of mathematical concepts and

principles; they must reason clearly and communicate effectively; they must recog nize mathematical applications in the world around them; and they must approach mathematical problems with confidence. In dividuals will need the fundamental skills that will enable them to apply their knowl

edge to new situations and to take control of their own lifelong learning.

Skill in whole-number computation is not an adequate indicator of mathematical achievement. Nor is it sufficient to develop skills apart from their applications or to

memorize rules without understanding the

concepts on which they are based. Students must understand mathematical principles; they must know when and how to use com

putation; and they must develop profi ciency in problem solving and higher-order thinking.

The NCSM position statement of 1977

responded to the "back-to-basics" move

ment with its overly narrow conception of basic skills. Now, as we look to the future, we recognize that the use of calculators and

computers and the application of statistical methods will continue to expand. Creative

problem solving, precise reasoning, and ef fective communication will grow in impor tance. To function effectively in the next

century, students will need proficiency in an enriched body of essential mathematics. The list that follows identifies twelve criti cal areas of mathematical competence for all students. It does not imply an instruc tional sequence or a priority among topics. In fact, the twelve essential mathematics areas are interrelated; competence in each area requires competence in other areas.

Twelve Components of Essential Mathematics

Problem solving

Learning to solve problems is the principal reason for studying mathematics. Problem

solving is the process of applying pre viously acquired knowledge to new and un familiar situations. Solving word problems in texts is one form of problem solving, but students also should be faced with nontext

problems. Problem-solving strategies in volve posing questions, analyzing situ

ations, translating results, illustrating re

sults, drawing diagrams, and using trial and error. Students should see alternate solutions to problems; they should experi ence problems with more than a single solu tion.

Communicating mathematical ideas

Students should learn the language and no tation of mathematics. For example, they should understand place value and scientif ic notation. They should learn to receive

mathematical ideas through listening, read

ing, and visualizing. They should be able to

present mathematical ideas by speaking, writing, drawing pictures and graphs, and

demonstrating with concrete models. They should be able to discuss mathematics and ask questions about mathematics.

Mathematical reasoning

Students should learn to make independent investigations of mathematical ideas. They should be able to identify and extend pat terns and use experiences and observations to make conjectures (tentative conclu

sions). They should learn to use a coun

terexample to disprove a conjecture, and

they should learn to use models, known

facts, and logical arguments to validate a

conjecture. They should be able to dis

tinguish between valid and invalid argu ments.

Applying mathematics to everyday situations

Students should be encouraged to take

everyday situations, translate them into mathematical representations (graphs,

May 1989 389

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tables, diagrams, or mathematical expres sions), process the mathematics, and inter

pret the results in light of the initial situ ation. They should be able to solve ratio, proportion, percent, direct-variation, and inverse-variation problems. Not only should students see how mathematics is applied in the real world but they should observe how

mathematics grows from the world around them.

Alertness to the reasonableness of results

In solving problems, students should ques tion the reasonableness of a solution or

conjecture in relation to the original prob lem. Students must develop the number sense to determine if results of calculations are reasonable in relation to the original numbers and the operations used. With the increase in the use of calculating devices in

society, this capability is more important than ever.

Estimation

Students should be able to carry out rapid approximate calculations through the use of mental arithmetic and a variety of com

putational estimation techniques. When

computation is needed in a problem or con sumer setting, an estimate can be used to check reasonableness, examine a conjec

ture, or make a decision. Students should

acquire simple techniques for estimating such measurements as length, area, volume, and mass (weight). They should be able to decide when a particular result is precise enough for the purpose at hand.

Appropriate computational skills

Students should gain facility in using addi

tion, subtraction, multiplication, and divi sion with whole numbers and decimals.

Today, long, complicated computations should be done with a calculator or com

puter. Knowledge of single-digit number facts is essential, and using mental arithme tic is a valuable skill. In learning to apply computation, students should have practice in choosing the appropriate computational

method: mental arithmetic, paper-pencil al

gorithm, or calculating device. Moreover,

everyday situations arise that demand rec

ognition of, and simple computation with, common fractions. In addition, the ability to recognize, use, and estimate with per cents must also be developed and main tained.

Algebraic thinking

Students should learn to use variables (let ters) to represent mathematical quantities and expressions ; they should be able to rep resent mathematical functions and relation

ships using tables, graphs, and equations. They should understand and correctly use

positive and negative numbers, order of op erations, formulas, equations, and inequali ties. They should recognize the ways in which one quantity changes in relation to another.

Measurement

Students should learn the fundamental con

cepts of measurement through concrete ex

periences. They should be able to measure

distance, mass (weight), time, capacity, tem

perature, and angles. They should learn to

calculate simple perimeters, areas, and vol umes. They should be able to perform measurement in both metric and customary systems using the appropriate tools and levels of precision.

Geometry

Students should understand the geometric concepts necessary to function effectively in the three-dimensional world. They should have knowledge of such concepts as

parallelism, perpendicularity, congruence,

similarity, and symmetry. Students should know properties of simple plane and solid

geometric figures. Students should visual ize and verbalize how objects move in the

world around them using such terms as

slides, flips, and turns. Geometric concepts should be explored in settings that involve

problem solving and measurement.

Statistics

Students should plan and carry out the col lection and organization of data to answer

questions in their everyday lives. Students

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should know how to construct, read, and draw conclusions from simple tables, maps, charts, and graphs. They should be able to

present information about numerical data, such as measures of central tendency (mean, median, mode) and measures of dis

persion (range, deviation). Students should

recognize the basic uses and misuses of sta tistical representation and inference.

Probability

Students should understand elementary no tions of probability to determine the likeli hood of future events. They should identify situations in which immediate past experi ence does not affect the likelihood of future events. They should become familiar with how mathematics is used to help make such

predictions as election results, business

forecasts, and outcomes of sporting events.

They should learn how probability applies to research results and to the decision

making process.

Climate for Learning

To learn the essential mathematics needed for the twenty-first century, students need a

nonthreatening environment in which they are encouraged to ask questions and take risks. The learning climate should incorpor ate high expectations for all students, re

gardless of sex, race, handicapping con

dition, or socioeconomic status. Students need to explore mathematics using manipu latives, measuring devices, models, calcu

lators, and computers. They need to have

opportunities to talk to each other about mathematics.

Students need modes of instruction that are suitable for the increased emphasis on

problem solving, applications, and higher order thinking skills. For example, cooper ative learning allows students to work to

gether in problem-solving situations to pose questions, analyze solutions, try alternative

strategies, and check for reasonableness of results.

To implement the jiew instructional

strategies, extensive professional devel

opment opportunities as well as new learn

ing materials will be needed, m

Polymer Geometry The question of the geometry of the interface between two linked but repelling polymers is closely related to

the mathematical problem of defining minimal sur

faces?surfaces that take up the least possible area

within a certain boundary. That connection has led to an unusual collaboration at the University of Massa chusetts at Amherst between polymer physicist Edwin L. Thomas, who studies the structure of polymers, and

mathematician David Hoffman, who is interested in minimal surfaces and computer graphics.

In one type of block copolymer, the two constituent

polymers form grains consisting of stacks of alter

nating, equally spaced layers. Because the stacks have no preferred orientation, layers in adjacent grains can

meet at any angle. When the layers meet at 90 degrees, electron microscope images show a pattern resembling a comb's regularly spaced teeth. That image seems to

correspond to a minimal surface known as Scherk's

first surface, discovered in 1835. It can be thought of as the smooth joining of two sets of parallel planes at

right angles to each other. Looking at the computed shadow, or projection, ofthat structure produces a pic ture resembling the polymer image, which is a two

dimensional view of the material.

Polyisoprene and polystyrene copolymers often form

interlaced networks having a tetrahedral geometry, re

sembling the way carbon atoms each link to four

neighbors in a diamond structure. Such a diamondlike

geometry strongly influences the copolymer's physical

properties. Whereas polystyrene by itself is stiff and

brittle and polyisoprene is rubbery, the combination ends up with the stiffness of one component and the

toughness of the other. Such synergistic combinations

may be useful in the production of superior composite materials. (Science News, 3 September 1988)

May 1989 391

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