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Higgins, Jon L., Ed.A Metric Handbook for Teachers.National Council of Teachers of Mathematics, Inc.,Washington, D.C.74132p.The National Council of Teachers of Mathematics, 1906Association Drive, Reston, Virginia 22091 ($2.40)

MF$0.75 HC Not Available from EDRS. PLUS POSTAGEActivity Learning; Curriculum; Elementary SchoolMathematics; *Instruction; *Instructional Materials;Learning Activities; *Mathematics Education;*Measurement; *Metric System; Secondary SchoolMathematics; Teaching Techniques

This handbook has been compiled to provide areference for teachers at all levels who are implementing the metricsystem in their classroom. It includes practical suggestions andrecommendations for teaching the metric system, as well as papersidentifying and discussing the fundamental mathematical andpsychological issues underlying the teaching of the metric system inthe schools. The articles--some reprinted from recent issues of the"Arithmetic Teacher," some written especially for thispublication--are organized under five headings: Introducing theMetric System; Teaching the Metric System: Activities; Teaching theMetric System: Guidelines; Looking at the Measurement Process; andMetrication, Measure, and Mathematics. (Editor/DT)

u S OE pARTmENT OF HEALTHEDUCATION & WELFARENATIONAL INSTITUTE OF

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PERMISSION 10 REPRODUCE THIS COPYRIGHTED tv, I ERIAL BY MICROFICHE ONLYHAS BEEN GRAN IFD BY

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A Metric Handbookfor Teachers

edited byJON L. IIAGGINS

A NI thellta tiCS Education Resource Series Projectof the ERIC Information Analysis Center for

Science. Mathematics, and Environmental EducationColumbus, Ohio

Published by the

NATIONAL COI! NC I., OF T EACH ERS OF MATHEMATICS

1901) Association Drive, Reston, Virginia 22091

1.tioal of Congt s+ Cululoging in Ptt lira 1 ton Data:

Higgins, Jon L comp.

A metric handbook for teachers.

"A mathematics education resource series project ofthe ERIC Information Analysis Center for Science, Mathematics, and Environmental Education, Columbus, Ohio."

Includes bibliographies.1. Metric system--Study and teaching--United States.

I. ERIC Information Analysis Center for Science,Mathematics, and Environmental Education. II. Title.QC93.H53 389'.152 74-13468

Pall,: 1,1 this ha ndho()1; hat appc,ned pre% lowly in issues of theArithmetic e, I, r cup% lighted by the National Council ofI. hers of .\[,,111,n1;0;(,. hhcc nuts/ not hepctmissiou of the Coumil. Also. -Ilistotical Steps toNsard Metrication"and "Metric Curriculum: Stole, .`n(picticc. and Guidelines" are adaptedhom copstihted matetii,ls pithh.hed IA the Agency for Instructional

Feletision :11111 11111,4 not he icpunlii«11 without their in.hitission."Actiitics tor Intiodming \kith (:otnepts to Teat icrs." .Metric Equip-

ment: II l to Imprmisc,- "Think 'Alen-lcLive '.letric." -Tun Basic Steps for`-;11ssItil Mehl( Nicasnicincitt." and "Metrication, 'Measure, and :11atlic-titatits" ttetr ptepared itur,tiont to a contract 111th the National Instituteof Education. I Depaitmcni of Health. Education, aud Nelfare.c,mtra(ons undeorking such motet[; under goulitnnmt sponsorship arecit«ttiv,tged ut u\pu.,,, heck their µulgment in professional and technic-31Inducts. Points of bete or opinions (III 110t, therefore, necessarily represtilt N.itional Institute of Edination position or policy.

Pinteil iu lhe [nifty/ clatrs if ,.1,11(?fra

Contents

Introduction

A Quick Guide to the Ilandhook

1. Introducing the Metric System

Activities for Introducing Nletric Conceptsto Teachers

Inching Our Way towards the Metric System

The Metric System: Past, Present--Future?

Historical Steps toward Metrication

2. Teaching the Metric System: Activities

Experiences for Metric Missionaries

Metric Equipment: How to Improvise

Think MetricLive Metric

Procedures for Designing Your ()n MetricGames for Pupil Involvement

Ideas

3. "caching the Metric System: Guidelines

Schools Are Going Metric

Metric: Not 1f, but flow

Jon 1.. 11n.,, ,,an, 7

Cerariho Ectuttoil 12

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Ifiirdyn Snvilam 36

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V7,1( AfrincCollimittce 72

iv' CONTENTS

Teaching the Metric System as Part of('omptilsor Conversion in the United States

Metric Curriculum: Scope, Sequence,and Guidelines

Vint tint J. Hawkins

tlurilsn A', Suydam

76

81

4. Looking at the Nleasurentent Process

Ten Basic Step; for Successful /our 1.. 89

Metric Nleasurement

Thinking about Measurement Lehi, /'. 93

Teaching about -About.' //iirii/i/ C. Trim Hi: 100

5. Metrication, Nleasure, and Nlatheinatcs

Metrication, Measure, and Nlatheinatics :Ilan R. Osborne 107

Tiatroduction

Clearinghouses of the Educational Resources Information Center (ERIC)are charged with both information 1,athering and information dissemination.As the growing movement to convert the basic measuring systems of theUnited States to the metric system became apparent, the ERIC Clearinghousefor Science, Mathematics, and Environmental Education commissioned apaper that would identify and discuss the fundamental mathematical andpsychological issues underlying the teaching, of the metric system in theschools. The result of that commission is the article "Metrication. Measure,and Mathematics'' in this '.and book.

As work on that paper progressed, it became obvious that a compilationof practical suggestions and recommendations for teaching the metric systemwas also needed. Indeed, it seemed that a theoretical paper discussing basicissues would be enhanced by includinL, it as a component of such a collection.The April 1973 and May 1973 issues of the Arithmetic Teacher -)rovicledexcellent sources of such recommendations, and the ERIC Clearinghousecontacted the National Council of Teachers of Mathematics to explain theconcept of the handbook and to seek permission to include NCTM materials.The enthusiastic response of NCTM resulted in the suggestion of a jointproject. The scope of the original project was greatly expanded, and ad-ditional short papers were commissioned to till gaps in coverage withinthe handbook.

Although the papers in this handbook present a remarkably unified setof recommendations, there are some contradictions in specific details, asmight be expected when seventeen different authors are represented. Theultimate resolution of these conflicts will depend on what happens in class-rooms, since successful classroom learning is the final test of educationaltheory. For this reason, the handbook is addressed to teachers. It is hopedthat it will prove to he both a useful and a frequently iiscd resource for teachers.

A quick guide to the handbook

This handbook has been compiled to provide a reference for teachers atall levels who are implementing the metric system in their classrooms. Hereis a brief summary of the articles in the handbook that may make the taskof finding specific information somewhat easier,

1. Introducing the Metric System

Are you interested in arbitrary units, fundamental metric units, and schemesfor subdividing units? The article ''Activities for Introducing Metric Con-cepts to Teachers" presents these ideas via sample classroom activities thatrequire inexpensive or homemade equipment.

Do you need to convince someone that the English system of measure-ment is cumbersome? Then read "Inching Our Way towards the MetricSystem." This article also presents tables of basic relationships betweenmetric units and an explanation of how the metric system was developed.

"The Metric System: Past, PresentFuture?" provides an insight into therole of measure in national affairs. A detailed history of interest in themetric system in the United States is given. A summary of this history isprovided in "Historical Steps toward Metrication."

2. Teaching the Metric System: Activities

Suggestions for introducing the metric system in the classroom are con-tained in "Experiences for Metric Missionaries." Conversion charts showingthe relative sizes of metric units and their English counterparts are alsoincluded.

Ideas for making your own metric equipment are expressed in "MetricEquipment: How to Improvise." The article also explains how to subdividea line segment into ten equal parts.

"Think MetricLive Metric" emphasizes the importance of estimatingmetric measures and provides a series of charts for recalibrating common

3

4 A METRIC HANDBOOK FOR TEACHERS

household measuring devices to the metric system. Sample test items forjudging your own to apply the metric system are also included.

Gaines can he exciting tools for teaching both metric facts and metricmeasuring skills. "Procedures for Designing Your Own Metric Games forPupil Involvement'' lists principles for making such games and gives anexample of a game involving weighing objects in kilograms.

The section concludes with samples of activities and worksheets relatingto the metric system that have appeared in the "Ideas- department of recentissues of the Arithmetic Teacher.

3. Teaching the Metric System: Guidelines

Six ideas for converting a school curriculum to the metric system arecontained in "Schools Are Going Metric.''

The NCI N.I Metric Implementation Committee sugg sts both general andspecific guidelines for teaching measurement and the metric system in thearticle "Metric: Not If, but How.

Placement of metric-system topics and skills in the elementary and sec-ondary curriculum is discussed in -Teaching the Metric System as Part ofCompulsory Conversion in the United States."

"Metric Curriculum: Scope, Sequence, and Guidelines'' expands the ideasof the two previous articles. A sequence of topics relating to the metricsystem is suggested for primary, intermediate, and secondary levels. Teach-ing guidelines are also presented.

4. Looking at the Measurement Process

What skills arc necessary for measuring? "Ten Basic Steps for SuccessfulMetric Measurement'' considers basic skills and suggests a natural sequencefor their development.

"Thinking about Measurement'' reviews the work of Piaget with respectto measurement and suggests specific teaching activities that are consonantwith Piaget's findings.

Actual measurements are approximations. The article "Teaching about'About suggests that a measurement is between hounds and developsan arithmetic for these bounds.

5. Metrication, Measure, and Mathematics

This article considers measurement as a mathematical function, and ex-plores several measure functions. Psychological difficulties and considerationsin the teaching of measure functions and the applications of measure functionsto metrication are considered. Teaching suggestions conclude the paper.

Introducing the Metric System

6

Activities for introducingmetric concepts to teachers

JON L. HIGGINS

The best way to learn about the metric system is to use it! Here are sonic sampleactivities that show you some of the advantages of the system. They are restrictedto activities that use simple materials. As you work through them, you shouldconsider how they might he adapted to use other materials, either commercial orteacher made, and if they are suitable to the age or grade level you teach.

Work Card #1: Arbitrary units

Find a textbook. Measure its length using your index finger as a lengthunit. How many fingers long is the book?

Now pass the hook to a neighbor. Have him measure the length of thehook using the length of his index finger as a unit. How long does he saythe hook is?

How do your two measurements compare?Is there a problem if the book is not a whole number of fingers long?

What might you do to help solve the problem?

Work Card =2: Parts of units

Use the same textbook you used for work card = I. Measure its widthusing the length of your index finger. What is its width to the nearest wholefinger?

Now measure the width of the textbook using as a basic unit the width ofyour thumb. What is the width of the textbook to the nearest whole thumb?

If you are restricted to whole-number measurements, which do you thinkis the better measure to use, lingers or thumbs?

Imagine that you measured the width of the textbook using the thicknessof your fingernail as the unit of length. Which would be the best measureto use, fingers. thumbs, or fingernails?

What are the disadvantages of using fingernail thicknesses as length units?Could you improve the process by using some combination of finger,

thumb, and fingernail units? What rules might you have to make to besure different peoPIC'ebuld get the same finger, thumb, and fingernail measure-ments for a given hook width?

7

8 A METRIC' HANDBOOK FOR TEACHERS

Work Card =3: Fractional units

It nut he more accurate to measure with smaller units, but most peopleget tired of eountinl, the big numbers that result. A good compromise isto use a big unit that is subilkidecl into smaller fractional parts. One can usethe big unit to measure most of the length a.al count only the fractional partsfor the last partial unit of length.

At the bottom or this work card are big units that have been divided intofractional parts. The name of the big unit is the awkward.

I low !tidily fractional parts make one awkward?('an You name one of the fractional parts? (You may want to make up

a name of your own for this subdivision. lie creative!)Measure the leagth, width, and thickness of a textbook, to the nearest frac-

tional awk ward.Pass the hook to your neighbor ',ind compare measurements. Do they

match? Why or v. hy not?

I I 1. L

,skward

Work Card -7:4: Computed measures

Imagine a square piece of paper one awkward on each side. How manyof them could you lay on top of the textbook? Make a guess to the nearestwhole square awkward.

Calculate the area of the top of the textbook in square awkwards bymultiplying the length in awkwards by the width in awkwards.

I low does your calculation compare with your guess?If your length and width measures included fractional parts of an awkward,

your area measure should include some fractional parts of a square awkward.If you use only the subdivisions shown on work card #3, what would bethe smallest fractional part of a square awkward?

Calculate the volume of a box that would hold the textbook by multiplyingtogether the length, width, and thickness of the book (all in awkwards).Your answer is the number of awkward cubes that would he required tofill such a box.

Construct a cubic awkward. Does your answer for the volume of thetextbook box look reasonable in terms of the cubic awkward? How couldyou check corn work`'

What is the smallest fraction of a cubic awkward von could possibly obtainif you use only the subdivisions of the awkward shown on work card #3?

Could we have 'mule a better choice of fractional parts when dividingthe awkward?

INTRODUCING THE M ETRIC'. SYSTEM 9

Work Card =5: Introducing the decimeter

Previous work cards have explored two problems: the necessity to agreeOn a standard unit of length instead of arbitrary units, and the advantagesand disadvantages of different schemes for subdividing units into fractionalparts. The bottom of this work card is ruled in a length unit that solvesboth these problems. 'the unit is a standard that is widely agreed on, andthe Unit has more convenient fractional parts than the awkward. Sec if youcan find out why its fractional parts arc easier to work with.

Measure the length, width, and thickness of a textbook with this ruler.Compute the area of the top of the book in square decimeters. Write

your answer in both fractions and decimals.Compute the volume of a box that would just hold the textbook.11 lengths, widths, and thicknesses are measured to the nearest tenth of

a decimeter, what will be the smallest fractional part of a square decimeterthat can be calculated? What will be the smallest fractional part of a cubicdecimeter that can he calculated?

(1

Decimeter

Work Card -=6: A bigger measure

Measure the height of your neighbor using the decimeter scale from workcard 5. (One convenient way to do this is to first mark his height on astrip of adding-machine tape: then measure the distance between the markand the end of the tape.)

Would a different length unit he more convenient for measuring theheights of different people'?

Suppose you invented a bigger length unit. but used the decimeter asthe fractional part of this new unit. Remembering what you know aboutcomputing with fractions and decimals, how many decimeters should youchoose to make one of the new length units'? Explain your answer.

-\ length unit that is ten decimeters long is called a meter.Using the decimeter s'.:alc on work Lard ±5, construct a meter on a piece

of adding-machine tape.Find the length ;Ind Width of your room in meters.Calculate the area of the floor of your room in square meters.

10 A METRIC HANDBOOK FOR TEACHERS

Work Card #7: Smaller measures

Measure the length and width of a business calling card. You may useeither the meter from work card #6 or the decimeter from work card #5.Which would be the better choice!

Would it be more convenient to define a unit o' length smaller than thedecimeter? Could you use your knowledge about he meter and decimeterto define this smaller unit? How might you do it?

A smaller length unit is ruled on the bottom of this s'ark card. I-low doesit compare to the decimeter scale on work card #5?

The name of this smaller unit length is the centimeter.Use centimeters to measure the length and width of the calling card.Use centimeters to measure the length and width of a paper clip.

0 I 2 3 4

Centimeter

Work Card #8: Relating measures

Measure a length of one meter on a strip of adding-machine tape. Nowremeasure this length using the decimeter scale from work card =5. Com-plete these equations using decimal notation:

1 meter decimeters: I decimeter -7:: meter

Use the ,:entimeter scale from work card =7 to measure a length of onedecimeter. Complete these equations:

1 decimeter = centimeters; I centimeter = decimeterI meter = centimeters; I centimeter = meter

If we want to measure even smaller lengths, it is convenient to give aname to the fractional part of a centimeter. This tiny length is called amillimeter. Make a table of equations relating the millimeter to the centimeter,decimeter, and meter.

If we want to measure bigger distances, we could invent a bigger lengthunit that has a meter as its fractional part. The most commonly used biglength unit is the kilometer. A kilometer is 1000 meters.: meter would bewhat fractional part of a kilometer?

Make a table of equations relating the l-Hlometer to the meter, decimeter,centimeter, and millimeter.

INTRODUCING THE NI TRIO SYSTEM 11

Work Card 9: Exploring volumes

Use light cardboard to make a box that is I decimeter long, I decimeterwide, and I decimeter high. Leave the top of the box open.

What is the volume of this box in cubic centimeters?Fill a one-quart milk carton with rice. Pour the rice into the cubic decimeter

box. is the volume of the box more or less than one quart?Use a one-cup measure and rice to estimate the volume of the box in

cups. Report the volume of the box as being less than cups andmore than . __... cups.

This box is a convenient measure of volume, and it has the advantageof being closely related t,,) the units of length you have just worke1 with.The name of the volume of the box is one /her.

Work Card = 10: Exploring weights

If we fill one cubic centimeter with water we can invent a small unit ofweight. This weight unit is called a gram.

The United States nickel weighs about 5 grams. Use a balance and a nickelto find the weight i n grams of one sheet of paper. ( I f the sheet of paper weighsless than the nickel, and you cannot cut the nickel up, what else can youdo to solve this problem? )

What is the weight in L.-anis of an en:ire ream of paper?If a liter box is filled with water, how much would it weigh in grams?One thousand grams is used as a big unit of weight, called the kilo.1,,rom.

What is the weight of a liter of water in kilograms?Put a plastic hag inside the liter box \ ou constructed for work card =Y.

Fill the bag ( to the rim of the box ) with water (always holding it over asink). You are holding a kilogram. Have someone place books in your otherhand until the books feel about as heavy as the kilogram.

What is the weight of each book according to ,our estimate?How could you improve the accuracy of \ our estimate?

Repcimcd From the .Apr'l :973 AR 11011 I lc rFAcul u. Copyright r 1973, National Council of feachers of Mathematics.

Inching our waytowards the metric system

GERARDUS VERVOORT

Is assistant professor of mathematics 'duration at Lai eheadl .1 ni ersit y (; erard us I "ern )ort's pro f es.Nional experience afro includes

teachint; in elementary Gml secondary schools. II e has worked withI mlians and liskitnos and has taught school in three di 11 crewcountries. I I e grew up in the Netherlands and, thus, is

14Seti to thinking ntetrie.

What is heavier, a pound of gold or apound of feathers?"

"They both weigh the same.- answersthe bright child in whom we have carefullynurtured logical thinking.

-\\ rong!.. we reply. "A pound of feathersis determined kv avoirdupois weight andmeasures 7,000 grains. A pound of gold isdetermined by troy weight and measures5,760 grains. Thus, a pound of feathers isheavier. Clear? Let us try once more. Whatis heavier. an ounce of gold or an ounce offeathers?-

"An ounce of feathers?""Wrong!"-They both weigh the same'?"-Wrong again! A pound of gold consists

of 12 ounces because it is determined bytro), weight. Therefore, an ounce of goldis equal to 480 grains. But there are 16ounces in an avoirdupois pound. Thereforean ounce of feathers equals 437.5 grains.-

It should come as no surprise that manypeople in North America have ceased allcritical thinking with respect to measure-ments. A full-page advertisement for a cer-tain small car in the 18 October 1971 issueof Newsweek boldly proclaims that it hasa fiftv-seven inch overall outside widthwhile it is a full live feet across on the in-side! I low many readers noticed the dis-crepancy'?

Anyone who feels smug and confident

12

regarding his knowledge f the NorthAmerican system of weights and measuresis invited to test his [nettle on the followingquestions:

1. Flow many cubic inches are there ina gallon?

2. What is the difference between a

liquid quart and a dry quart?

3. Flow many square feet are there inan acre?

4, A common aspirin tablet is fivegrains. I-low many scruples does that repre-sent?

5. What is the number of pennyweightsin a troy ounce'?

There will be few who can answer allof these correctly. Yet the list of questionscould have been made much longer andmore difficult by including references torods, furlongs, square perches, poles,

chains, cords, fathoms, cables, nauticalmiles, leagues. pecks, gills, drams, hogs-heads, and barley corns. And it must nothe overlooked that though a bushel gen-erally represents 60 pounds avoirdupois,it is equal to only 48 pounds of barley, 32pounds of oats, or 56 pounds of rye orIndian corn. And do not forget the regionaldifferences. In Massachusetts a bushel ofpotatoes is 60 pounds, but only 56 poundsin North Carolina or \Vest Virginia.

INTRODUCING THE METRIC SYSTEM 13

Is it any wonder that 14 countries arepresently preparing to Go NIETRic and jointhe 114 countries and territories that haveadopted the metric system already? In-creasing world trade and the fact thatBritain is in an advanced stage of change-over from the inch-pound to the meter-gram system. makes a similar change man-datory for the economic survival of thefew remaining nonmetric countries. Mostthink as Canada:

The Government believes that adoption of themetric systern of measurement is ultimately in-evitableand desirablefor Canada. It wouldview wi'h concern North America remaining asan inch-pound island in an otherwise metricworlda position which would he in conflictwith Canadian industrial and trade interests andcommercial policy objectives. The Governmentbelieves that the goal is clear, the problemslie in determining how to reach this goal so asto ensure the benefits with a minimum of cost.

If such governments arc correct in theirassessments, then the need to begin thisprocess of change as quickly as possibleis obvious. The longer the decision is de-layed, the more the eventual cost of thechange will he increased.

The implications for the educationalsystem are clear. The children presentlyin school will be in their early thirties inthe year 2000. Presumably. the wholeworld will he metric by that time. Inches,pounds, and yards will have gone the wayof the fountain pen. the kerosene lamp.and the log cabinpicturesque memoriesof the past. surviving in a few standardexpressions and in museum exhibits, butotherwise of historical interest only.

In preparation for that time, there is animmediate need for greater emphasis onteaching the metric system and a conse-quent need for retraining teachers andrevising the textbooks. This is urgent al-ready because of the years that elapsebetween the introduction of new texts andthe graduation of the students who haveused them.

As soon as primitive men learned tospeak and communicate, a need for ex-pressing quantities must have arisen. No

doubt first expressions were vague andinexact, but they served a purpose, justas similar statements of measurement servea purpose today. We are still told to gatheran "armful" of wood, and to add a "hand-ful" of flour or a "pinch" of salt to acertain cooking recipe. The grocery storemay advertise that a "truckload" of water-melons has arrived just in time for theweekly special. The term truckload servesa purpose because no one, except the storekeeper, cares whether that means 600watermelons or 1000.

All these measurements are easy tovisualize and often directly related to phy-sical experiences. A nomadic Eskimo reck-ons distances by so many "sleeps." AGerman farmer may explain that he ownssix "mornings" of land, meaning the landarea that can he plowed by a man in sixmornings. We do comparable things in

North America when we measure distanceby stating that it is a "three-hour drive"or when we measure areas by "city blocks."Sometimes such measurements survive inour language even though they can nolonger he easily visualized. Electoral dis-tricts are sometimes called "ridings" fromthe distance a man could cover on horse-back. And just as primitive man developednew measures as the need arose, so do we.We talk about a "pack" of cigarettes anda "roll" of paper towels.

However, such ine;:act measures werenot sufficient for trade or barterthey lefttoo much room for disagreement becausethey meant different things to differentpeople. Even where agreement existed, it

still could he very confusing. For example,a last (load) of herring was 12 kegs, buta last of gun powder was 24 kegs. A lastof brick was 500 bricks, but a last of tileswas only 144 tiles. A last of wool was12 sacks.

If one goes to a market place in Europe,one can still buy goads by the ell. An ellof cloth is a length of cloth stretched be-tween the hand and the shoulder (thismeasure survives in our word elbow), butwhen purchasing by the ell, watch the

14 A METRIC HANDBOOK FOR TEACHERF;

salesman closely to see that he indeedstretches his hand and arm completelywhile measuring Your purchase. Preferablybuy from people with long arms, and underno circumstances buy elastic that way.

To make trade possible, local barons andchieftains often established certain stan-dards of measurement. Their foot was al-ways a popular standard, So was theirthumb. A certain Anglo-Saxon king definedthe yard as the length of his girth. Picturethe foot, thumb, and waist measuring cere-monies. Imagine all the resulting confusion.For not only did these measures differ fromplace to place, but they changed with theadvent of any new ruler. And lift expect-ancy was rather short in those days,

As long as trade occurred primarily atthe local level, the situation was not dis-astrous. People did not question why clothshould be measured by the ell, land by rods,and a horse's height by hands. And beforethe Arabic numerals were used, convert-ing from one measure of length to anotherwas difficult regardless,

With the grow ing acceptance of thedecimal system of numeration, the begin-ning of science and industry, and the de-velopment of more powerful national gov-ernments ( who were interested in the nowof goods for purposes of taxation 1, thesituation changed. Voices became adamantin favor of a more rational system of Mea-surement; one that not only would he uni-versal, but in which the units of length,area, and capacity would he related in a

simple manner.If the new system was to he truly uni-

versal, with all measures related as muchas possible. then the selection of a basicunit was important. Several possibilitieswere considered. The time of the swing of apendulum is directly related to its length.The length of a pendulum that would de-scribe one complete swing per second wassuggested as the fundamental unit of thenew linear measure. But that would hardlyhe universal, critics pointed out: a pendu-lum swings faster at the north and southpoles than it does at the equator. Moreover,

a measure defined in that Way would pre-suppose a definition of a second which wasin itself a questionable measure.

A sector of the equator was also sug-gested as a unit. But the length of theequator would be difficult to me.,:wre. Be-sides, few countries touch the equator;thus, the new measure would not he trulyuniversal.

Finally, a third proposal was agreed on:a portion of a meridian would be used asa general standard. Although few countrieswere on the equator. every nation was onsome meridian. (It was generally acceptedat that time that every meridian was of ex-actly the same length as any other meridian,a belief that was later proven wrong.) Butwhat portion of the meridian should heused? One millionth. a ten-millionth, a hun-dred-millionth? Practical aspects of dailylife, as well as trade and commerce. had tohe taken into consideration. Since the ap-proximate circumference of the earth alonga meridian was already known from astro-nomical calculations, and because dividingthat length by 40 million would yield alength of about one yard. that was the unitdecided on for the basic measure. The newmeasure was called the meter (in French.metrefrom the Greek metros. "mea-sure-). In turn the basic measure was in-creased or decreased by powers of ten toestablish other linear measures. Greek pre-Ike', to the term niter were used to denotemultiples of the unit, while Latin prefixesindicated subdivisions.

The result was as follows:kilometer = I(NN) meters

I hectometer = 100 metersdekameter = to metersmeter = I meter

I decimeter = O. t meterI centimeter = 0.01 meterI millimeter = 0.00I meter

For a unit of area, the square dekameterwas decided on. A square meter wouldhave been too small for practical purposes;a square hectometer, too big for a landwhere fields were small. A square 10 metersby I() meters roughly equaled the size ofa woman's herb and vegetable garden, thus

INTPODUCING THE METRIC SYSTEM 15

making it easy to visualize. The new unitof area was called an are.

At first glance one would expect an ex-tension of this new measure by the properprefixes to create the whole increasing anddecreasing sequence. But a square equalin area to R) are would have a length of10 \ rio meters (approximately 31.6 meters),thus upsetting the simplicity of the system,Hence the only acceptable extensions ofthis measure are the following:

I hectare = tOO oresI are = I areI centiare 0.01 are

Of course one can always speak of a squarekilometer or a square meter if the needsrequire it.

The basic measure for volume (capacity)posed no great difficulty. Reason demandedthat it be defined in terms of the meter.One cubic meter was clearly much toobig ( approximately 250 gallons): a -ablecentimeter, too small. Hence the only rea-sonable choice was the cubic decimeterwhich is equal in capacity to about onequart. The new measure was called a liter( from the French litre). Again the derivedmeasures followed the same pattern as themeter:

I kiloliter1 hectoliterI dekaliterI liter

deciliterI centiliter1 milliliter

11X)0 liters1(X) liters

IO liters1 liter

= O. I liter= 0.01 liter= (1.001 liter

Convenient though the liter was for pur-poses of measuring liquids, it was not satis-factory in all cases. For instance, firewooda cubic meter would appear a lot morereasonable. It was adopted as such andcalled the .s.tere (from the Greek stereo,"solid" ). The stere was used nearly ex-clusively for wood: and, as a result, nonames for powers of the stere were everadopted because there existed little needfor them.

To us, living in the second half of thetwentieth century, the unit of weight (moreproperly, mass ) agreed on is surprising be-cause it is so small. Rut at the time the unit

was selected, relatively few goods were soldby weight. Notable exceptions were pre-cious metals and spices, which were sold insmall quantities, of course, but whichplayed a very important part in the eco-nomic structure of the country. And thescientists themselves often dealt in verysmall quantities in their laboratories. Atany rate, the unit of mass selected was themass of one cubic centimeter of water atits greatest density. This was called thegram (French gramme). Again the usualderivations were agreed on:

kilogramhectogramdeka gra mgramdecigramcentigrammilligram

= 1000 grams= 100 grams

In grainsI gram

0.t gram= 0.01 grain= 0.001 grant

For the measure of angles, the traditional90 dsgree angle was called a grade. It wasdivided into decigrades, centigrades, andmilligrades. ( It is for that reason that theterm centigrade, as applied to temperature,is incorrect. It is more properly calledCelsius, after the Swedish scientist AndersCelsius who created that particular tem-perature scale.) The renaming of an Ilesnever caught on, however, due to thecumbersome fractions involved. For in-stance the traditional 60 degree angle be-came 66=3 centigrades. It is clear that thischange was no improvement.

The metric system originated in Franceduring the period of the French revolution.What hampered the acceptance of themetric system in non-French countriesmost, however, was the excessive zeal dis-played by the metric creators in other areas.They began an entirely new calendar start-ing with the year one. They fashioned anew "week" of ten days duration, thusdoing away with the Sabbath. As a result,the whol:, metric system came to be asso-ciated in the eyes of many with a "godlessatheism," a system "conceived in sin andborn in iniquity," as some put it. Combinethis with a common veneration for mattersold and familiar, and with the distastes ofthe English-speaking world for anything

16 A METRIC HANDBOOK FOR

French that resulted from the Napoleonicwars.

How lon!, lasting and extreme this feelingcould be was expressed by the periodicalcalled The International Standard that waspublished in the I SSOs by the Ohio Auxil-iary of the International Institute for Pre-serving and Perfecting Weights and Mea-sures. The president of the Ohio group, acivil engineer who prided himself on havingan arm exactly one cubit in length. hadthis to say in the first issue:

We believe our work to be of God: we areactuated by no selfish or mercenary motive. Wedepreciate personal antagonism of every kind,but we proclaim a ceaseless antagonism to thatgreat evil. the French Metric System. . . . Thejests of the ignorant and the ridicule of theprejudiced fall harmless upon us and deserveno notice. . It is the battle of the Standards.May our banner he ever upheld in the cause ofTruth, Freedom. and Universal Brotherhood,founded upon a just weight and a just measure,which alone are acceptable to the Lord.

A later edition combined religion andchauvinism beautifully in the song entitled"A Pint's a Pound the World Around."Following are some of the verses and thechorus:

They bid us change the ancient "names",The "seasons" and the "times".And for our measures go abroadTo strange and distant climes.But we'll abide by things long dear,And cling to things of yore

Chorus:

Then swell the chorus heartily.Let every Saxon sing:"A pint's a pound the world around",'Fill all the earth shall ring."A pint's a pound the world around",Fur rich and poor the same:Just measure and a perfect weightCalled by their ancient name!

TEACHERS

Now Great Britain has discarded theinch-pound system and Canada has de-clared its intention to go the same way.The time for decision in the United Stateshas come. On 6 August 1971 Mr. Clai-borne Pell, the Senator from Rhode Island,introduced a bill (S.2483) to Congress"to provide a national program in order tomake the international metric system theofficial and standard SVStent of measure-ment in the United States and to providefor converting to the general use of suchsystem within ten years after the date ofenactment of this Act." The bill has beenpassed by the Senate. The ultimate decisionto go metric appears inevitable. Teacherswould do well to start acquainting theirstudents with the system more thoroughlythan in the past. THINK METRIC- should hethe slofan in the teaching of measurementfor the child who will spend most of hisadult life in the twenty-first century.

References

Arnold, C. J. "An Answer to 'Argument,; AgainstUniversal Adoption of the Metric System.'Sch,)ol Science and Mathematics 51 (April1951): 310-15.

Donovan, F. Prepare Now for a Metric Future.New York: Weybright & Taliey. 1970,

Drews, R. M.. and Chauncey D. Leake. "TheMetric SystemPro and Con." Popular Me-chanics 114 (December 10601: 138-39.

Herbert, E. W. The Concise Afetric Conrerter,Winchester, England: Prentice Press, 1070.

Read, Cecil B. "Arguments against UniversalAdoption of the Metric System," SchoolScience and ,Vathematics 50 (April 1950).287-306.

Swan. Malcolm D. "Experience Key to MetricUnit Conversion," Science Teacher 37 (No-vember 1970): 69-70.

White Paper of Metric Conutrsion in Canada.Ottawa. Ont.: The Queen's Printer for Canada.1970.

Reps inlet! Ii.nn the April 1973 Awl ti,11 uc 1 i Acin it. (.01,)light 1973, Natiorml Council a rettehers of tslathematics.

The metric system:past, presentfuture?

A R. 'F U R F . EIALL RI3FRG

.-Is projesmr of mathematics at UnircrAity

in l'alpariliNo, Indiana, Arthur I ailerl,crg teacher mathematicsc,,(11,111 ceunses ,01,i scc,,n,hu '01 tem hers.

ICUS' the relit,r of the .VCT:Ils Thinly -first learbool,,Tories for the Miithemitties Classroom.

The wife's query in an old cartoon getsright to the point: ''Well, why on earthdon't we adopt the metric system if its somuch better?" The husband's response isequally direct: "Why. just because wenever did, my dearl-

It is unlikely that any simpler explana-don, enigmatic vet revealing, could begiven to summarize the introduction of themetric system in the United States. Oneshould not be surprised that this cartoonfirst appeared over twenty -1\e years ago(Committee on the Nletric System 1948,p. 4 ). It could have appeared twenty-fiveor flits years before that And twenty -fiveyears from now, will proponents for theincreased use of a system still uncommonin the United States point in similar fu-tility to this same cartoon? There is evi-dence to the contrary.

This brief paper will endeavor to giveteachers of mathematics some of the rele-vant background on the present status ofthe metric system in the United States andother countries, While we will include somereference to the problems that must hefaced in metricaliwt (the word used by thef3ritish to describe the process of convertingor changing over to the metric system), thispaper reflects the official position of theNCTM which -encourages the universaladoption of the metric system of measure."In 1945 the Twentieth Yearbook of the

17

NCTM was &voted exclusively to docu-mcnting the desirability of officially adopt-ing the metric system in this country. Inthree of the last four years, NCTM Dele-gate Assembly resolutions have focused onthe metric question: these have been ac-cepted by the Board of Directors andreferred to the appropriate committees. The1972 resolution urges "that the NCTMcontinue to support the adoption of themetric system and encourage that this hea system to be taught by teachers of allgrades, along with other systems of mea-surement, beginning in the 1973-74 schoolyear.-

Early measurements

In I590 J. W. I.. Glaisher said, "1 amsure that no subject loses more than mathe-matics by an attempt to dissociate it fromits history.- Certainly we show no dis-respect to mathematics if we restate this bysubstituting measurement in general, andthe metric system in particular, in thisdictum. The history of measurement in-cludes practical origins, theoretical aspects,cultural sidelights, and pedagogical impli-cations.

We teach our children that the process ofmeasurement assigns a number to somephysical characterization of an object. Itmay be a characterization of length, vol-ume, capacity, mass, or weight. To do this

18 A \tFTRIC HANDBOOK FOR TEACHERS

we must begin with a unit that exemplifiesthe same characteristic that we wish tomeasure. By determining how many ofthese units are "contained" in the objector quantity to be measured, we arrive atthe appropriate number, or nicaviire, ofthe object.

The earliest units of length found inrecorded history clearly indicate that manfound it convenient and simple to use hisown person for defining, and indeed sup-plying, measuring units. The interestingstory of the historical development of mea-sures and weights cannot be detailed here.But certainly our children should hear ofthe cubit and the fathom, the thousandpaces, and the "three barleycorns roundand dry from the middle of the ear."

In early times the relative size of theobject to he measured determined thechoice of a larger or smaller unit as themeasuring stick. If one measures the lengthof an object with some particular unit, it isunlikely that the unit measure will be con-tained an exact whole number of times inthe given length. Eventually, simple round-ing off of such a measure could not hetolerated and the use of a "smaller unit"was needed to give a more careful reading.The seemi igly natural relations betweencertain of the different basic units un-doubtedly sufficed for a time ( hand = 4digits. yard = 3 feet), but the basicprinciple of subdividing a given unit intoa certain number of equal subunits wasthe inevitable refinement.

Although each person had his ownbuilt-in measuring system, the units variedfrom individual to individual. If barter orpurchase involved a specified number ofsuch units, unequal sizes of the unit pre-sented obvious difficulties. It was naturalthat persons of the same community wouldagree on certain standards reasonably ac-cessible to everyone. As commerce grewand man become more mobile, the needfor standards acceptable to larger groupsand geographical areas became obvious.Again, the early development of standardunits of various measures is a story too

involved to give in any detail here (Ameri-can National Standards Institute, pp. 7-14). For our purposes, we pick up thestory at two related points in history.

The Jefferson plan

The United States Constitution, asadopted in 1788, provided that CongressshouJ.I have the power "to coin money, ...and fix the standard of weights and mea-surements." Prior to this, under the leader-ship of Thomas Jefferson and RobertMorris, the Congress of the Confederationhad adopted the dollar as the fundamentalmonetary unit, based on the Spanish pieceof eight, So that parts and multiples of thefundamental unit would be in an easily cal-culated proportion to each other, the deci-mal ratio was clearly favored. So in 1786a completely decimal system of coinagewas approved by Congress, and by 1792the coinage system was finally implementedwith the establishment of a mint.

In 1790 Thomas Jefferson, as Secretaryof State, was requested to prepare a similarplan for a unified system of weights andmeasures. His response emphasized severalfeatures. The first was the need for aninvariable standard of length. This he basedon the pendulum principleessentially, acylindrical iron rod of such length that aswing from one end of its arc to the otherand hack again would take two seconds.This based the standard on the motion ofthe earth on its axis. For linear measure,this rod (about 58.7 inches) was to hesubdivided into live equa: parts, each tohe one "foot." As a second feature, thesystem suggested was, like the coinage sys-tem, decimal in nature. The basic foot wasto be successively subdivided, each timeinto ten parts. forming inches, lines, andpoints. Similarly. 10 feet were to equal I

decad: and derived larger units, all basedon successive multiplication by ten, led toroods furlongs. and eventually to themile ( 10.000 feet).

The ounce was to be the basic unit ofweight, derived from the weight of a cubicinch of rainwater. Aug yin, the multiples

INTRODUCING TIIE rslETRIC. SYSTEM 19

and subdivisions were all decimally ar-ranged, with a pound equal to ten ounces.(A cubic foot of water would thus be di-ided into one hundred new pounds of

ten ounces each. )Jefferson's plan was discussed by Con-

gress over a period of six years, but no lawswere passed as a result of his work. Al-though Jefferson used sonic of the samenames (inches, feet, ounces, pounds) forhis new units, which were not too far re-moved in size from the old units they wereto replace, the division of the foot into teninches and the pound into ten ounces mayhave been a part of the reason for hesita-tion on the part of Congress. Another rea-son, undoubtedly, was the discussion of anew measurement system that was con-currently taking place in France.

It should he noted that Jefferson's planinvolved three important principles: (1 )the standard unit of length should be basedOn some unchanging, absolute standardfound in the physical universe; (2) thebasic units of length, v,Ilutue, and weightshould be directly related to each other;and ( 31 the specifically named multiplesand subdivisions of the standard unitsshould be decimally related. In presentinga plan based on these ideas, Jefferson wasfollowing the same principles previouslyadvanced by scientists and mathetna!iciselsewhere in the world. Our attention turnsto an earlier time, across the ocean.

The beginning of the metric system

By the late middle ayes a complex ;is-sortment of measuring systems had been inuse throughout all of Europe. Intenseprovincialism and the pressure of traditioncontributed to this, as did the custom togive measurements in the units unique tothe object to he measured. Thus, land wasmeasured in rods, horse's height in hands.depth of water in fathoms, diamonds incarats, and so OIL For many years the stateof trade and technology was such thatsociety could function satisfactorily evenwith such variations.

But scientific advances made during the

seventeenth and eighteenth centuries indi-cated the need for a change. As men likeNewton (in physics) and, later, Lavoisierin chemistry ) needed increasingly more

accurate measurements to investigate andto substantiate their theories, existing sci-entific instruments were improved on, ornew ones were invented. The need forinternational standards of measurement incommunicating the results of research andstudy to scientists was obvious.

A final impetus for a reform of the sys-tem of measurement came at the time ofthe French Revolution when reminders ofthe feudal system and of kings who ruledby divine right were to be discarded, Widevariations existed in measurements, withaccompanying errors, frauds, and disputes.Fortunately, the change to a new systemwas made with serious regard for broadimplications for the future. The three prin-ciples listed previously in connection withJefferson's work were advanced in seekinga reform, with the added provision (in 1790)'hat the Royal Society of London was in-

ted to join with the Academy of Scienceof Paris in deducing an invariable standardfor all measures and all weights which couldthen receive uniform acceptance.

The French Academy began work im-mediatelyfortunatelv, for the Britishnever accepted the invitationand pro-posed, first of all, a fully decimal-basedsystem. Shortly thereafter they spe,:ifiedthat the unit of length should be equal toone ten-millionth of an arc representingthe distance between the North Pole andthe equator ( a quadrant of the earth'smeridian). This standard unit of length wasto be called the metre ( we write meter).derived from the Greek metro,' "a mea-sure.- The basing of the standard on thelength of an arc of some part of a greatcircle of the earth was first proposed in1670, along with its decimalization, by theFrench iff-)11, Gabriel Mouton. He alsosuggested an equivalent definition based onthe length of a pendulum that would heat3,959.2 times in half an hour at Lyons.France.

A METRIC HANDBOOK FOR TEACHERS

Preliminary calculations had alreadybeen made in 1739-40 when the and seg-ment of the meridian passing through Parkwith both ends terminating at sea level hadbeen measured. This segment began onthe northern coast of France at Dunkerqueand ended at the Mediteranenn Sea at Bar-celona, Spain. It was the longest length ofan accessible arc available in Europe, ap-proximately 9 degrees and 30 minutes. Thepreliminary calculations made in 1740 sup-plied a provisional form of the meter whichwas to be used for almost ten years. In1793 a provisional kilogram was alsoadoptedthe weight in a vacuum of I

cubic decimeter of distilled water at thefreezing point.

From 1792 to 1798 a second survey ofthe arc was accomplished, though onlyunder the most adverse circumstances. Alsoat this time, additional experimentation wascarried out in the weighinL, of water atvarious temperatures, since temperature af-fects density. Thus in 1799, the provisionalmeter and kilogram were replaced by thenewly established standards. (The pro-visional meter was L.3 millimeter too long.)A standard meter and a standard kilogramwere constructed of platinum and depositedin the Institut National des Sciences et desArts.

The activities in France during the 1790scoincided with the placing of Jefferson'splan before the U. S. Congress. Specialcommittee report.; to Congress spoke ofthe plans of the French, and although theynoted the desirability of uniformity in mea-sures and weights of all commercial nations,they did not recommend a change in theexisting measures in the United States. In1795, copies of the French provisionalmeter and kilogram were sent to the U. S.Government in an attempt to obtain trueinternational uniformity. But the prevailingpolitical conditions of that day (the UnitedStates had refused to take sides in a disputebetween the British and the French at thattime ) did not offer a favorable climate inCongress. This, together with the Jeffersonproposal and the traditional viewpoint that

advocated no change, resulted in no actionwhatsoever by Congress.

To accompany the establishment of themetric system in France, all Europeancountries were invited in 1798 to sendrepresentatives to Paris to learn of the sys-tem so that it might be accepted ultimatelyas an international standard. Althoughnone of the nine countries that respondedto the invitation adopted the system at thattime, the educa!ional impact of this effortwas to be felt at a later time.

France itself did not achieve immediatesuccess with its new system. Mandatorythough it was, it could not he enforced,since secondary standards had not beendistributed to the various governmentalagencies throughout the country, let alonethe commercial and household users. Thesituation was further complicated in 1812when Napoleon Bonaparte issued a decreeestablishing a ..system of measures termedust(e//c, based on the metric system butusing old unit names and ratios rather thanthe decimal system. The confusion resultingfrom the increase in the number of mea-sures was so great that in 1837 an act waspassed abolishing the 'allelic system andreturning to the original decimal metricsystem. 13y 1840 the mere possession of oldstyle weights and measures was punishableto the same degree as their illegal use. Thus,the conversion was to be effected.

The Adams report

During the early1 800s, the Congress ofthe United States was preoccupied withmatters pertaining to the growth and ex-pansion of the country to such a degreethat it still did not make provision for theuniformity of weights and measures pro-vided for by the Constitution. In 1816President Madison reminded Congress ofthis fact. Hence, a year later, PresidentMonroe's Secretary of State, John QuincyAdams, was asked by the Senate to pre-pare a new statement concerning the regu-lations and standards which could beadopted in the United States. The ReportUpon Wei.vins and Measures, issued by

INTRODUCING THE METRIC SYSTEM 21

Adams in 1821, has become a classic inthe history of the metric controversy inthe United States. Its exhaustive treatmentof the advantages and disadvantages of boththe English and metric systems to theUnited States in 1821 was such that bothproponents and opponents of the metricsystem were to turn to it for years to conicin support of their positions. While thetheoretical advantages of the metric systemwere described by Adams, the practicabilityof the system remained much in question.The altering of the system by Napoleonwas an indication of the inability of thesystem to gain proper acceptance. Adamstherefore recommended that, while theUnited States should confer with the gov-ernments of France, Spain, and GreatBritain in the attempt to develop the prin-ciple of uniformity in weights and mea-sures, any action that would he taken toachieve uniformity in the meantime shouldbe within the framework of the BritishStandards.

The recommendations of the Adams re-port were a realistic response to the con-ditions existing in the United States in

1821. Unfortunately, at the same time,they were to preclude further considerationof the metric system for the next fortyyears. During those years the countrywould experience such great developmentsin transportation, commerce, and industryas to give it a feeling of independence fromtie need for international uniformity in any

Uniformity achieved

While no action of any sort was im-mediately taken by Congress in responseto tne Adams report, within ten years an-why.- clause of the Constitution was tohave its effect. In 1$30 the Secretary ofthe Treasury was directed to make a com-parison of the standards of weights andmeasures used in the principal customhouses of the United States. The widevariation in these standards, which weretic.ed for purposes of taxation, clearly in-dicated a direct violation of another section

of the Constitution, that "all duties, im-posts, and excises shall be uniform through-out the United States." The Treasury De-partment felt it had sufficient authority tocorrect this situation without further legis-lation, and, therefore, adopted the yard,the avoirdupois pound, and the Winchesterbushel as the official units. The unofficialstandard of length was based on a copy ofthe Troughton scale, an 82-inch bronze barwith an inlaid silver scare obtained from aLondon instrument maker in 1814. Thedistance between the 27th and 63d inches,representing 36 average inches of the bar,was taken to be equal to the British yardat 62 degrees Fahrenheit. During the periodfrom 1836 to 1856, Congress providedthat complete sets of standards of weightsand measures should be supplied to all

states and territories, and the individualstates adopted these as their standards ofweights and measures. Uniformity of mea-surements had been secured throughoutthe country!

But interest in the metric system was notcompletely dead in the United States. Theactivities of Alexander Dallas Bache, agreat-grandson of Benjamin Franklin, re-flected the continuing concern for a simpler,but more uniform system of weights andmeasures. In 1843 Bache was appointedSuperintendent of the Coast Survey, andas such was responsible for the Office ofWeights and Measures. In various reportsto Congress, he noted that the current ar-rangement of weights and measures was"deficient in simplicity and in system" andargued for the universal uniformity ofweights and measures.

The attention of Congress to the prob-lems of the Civil War period gave lowpriority to these and other plans for theadoption of the decimal system. In 1863the National Academy of Sciences wasestablished by an act of Congress. Its firstpresident was the same Bache, who was tosee that its first established committee wason Weil2hts. Measures. and Coinave. .\,tanding committee of similar name "ascreated by the !louse of Representatives

A METRIC HANDBOOK FOR TEACHERS

in 1864. This committk..k.. initiated legisla-tion that, in 1860, made it lawful through-out the United States of America "to em-ploy the \veights and measures of the metricsystem.- While a date by which the use ofthe system should become mandatory' wasnot included, it was anticipated that aftera short period ;1 further act would thedate for its excluske adoption.

Shortly thereafter steps were takenabroad for the establishment of an im-proved international metric system. In 1875the "Treaty of the \leek:C. was signed inParis by seventeen nations, including theUnited States. This pros ided for the estab-lishment of a permanent InternationalBureau of Weights and Measures withheadquarters near Paris, and for the fabri-cation of new international prototypes. In1890 the United States received prototypemeters No. 21 and No. 27 and prototypekilograms No. 4 and No. 2P. Three yearslater, by administrative action of the Super-intendent of Weights and \leasures, thesewere declared to be the nation's funda-mental standards of length and mass-, andthe units of the English customary systemwere defined by carefully specif ing whatfraction of a meter would constitute a yardand \\ hat fraction of a kilogram wouldconstitute a pound.

While the preceeding account summa-rizes the positive action in establishing na-tional standards of measure in the UnitedStates, it by no means covers the entirestory of the proponents and opponents forthe adoption of the metric system in thiscountry.

The metric controversy

The history of the metric system con-troversy' in the United States has recentlybeen published as one of the substudics refthe U. S. Nletrie Study (Treat 1971).

Perhaps the longest running debate in the historyof this country k whether the United Statesshould convert to the metric syst,m1. In thecourse of almost two centuries doiens of ingil-ments have been advanced, attacked. and defended with a passion inspired by a topic uithimplications that are both intensely practical and

intellectually stimulating We Simone 1471, p.351.

The substudy report records, in fascinat-ing detail and with complete documenta-tion, this debate, which has been held notonly in the COMM ittee rooms of Congress,but also in educational institutions andpublications, scientific societies, engineeringorganizations and trade journals, hoards ofcommerce and trade, and in the publicpress.

Some idea of the focus of interest andtime element involved is obtained front thefollowing listing of the major periods ofactivity treated by historian Charles F.

Freat.

I . The period of consolidation ( 1786-I 806 )

2. The educational mmcink:nt (1886-(88) )

3. The movement to introduce the metricsystem through government adoption( 189(1-1914 )

4. The propaganda period ( 1914-1933 )

5. The comprehensive study phase( 1934-1908 )

A summariiins,! paragraph for one of theseperiodsit happen; to he that of 1914-1933. although, as indicated. it would doabout as well for ;my of the previousperiodsis the follm%

For the most part, the arguments used by bothsides during this period were simply moderniiedversions of those that had been advanced fordecades. The proponents continued to insistthat the metric ssstern 1/4 superior. that itshould be adopted in the interests of interna-tional uniformity. that the costs and difficultiesin\ olvcd in adopting it uould be surprisinglyslight. and that the eventual displacement of:ill other systems by the metric system uds in-esitable. Furthermore, it via, said. the main-tenance and improvement of our foreign tradedepended upon metric adoption. I he opponentsof the system claimed that the r sited Stateshad already achieved greater uniformit\ ;Indstandardization using the ,..nsion,,,r\ 1:),Qh,hSystem than was enjoyed b :my other nationon earth. that the site of our foreign tradein no %% ;1y related to our ',ysteni of \\ eights :Indmeasures. and that changing over to the metrics,tem would be confusing. co,11, and notproductive of corresponding benetit,. In

INTRODUCING THE M FTRIC SYSTEM 23

tion, the opponents claimed that what appearedto be a popular clamor for metric adoption wasreally an artificial demand that had been gen-erated by insidious prometric propaganda(Treat 1971, pp. 227-28).

Recent developments

Events over a period of ten years addeda sense of urgency, lacking befoi-e, That

were to culminate in 1968 in the mostsignificant investigation requested by theU.S. Congress.

1957 The launching of the Soviet Union'sfirst Sputnik.

This created new interest in scientificeducation and research.

195$ The U.S. House of Representativescreated a standing Committee on Scienceand Astronautics.

The committee was given jurisdictionover standardization of weights and mea-sures and the metric system.

1959 7'he customary standards were of-ficially defined in terms of metric units.

Countries using the customary units, in-cluding the United States and the UnitedKingdom. defined the yard to he 0.9144meter (1 inch = 2.54 centimeters) and theavoirdupois pound as 0.45359237 kilo-gram.

1960 The Eleventh General Conferenceon II.eights and Aleasnres (of which theUnited States is a member) redefined themeter.

The "meter bar" was abandoned as theinternational standard of length, and awavelength of light was substituted(1,650,763.73 wavelengths of the orange-red line produced by krypton 86 was de-fined as 1 meter), This new definition wasa return to the original concept underlyingthe metric system, namely. that an im-mutable standard he found in nature. Thenew determination is accurate to I partin 100,000,000 and has the advantage ofbeing reproducible in scientific laboratoriesthroughout the world.

1960 The Systi'me late rnathmal d'Unitts(,SI) was established.

Adopted by the General Conference onWeights and Measures, it became the of-ficial system, with the basic units of meter,kilogram, second, ampere, kelvin, candela,and other units based on these.

1965 Great Britain announced its inten-tion to convert to the metric system withinten years.

This was done after repeated requestsby industry that the conversion be made,although it previously had been held thatany conversion would have to be accomp-lished simultaneously with a conversion bythe United States.

1968 Du, U.S. Congress directed theSecretary of Commerce to undertake theU.S. i1letric .S.tudy.

The purpose of the study was to evaluatethe impact on America of the metric trendand to consider alternatives for nationalpolicy.

hi 1971, the Report of the U.S. MetricStudy was transmitted to the Congress ofthe United States. The study was conductedby the National Bureau of Standards of theDepartment of Commerce. The reportrecommended "that the United States

change to the International Metric Systemdeliberately and carefully," and that "theCongress, after deciding on a plan for thenation, establish it target date ten yearsahead, by which time the United Stateswill have become predominately, thoughnot exclusively. metric."

One of the most important recent de-velopments to give urgency to metricationis the increase in internationalized engineer-ing standards. Engineering standards (notto be confused with "measuring standards")are "norms" regulating size, weight, com-position or configuration of products. andstandardization of practices. The Interna-tional Organization for Standardization(referred to as ISOa similar agency, IEC,is concerned with electrotechnical matters)is a nongovernmental body "to promote thedevelopment of standards in the world witha view to facilitating international exchangeof goods and services and to developing

24 A METRIC HANDBOOK FOR TEACHERS

cooperation in the sphere of intellectual,scientific, technological, and economic ac-tivity (American National Standards In-stitute, p. 36)." The American NationalStandards Institute ( ANSI ) is the U.S.member of both international orgnniza-tions. The great majority of standards stillremain to he developed. Thus, by express-ing itself in metric units the United Stateshas the opportunity to influence interna-tional standards negotiations with its ex-perience and technology.

Closely related to such internationalstandards is the matter of world trade. Arelatively slight drop in our exports of"measurement sensitive" products couldmean the difference between a favorableand an unfavorable balance of trade forthe United States. The decision of GreatBritain to go metric leaves the UnitedStates as the only industralized nation thatis not committed to metric conversion.Canada ht-.s the commitment, but hasdelayed in-a-,;cm..:ntation because of theclose trade relations with the United States.

The conversion act of 1972

On 18 August 1972, the U.S. Senatepassed on a voice vote the "Metric Con-version Act of 1972" (S. 2483 ). However.action was not taken by the House duringthe final weeks of the ninety-second Con-gress, so new action will he required inthe next Congress.

Although chapter after chapter of thehistory of the efforts to convert the UnitedStates to the metric system has ended withthe words "hut no action was taken byCongress," there are strong indications thatthe last chapter is about to be written.Even though many important investigationsand studies have been made at the requestof Congress over the nearly two hundredyears since the beginning of our country,the U.S Metric Study carried out from1968-1971 is something different. No pre-vious study was preceeded by so much dis-cussion and careful consideration of theoverall objectives that such a study shouldmeet. No previous study was required to

be conducted with such breadth and depthin giving every sector of society an op-portunity to express itself in public hearingsand special investigations.

Though Congress will need to beginagain on a new metric conversion act in1973, sonic idea of possible legislation canhe had by examining the 1972 Senatebill which reflected recommendations thatemanated from the U. S. Metric Study.

1. it declared that it would be the policyof the United States to encourage the sub-stitution of metric measurement units forcustomary units in all sectors of the econ-omy with a view toward making metricunits the predominant, although not ex-clusive, language of measurement within aperiod of ten years. (Note that the metricsystem is not to he the sole official, stan-dard system of measurementa footballfield may very well remain 100 yards long.)

2. It encouraged the development of en-gineering standar,l based on metric unitswhere it would result in simplification, im-provement of design, or increase in econ-omy.

3. It recommended the establishmentof an eleven-member Metric ConversionBoard to oversee and implement the con-version process.

4. It recognized that federal procurementpolicies would have to he used to en-courage gereral conversion to the metricsystem.

S. It called for programs for educatingthe public to the meaning and applicabilityof metric terms and measures in daily life,for assuring that the metric system of mea-surement becomes a part of the curriculumof the nation's educational institutions, andthat teachers be appropriately trained toteach the metric system.

The following paragraph summarizesthe U.S. Metric Study Report:The cost and inconvenience of a change to..netric will be substantial even if it is carefullydone by plan. But the analysis of benefits andcosts made in (this report] confirms the intuitivejudgment of U. S. business and industry thatincreasing the use of the metric system is in

INTRODUCING THE METRIC SYSTEM

the hest interests of the country and that thisshould be done through a coordinated nationalprogram There will he less cost and morereward than if the change is unplanned andoccurs oscr a much longer period of time (DeSimone 1' I7 p.

NktriCntiOn is indeed "a decision whosetime has Mc:- for Inierie-J. and pr-tieuktrly for its cducatiomil institutions!

ReferencesAmerican National st,IndArds Institute. Ahtvur-

S.istcnis and S!ou iiir,l.c ()reanizations. New

25

York: American National Standards Institute,11.d. (updated to 19701,

Committee on the Metric System. The 111ctric-5ysrcin of Iiicirtiity and Ste/wires. TwentiethYearbook of the National Council of 'leach-ers of Mathematics. New York: Bureau ofPublications, Columbia University, 1948.

De Simone, Daniel V. A Aletric America: ADecision It'hose Thule MIS COW. U.S. MetricStudy. Washington, D.C.: GPO, 1971.

Treat, Charles F. A History of the Metric SyPen,Contrm.ersr in the ('aired Stales. U.S. MetricStudy, tenth substudy report. Washington,D.C.: GPO, 1971.

Commonly listed advantages of the metric system

I. The metric system is :u simple. logicaltyplanned system. Its decimal basis conforms toour numeration system.

2. The meter, which establishes the basis forthe entire system, is zilss ay. reproducible fromnatural phenomena and, therefore. is immune todestruction: it is international in character andwell suited for precision work. The coordinationof measures of length, area, VOLUM: and Mass,combined with decimalization, facilitates com-putations.

3. Once the system of prefixes has beenlearned, the uniformity in names for all types ofmeasures makes for greater simplicity and easein changing to more convenient-sized units forspecitie purposes.

4. Although many of the claims of the amountof time that can he sased in learning fractionsare probably exaggerated, less competence in themanipulation of common fractions WOOLd berequired of many students. and sonic restructur-ing of time spent on such computation Imo( heprofitably accomplished. Ultimately. when onlythe metric system would he taught. time couldhe saved in the learning of conversion unitsI I ft. 12 in.. I yd. 3 ft., 1 rd. 161/2 ft.,1 1111. 5251) ft., and so on -I and in the learn-ing of a second new system for use in scienceckisses.

:. For ordinary work, familiarity ith thenutter. gram, and liter would he sufficient. How-ever, for the engineer and the scientist, the newInter national System of Units (SI), based on themetric units. eliminates conversion problems incalculations \kith derived units. That is, althoughpower is universally defined as work per unit

time, it is variously expressed in Btu,:sec, therms./day, ft-lb/hr, horsepower, calories 'sec, watts, andso on. The use of Iwo different terms, kilo.wrantand newton. for units of mass and force respec-tively, eliminates much of the confusion studentshave concerning these concepts.

6. Greater participation of the United Statesin the setting of international engineering stan-dards would he possible since the metric systemis the universal language of measure. Exports,foreign trade, and competition with other nationsis ill depend more and more on production undermetric standards.

7. A common measurement language that isused by scientists, engineers, and industrial work-ers would improve communication and reducebarriers between different sectors of society.

S. The necessity cif Conversion would offerfringe benefits. During the adjustment to thenew measurements, there would be opportunitiesto eliminate superfluous varieties in sizes ofproducts, parts, and containers, and to makeadditional worth-while changes in engineeringstandards, construction codes, and products de-sign.

It should be noted that men of honest persuasionhave often listed statements contrary to thosegiven here as advantages of the customary sys-tem. A comprehensive listing of both prometricand procustomary arguments is given in A MetricAmerica. (Report of the U.S. Metric Study, 1971.Order by SD Catalog No. C 13.10:345, S2.25,from Superintendent of Documents, U.S. Govern-ment Printing Office, Washington, D.C. 20402.]

Listed by AR 111UR E. Hit FOE RG,

Vo/parfliS0 University, Valparaiso, Indiana

Iistorical stepstoward metrication

N't A R 1 1. N ti. S 1.] 1' I) A NI

670 Gabriel Nlouton (a Lyons vicar generally regarded as the founderof the inetric Nstcni) proposed a decimal system of weights andmeasures. defining its basic unit of length as a fraction of thelength of a great circle of the earth.

1740 Preliminary calculations were made with a provisional form of ameter.

1790 A metric system of int.asurement was developed by the FrenchAcademy. The need for a uniform Ny.naem of eight.,; and measure'seras //Wed Und dius.sed in the U.S. Colf,.:res.V. bla 110 actionwas laken.

1795 France officially adopted a decimal system of measurement.

1798 A meeting was held in Paris to disseminate information about themetric system.

1799 -Inc provisional meter and kilogram were replaced by newly estab-lished standards.

-I 821 1 document was i.,wied hr It/hr, Orti/icy Adams e.vhau.ytirelythe adranta.tfes ant' di.Vadri011cf:,,C.,; of both the English and metricvvstents; Adams concluded that "the time was not right.-

1840 France made use of the metric system compulsory.

1866 1.e:4/via/ion made it "lawful throughout the U,S. to enipioy theweit;hts. and measures of the metric .sycion.- The Ny.,aem was notmade mandatory, although this way wain:paled.

1875 The -Treaty of the N1eter$ setting up well-defined metric standardsfor length and mass. vis signed in Paris by seventeen nations,including tile U.S. The International 13ureati of \\eights and Mea-sures was established.

1880 Most of Europe and South America had gone metric.

1890 The U.S. receiced prototype meters and Ailo,,,,ram.v.

:ViLmted front ninteri;t1, publisheLl And coprighted by the Agency far lintructionnlTelevi,[on. 197.1. t:sed

16

INTRODUCING THE N1 sYsTEm 27/41893 The meiric prototypes were declared by the Superintendent of

li'eMits and ,\leasure.s to he the "fundamental standards- for theC.S,.. other I I I eamres were defined in terms of the standard meterand kilogram. (Thus, the vard is legally defined as a fractionalpart of a meter and the points/ as a fractional part of a kilogram.)

1() ..Ippro.kimate/y forty bilk on metrication were introduced in Con-;Tess, Ina no action was taken.

1959 cuqumary unite were officially defined in terms of metric. units.

96t The meter \Aas redefined in terms of a wavelength of light. Themodernized metric system, the International System of Units(Ss.stt:nic International d'Lnit.L's), referred to as SI, was established.

I 965 Great Britain announced its intention to convert to the metricsystem.

1968 The I'.S. C'ongress directed the Secretary of Commerce to under-take the three-year U.S. Metric ,Study, to evaluate the impact ofthe metric trend, and to consider alternatives for national policy.

1971 a result of the metric .sludy, it eras recommended that the 11.S.shan't' to predominant use of the metric system through a co-ordinated national program.

1972 The Metric Conversion Act was passed by the Senate, but no actionwas taken .).1. the 116 (se, .co new action is required ht' Congress.

Teaching the Metric System: Activities

Rep, hued from the .is Ivird A1,11'1\11 ur r, CCII,Slight t' 1'171, National Council of .1 c.icliers of Ntithematics.

Experiences for metric missionaries

LOTTIE VIE:TS

,-is assistant professor I PI education at Kansas .State e COUCKe,

l'ittshurg, Louie rids teaches methods for clementury schoolmathematics. She preciously serrecl as el(ThelIlary SIIIWITLS'Or in the

!Ionic(' Mann 1.aboran,ry School at Kansas Stale College and has fort.ehtwale levels one throuKh co,ht in Kansas schools.

For the past 180 years, the United Statesof America has been inching along towardthe almost universally adopted metric sys-tem of weights and measures. Certain sec-tors of our societyscience, medicine, en-cineering. and athleticsare in variousstates of transition at the present time.Some important developments and majorc .'visions of the United States governmentregarding metrication, including a conver-sion date projected by the National Bureauof Standards, are shown in figure 1.

A Gallup Poll, prepared in August 1c/71and published in October of that year,found that only forty-four percent of theadults in the United States knew what themetric system was and of these, only forty-two percent were in favor of adopting it.

The survey also showed that among thosewho had attended college, eight in ten knewof the existence of the metric system; andof those who were aware of the system,their opinions favored adoption live to four.The former Secretary of Commerce, Mau-rice E. Stuns, made this comment followingthe survey:

The poll confirms our own imestigations v,hichrevealed that the more people know about metri-cation the more they favor its adoption. Whatis now needed is intensive education of thepublic on the significant benefits to be derivedfrom he system.'

The evidence, therefore, is clear. Thereis an immediate need for education to

I. George Gallup, "Nletlic Support Groins,'' Kan-sas City Star I October 3, 1971); 5A.

1894 Metric System ockpre-i of U S wcr Depairren far medical work

1902 Metrc System adopted by US Heath Department:926 EV! to Mange to metric system fads

'932 Metric System adopted byAmerican Athletic' Union

1908 Rassage of US Metric Study Act

.1971 Report of the US Metric Study delivered to Congress1983 Rircec tea date of the changeover to

metric system

' [893 Meter and kilogram made standard

1875 Internotional Bureau of Weights and Memories eslablisr?ed

:856 congress authorizes use of Ine metric system in me United States

1521 John Q',incy Adams advocates Ire odoption of Ihe metric system

1790 George Washington mks for uniformity in weights and measure

Fig. I

31

A METRIC HANDBOOK FOR

prepare the public for metrication. Thiseducation can he done in large part throughelementary school children who can carrythe language. measurement experiences,computation exercises, and problem solv-ing involving the metric system into homesin our country. It remains the responsibilityof the school to give pupils the opportuni-ties for measuring and estimating measuresthat Cone 1 1964 ) found necessary in hisstudies. It is the purpose of this article tosuggest experiences that will aid teachers inaccepting the challenge of educating chil-dren who can be metric missionaries in thisperiod of convetsion to another system ofweights and measures.

The initial emphasis in instruction shouldbe placed on teaching the fundamentalunitsmeter, gram, and literand theprefixes that indicate the multiples and sub-multiples of ten. Charts and models show-ing basic terminology and the interrelation-ship of the basic units should be givenpermanent space for display for referencepurposes in the classroom. The picture infigure 2 shows children making use of suchmaterials.

Fig. 2

To accomplish this beginning instruction,each child should be helped to develop akit of the units of measure that includesmaterials like the following: string or rib-bon the length of a meter, soda straws of adecimeter length strung with yarn to makea meter (can also be folded for decimeterlengths), centimeter lengths cut from soda

TEACHERS

straws, centimeter grid paper for construct-ing the cubic centimeter and the liter.Heavy cardboard meter sticks can be con-structed for home use. These can be kept athome with the usual yardstick and used asa measuring instrument for homework as-signments and routine household measures.

Children should be given guidance in de-veloping references or approximate meas-urements such as the following:

I meter Height or arm stretch of a kinder-garten child

Distance of the chalkboard from thefloor

I decimeter Length of a piece of chalkDistance across face over eyes of a

child1 centimeter Width of a finger

Length of a bean or an eraser on apencil

1 millimeter Thickness of hem y tagboardI grain Weight of a paper clip2 grants Weight of a sugar cube5 grams Weight of a nickelI liter Capacity of gasoline tank of small

lawnmower

Charts and displays like those in figures 3through 7 show common equivalents thatwill be helpful for further reference whileaccommodating the English system ofweights and measures.

Fig. 3

TEACHING THE METRIC SYSTEM: ACTIVITIES 33

I yd

Units ofLength

yard

77-

meter

Fig. 4

Conversion charts from pounds to kilo-grams can he helpful for reference in indi-vidual graphing of weight in kilograms. The

I oz.

Units ofWeight

T

ounce gram

2 lbs

I lb.

Units of

Weight

Fbund Kilogram

Fig. 6

graphs could he started at school and com-pleted in the home at regular intervalsthroughout the school year. The classroomgraph in figure 8 shows the number of pu-pils in weight ranges to the nearest kilo-gram. It should provide assistance to pupilsin preparation of their individual graphs.

1.1qts_

3qts

2qts

I qts

Units ofLiquid Measure

gallon quart liter

Fig. 5 Fig. 7

34 A NI ETRIC HANDBOOK FOR TEACHERS

Numberofstudents

14

12

10

8

6

4

2

22-24 25-27 28-30 31-33 34-36 37-39Weight to nearest kilogram

Fig. 8

Children enjoy using an illustrated meas-uring tape mounted on the classroom wallto check their heights in centimeters forlater use in making graphs. The picture infigure 9 shows children measuring theirheights. Charts of average heights that aredisplayed in the classroom for comparisonpurposes also appear in the picture. Eachchild can also be helped to prepare one ofthe measuring tapes for his home use.:Adding machine tape is suitable for thisproject.

Fig. 9

The reference measurements listed ear-lier could be put to use in estimating forsolving problems like the ones that follow:How many meter lengths of fabric would heneeded to make each girl in the first grade a longdress for the school program?There are grams of sugar left in the box.How many people can be served at a tea ifeach person uses one sugar cube?

Homework assignments might requirepupils to locate labels indicating weightsand measurements in metric language. Thefollowing are examples:I box gelatin 85 grams1 box soda crackers 198 gramsI can soup 298 grams

1 bottle vinegar 355 cubic centimetersI can vegetable juice 177 milliliters

Other assignments could include makingmeasurements of parts of buildings, oritems in and around the home, and findingperimeters, areas, and volumes in metricmeasurement. Finally, children will betaught to make conversions from one sys-tem to another.

These methods and materials cannot beexpected to be effective unless the stagesof children's learning about measures andthe theory of measure, among other factorsof teaching and learning, are recognized

TEACHING THE

and observed by teachers. However, thesuggestions given are the results of early,concentrated efforts to provide immediateand useful ideas for meeting the impend-ing challenge of teaching for the metricchangeover. Teachers cannot afford toignore or delay the privilege and duty ofteaching today's children for tomorrow'sworld of metrics as Congress prepares togive the green light for conversion. Teach-ers must assume the obligation to preparethemselves, their pupils, and, subsequently,

METRIC SYSTEM: ACTIVITIES 35

the public to meet our country's commit-ment to go metric.

References

Corte, Clyde B., Teaching Mathematics in theElementary School. New York: Ronald PressCo., 1964.

De Simone, Daniel V., (a ) Education. U.S.Metric Study, sixth substudy report. Wash-ington, D.C.: GPO, 1971.

tb). A Metric Anzerica: A DecisionWhose Time Has Come. U.S. Metric Study.Washington, D.C.: GPO, 1971.

"Give me 40 liters"

etric equipment: how to improvise

JON L. HIGGINS

Many commercial companies arc nowmaking metric equipment for classroomuse, but until a large number of industriesconvert to the metric system, metric mea-suring tools will probably be more expen-sive than their English counterparts. Be-lieving, that the best way for children tolearn the metric system is for every childto have metric measuring tools for makingmeasurements, teachers may face sizablepurchasing problems. There is no substitutefor sturdy, durable, and accurate measur-ing equipment, but if a school cannotafford to buy such equipment immediatelyfor every student, there is no reason todelay teaching the to..-:tric system. Buy afew good pieces of equipment to use asstandards, and then look around for sub-stitutes that each student can use in themeantime. You may be surprised to learnthat you are already living in a metricworld!

Dividing by ten

One of the basic difficulties in impro-vising metric equipment is the problem ofdividing by ten. The problem of dividinga given unit of length into ten equal partsdoes not have an obvious solution to chil-dren. If one takes a strip of paper andfolds it in half, in half again, and in halfonce again, the creases will divide thestrip into halves, fourths, and eighths. (Isthis why the English system was dividedinto eighths and sixteenths?) But how canone divide a length into tenths?

36

The answer lies in a nice piece of geom-etry that older children might enjoy. Startwith a strip of paper representing a lengthyou wish to divide into tenths. Lay thestrip on a sheet of paper and draw a linefrom the beginning of the strip outwardat an angle; any angle will do (see fig. I ).

Strip to be subdivided

Fig. 1

Now take any convenient length unit,and mark ten of them off on the angledline, beginning at the vertex formed by theline and the strip. You may need to ex-tend the line in order to mark off tenlengths, or you may not use all of the line(see fig. 2).

Strip to be subdivided

Fig. 2

TEACHING THE

Draw a line from the end of the tenthunit to the other end of the strip, as shownin figure 3. Then draw parallel lines fromthe ends of the other lengths. These par-allels will intercept the strip, marking offten equal lengths and thus subdividing thestrip just as we want it.

\ Parallellines

\\

Strip to be subdivided

Fig. 3

Children may he able to estimate paral-lel lines by simply sliding a straightedgesideways. If this does not seem accurateenough, they can construct parallels witha compass and straightedge as suggestedby figure 4.

I. Set compass here 2 Mark arc of\ equal radius here

3 Place straightedgeon length mark

and touching arc;then draw the line

Fig. 4

For children not adept at using a compass,the following method may he somewhateasier. Perpendicular lines are extendedfront both ends of the segment to be sub-divided. A straightedge with ten equal sub-divisions is turned until its ends lie on theparallel lines, and the position of theendpoint of each subdivision is marked.(The total length of the ten arbitrary sub-divisions on the straightedge must be longerthan the segment to be subdivided.) Thenthe straightedge is slid down slightly and

METRIC SYSTEM: ACTIVITIES 37

the positions are marked again. Corre-sponding pairs of marks can be joined withlines that, when extended, will interceptthe original segment, dividing it as desired.Figure 5 illustrates this procedure.

Fig. 5

Line segmentto be subdivided

Length units

Metric measuring tapes can be made bycopying metric scales on strips of adding-machine tape. You may want to maketapes that are as long as 5 or 10 meters tohelp children visualize longer lengths, oryou might want to use the tapes for theconstruction exercise discussed in the pre-vious section. Children can learn aboutunits and subdivisions in the process ofcopying and making their own tapes.

Do you own a set of Cuisenaire rods?These rods are one square centimeter incross section, and their lengths range fromone to ten centimeters. The notions de-partments of fabric stores often carry mea-suring tapes that are marked in metricunits. If you need smaller units, rememberthat a dime is one millimeter thick.

Area units and volume units

Look for graph paper ruled in squaresthat are 2 millimeters on each side. Rulea piece of paper into grids of one squarecentimeter. You may have a machine inyour school office that will make trans-parencies for overhead projectors; if so,make several transparencies using yourruled paper. These plastic sheets can he

38 A METRIC HANDBOOK FOR TEACHERS

laid over objects and the squares countedto estimate areas in square centimeters.

Building boxes in metric units to illus-trate metric volumes is a good exercise ingeometry as well as metric measurement,Figure 6 illustrates some different patternsfor building boxes. Flow many others canyour students think of'?

j L

L

Fig. 6

Weighing and weight units

Because the gram is a very light unit ofweight, many older balances are not sensi-tive enough to be used successfully withmetric weights. Fortunately, it is fairly easyto build an inexpensive balance that is

easy to use. The main components of thebalance are shown in Figure 7. The heart

Fig. 7

of the balance is a wooden stick, about 40cm long. Three cup hooks arc screwed intothe stick, one in the middle on one sideof the stick, and one at each end on theother side of the stick. A small nail is

driven into the stick opposite the middlecup hook to serve as an indicator. Weigh-ing pans made from small pie tins are hungfrom the cup hooks at the ends of thebalance (tins from frozen pot pies workwell for this). The entire balance is hungfrom the center cup hook. A nail on a

wall, a string front the ceiling, or a woodenupright stand will all make suitable sup-ports for hanging the balance,

To use the balance, hang it front itssupport. You will need to adjust it so thatthe suck is horizontal when the pans areempty. One easy way to do this is tofasten small nails or paper clips with tapeto the light end of the stick. When theempty balance is adjusted, mark the posi-tion of the indicator nail below the centercup hook. When weighing objects, put theobject in one pan and add metric weightsto the other pan until the indicator nailreturns to the marked position.

Sets of weights can be made by be-ginning with nickels. A nickel weighs fivegrams. Paper clips arc inexpensive anduniform weights; however, sizes of paperclips vary from brand to brand, so theirweight should always be checked againstthe five-gram nickel. Cubes of sugar weighapproximately two grains and can be usedfor weights, if you don't mind losing a fewto nibblers!

For larger weights, check your pantryat home. Many cans of soups and vege-tables are now marked in both ounces andgrants. Metric weights are usually found insmall print on the sides of can labels orthe small side panels of boxes. A quicktrip to one pantry turned up a 305-gramcan of chicken noodle soup. a 425-gramcan of beans, and a 907-grant box of pan-cake mix.

Temperature units

Thermometers may be recalibrated tothe Celsius temperature scale. Put thethermometer in a pan of ice and water,and mark the lowest point that the mer-cury reaches. ( Dip only the bulb of thethermometer in the water; if tile stem isdry you can mark the level of the mercurywith a grease pencil or a small piece of:ape.

Now place the bulb of the thermometerin a pan of boiling water and mark thehighest level the mercury reaches. Thetwo marked points are the 0 (freezing) and

TEACHING THE METRIC SYSTEM: ACTIVITIES 39

100 (boiling) points on the Celsius scale.Lay the dry thermometer on a piece ofpaper and copy the two marks. The dis-tance between these marks must now besubdivided into 100 equal pieces. Thistask is one of subdividing lengths, and thegeometrical procedure discussed earlier cannow he used.

Accuracy

Making your own Celsius thermometersacrifices the accuracy in favor of lowcost. The temperature of the ice and watermixture may not be exactly the tempera-ture at which water freezes, and the tem-perature of the boiling water depends onthe air pressure of the room. However,

you may find that you gain new attitudestoward measurements and measuring in-struments when you make or adapt themyourself. The mystery of a thermometerscale tends to disappear when it is treatedas a special length scale.

In a similar fashion, constructing weightunits out of familar everyday objects maymake the weighing process seem muchinure practical than if one relied exclusivelyMI brightly polished brass weighis. Con-structing units of length, ,urea, and volumequickly impar,s the idea that the metricsystem is a constructed system. In the end,you may discover that saving money is theleast important reason for improvisine,metric equipment.

Think metriclive metric

RICHARD J. SHUMWAY, LARRY A. SACHS,and JON L. HIGGINS

Suppose we accept the premise that weshould learn the metric system. What kindsof knowledge would be needed? How couldone best acquire such knowledge? Oneplace to begin is by looking at yourself.Try your hand at the following sampletest items. Can you make reasonable esti-mates or guesses?

I. The height of a man playing center ona typical high school basketball team is

approximatelya) 6 inb) 240 mc) 2 md) 78 cm

2. Nty car was a little low on oil so the gasstation attendant recommended that I adda can of oil containing

a) 2 ml/0 I 1

c) 10 ml(1) 20 1

3. The diameter of a coffee cup is abouta) I cmh) 8 cmc) 20 cm(1) 50 cm

4. In order to hake a pizza, one should setthe oven temperature at about

a) 100 Cb) 400.'Cc) 220 Cd) 600 C

40

5. A good weight for a college-age girl ofaverage height would be about

a) 130 gmb) 40 gmc) 150 kgd) 55 kg

6. The length of a car is approximatelya) 5 mb) 15 nunc) 26 md) 3 cm

7. The capacity of a classroom aquariumis usually about

a) 10 1

h) 200 mlc) 40 I

d) 25 ml

8. A good temperature to set your homethermostat at for comfortable living wouldhe

a) 90 Cb) 32 C'c) 70Cd) 2(1 C

(Answers for these items can be found atthe end of the article.)

Living metric will certainly requireeveryday knowledge of the type reflectedin these test items. It probably will notrequire extensive ability to compute unitconversions accurate to three decimalplaces. Decisions will need to be made more

TEACHING THE METRIC SYSTEM: ACTIVITIES

rapidly than the time required for com-puting equivalents; however, most every-day decisions do not require a high degreeof precision. Thus, except for a very fewtechnical jobs, living metric will requirefamiliarity with the system so that reason-able estimates can he made,

How can you acquire this familiarity?We propose that you go "cold turkey."That is, don't even think metric, just livemetric. Two rules are important: (I) allmeasure must be metric, and (2) you must"get your hands dirty:" i.e.. you must bewilling to measure in order to learn mea-sure.

To quote an old adage, education beginsin the home. Start with your thermometer.Make ten labels that you can stick overthe scale of your thermometer, one labelwith each of the following numbers: 40,30, 20, 10, 0, 10, 20, 30, 40, 50.Table I shows over which number to puteach label.

Table 1Temperature conversion

Celsius

40 40.030 22.020111 14.00 32.0

111 50.020 68.030 86.0.40 104.050 122.0

Here is the computer program used togenerate the table:

I)) FOR C' 40 TO 50 STEP I))20 LET F = 9*C',5+3230 PRINT USING 40. C,F40 IMAGE

50 NEXT C

For example, the 0 should he pastedover the 32 mark on the Fahrenheit scale,the 10 over the 50 mark, and so on. Besure you cover up the old numbers com-pletely so that you can't see the tempera-

41

tune in Fahrenheit. Now take a thin stripof paper and cover up the subdivisionmarks from the old scale. You can estimatethe appropriate subdivisions between thenew Celsius numbers.

You may he uncomfortable about read-ing temperatures in Celsius degrees at first,but if you make yourself live metric, youwill soon feel just as comfortable withnumbers like 10, 20, and 30 as you usedto feel with numbers like 50, 70, and 90.You have to be ruthless, though, and coverall the old Fahrenheit scales so that youforce yourself to live metric.

Now to the bathroom scale. Maketwenty labels for the numbers 5, 10, 15,20, ... 90, 95, 100. Peel back the materialaround the window of your scale so thatthe window may he removed (see fig. I ).

Fig. 1

Then, using table 2, paste the labels overthe appropriate numbers.

Table 2Weight conversionbathroom scale

g lltc k g lhs

5 11 65 4310 70 54IS .33 75 65

44 80 7625 55 85 8731) 66 90 9835 77 95 20940 88 (IX) 22045 99 105 23150 110 III) 24355 121 115 254

132

For example, the 5 would be pasted onthe scale at 11, the 10 at 22, the 15 at 33,

42 A METRIC HANDBOOK FOR TEACHERS

and so on. Again, give yourself no 5111-pathy. Cover up the old numbers and theold subdivision marks completely. Don'tuse table 2 to convert back to poundsagain--put it away! Once your scale is

labeled, you no longer need table 2. Thebest way to get used to kilograms is tohave to use kilograms.

Now relabel those old yardsticks! Pastenew numbers at the positions shown intable 3.

Table 3Length conversions

n rot

-1

811 7 51S 3

19 523 527 I

31 I

35 3 8

III

")1

311

-11)

5)1

70tiU

90

Cover all the old numbers and scaledivisions. Estimate the subdivisions be-tween the new centimeter numbers.

You may prefer to destroy your oldyardsticks and make new meter tapes. Twometers is an ideal length for many pur-poses. and buckram (available at mostfabric stores) is a near-perfect, yet inex-pensive, material. 13ack ram usually comesin three-inch widths (fabric stores haven'tcompleted their metric changeover) thatcan easily he torn lengthwise into strips.Make three strips, each about 2.5 cm wide.Cut them into two-meter lengths. Using ameter stick and a felt-tip marker, dimen-sion and label your tape measure withdivisions and numerals as are desirable andpractical. (A scale of centimeters is usuallyfine, with possibly some millimeters shownat the beginning. )

Again, remember that the point of livingmetric is to make a cmiiplete change.Change at/ your yardsticks, or destroythem, or lock them away forever! You'llfind that forcing yourself to live metric

will eventually make thinking metric a verynatural thing to do.

There are many other changes you canmake in your home. Use the informationin table 4 to begin labeling other thingsin your home. Be a little goofy about it.You can make it fun!

Don't forget to relabel the tools you usefor your hobbies. Sewing, woodworking,photography, travel, and so on, are allideal vehicles for learning to live metricbecause the sizes of things are familiar to

Living metric in the classroom

When you have relabeled your home,start relabeling your classroom and school.You may want to accelerate the pace atwhich the metric system is used by creatingspecial measuring activities. Use the newmeter stick or tape measure to becomefamiliar with some common lengths in

metric. Are you 1.6 m tall with a waist of60 cm? Is your pencil 11.5 cm long andyour notebook paper 22 cm wide? As youbecome more comfortable with metriclengths, estimate before you actually mea-sure. You may be surprised at how quicklythe accuracy of your estimates improves.

Do you have a juice or Kool-Aid breakin your class'? Turn it into an experiencewith metric volumes. Let the children orderthe amount they want in metric measures.An order for ten milliliters receives lessthan a tablespoon full; a request for twoliters gets a half-gallon pitcher full!

Besides requiring children to practicetheir metric knowledge and adding a littlehumor, this is an excellent opportunity toemphasize the volume/mass relationship.Try using styrofoam cups that have a verysmall mass (less than 5 g). If a child re-quested ISO ml, how could you check howaccurate the dispenser of the juice waswith his estimate? If you were thinking"weigh the juice and cup," you're right!Since juice is mostly water and one milli-liter of water at 4 degrees Celsius weighsone gram, the weight of the juice in grams

TEACHING THE METRIC SYSTEM: ACTIVITIES 43

is a good approximation of its volume inmilliliters.

After von have introduced a series ofindividual activities for practicing metricskills, it's interesting to get the class to-gether for a metric countdown. Prior toits start, each child should have measuredand labeled several items for use in thecountdown. Divide the class into two teamsand proceed as in a spelling bee, wherethe object is to guess the metric measure-ments of different items. Two modifications

make for a better game. First, accept "ballpark" answers as correct. If one playerguesses the width of a desk to be 80 cmwhen it is actually 88 cm, for example, heshould certainly be given credit. A guessof 50 em, though, is probably not closeenough. One persoi, can act as judge,basing his decisions on the ability level ofthe group. Second, if one person on eachteam misses an item, it is best to announcethe correct measurement and go on to thenext item.

Table 4Other metric conversions

ni.stanccv In Places Yon frccel kitchen Rot1111 1E111016W

5 km 3 mi Intl em 3.5 0/ 0.22 lb 1.0 m 3.3 It10 km 6 mi 2011 gm 7.1 (v 0.11 111 1.5 in 4.9 II15 km 9 in: 300 gm 10.6 iv (466 Ili 2.0 m 6.6 it20 km 12 nu 400 gm 14.1 (v 0.88 lb 2.5 ni 8.2 ft

25 km 16 im 5(5) gm 17.6 of 1.10 lb 3.0 in 9.8 ft

7)) km 19 mi 6110 gill 21 . ' of I .32 lb 3.5 m 1.5 it

35 km 22 mi 700 Lull 2-1.7 of 1.54 lb 4.0 in 3.1 ft

40 km 25 nit 500 gill 28.1 of I .76 lb 4.5 in 4.8 ft

45 km 28 mi 900 gill 31.7 oz 1.98 lb 5.0 in 6.4 ft

50 km 31 mi 000 gin 35.3 of 2.10 lb 5.5 in 8.0 ft

100 km 62 mi 100 gill 38.8 of 2.43 lb 6.0 in 9.7 ft150 km 93 rin 200 gm 42.3 of 2.65 lb 6.5 in 21.3 ft

200 km 124 mi 3(X) gin -15.9 m' 2.87 lb 7.0 in 23.0250 km 155 mi 100 gm 49.4 of 3.09 lb 7.5 in 24.6 ft300 km 186 no 500 gm 52.9 oz 3.31 lb 8.0 m 26.2 ft;50 km 217 mi 600 1111 56.4 oz 3.53 lb 8.5 m 27.9 ft

400 km 2.49 mi 70(1 gm 60.0 oz 3.75 11, 9.0 m 29.5 ft450 km 250 lin 51X) gm 63.5 0, 3.97 lb 9.5 m 31.1 Ii5(10 kill 311 mi 9)0 gin 67.0 0/ 4.19 lb 10.0 in 32.8ft

1000 km 621 mi 21100 gni 70.5 oz 4.41 lb 10.5 in 34.4 ft

1500 km 932 mi 11.0 m 36.1 It

2000 km 1243 mi2500 km 1553 mi Atrchor rwtt(ctrriti3000 km 186-1 nu3500 km 2175 mi 2.5 in 2 tetb.poon

rub Nil

4000 km 1455 mi 5 m tetNpoon I 1.1 (4N15 in tabk".poon 2 2.1 LIK60

.Specdotttchtt 80niin

4 cup3 cup

3

43.2 qt,4.2 qt,-;

110 in 2 cup 5 5.3 qt,.;2n km hr 12 mi hr 161 in 2 3 cup 6 6.3 (4ts4)) kin hr 25 no hr 150 m 4 cup 7 7.4 LIN60 kin hr 37 mi hr 210 in I cup 8 8.5 1)1550 km hr 50 nu hr

100 km hr 62 in hr9

109.5 L'itt,

10.6 LIN12n km hr 75 mi hr110 km hr 57 mi hr WinA ittl; Giasi am/ Pitcher Ga.% nIllk

1)5) 3.1 1 oz 10 2.6 ga(): t'l; l CIIII,C0111f le' 2(X) 6.5 t 5.3 ga

3)))) Ill 10.1 1 of. 30 7.9 ga100 (' 21211' 4)5) in 13.51 of 40 10.6 ga25'(' 257" 500 in 16.9 I oz. 13.2 ga

150-C 30-1' 60(1 m 20.31 oz 611 15.9 ga175'(' 347 7(X) rn 23.7 I of 70 18.5 ga200 392 SIX) m 27.1 I oz. 80 _21.1 ga225 (' -137- 900 mi 30.11 of 90 23.8 ga2501(' 182' IINi0 nil 33.81 of 100 26.4 ga

44 A METRIC HANDBOOK FOR TEACHERS

The list of activities and games thatwill encourage children to live metric isendless. Once you have converted yourhouse and classroom to the metric system,get out your plan book and convert yourown time-tested classroom activities to

metric. Once you do, you'll lind that think-ing metric and living metric is not onlyeasy, but enjoyable!

Oh, yes, the answers to the sample testitems are c; 2, b; 3, b; 4, c; 5, (1;

6, a; 7, c; 8, (1.

Reprinted (mitt the May 1974 At/111041 I W Il ACM H. Copyright r 0474, Nation:It Council of Feitehers or matheomoes.

Procedures for designing your ownmetric games for pupil involvement

CECIL R. TRUEBLOOD andMICHAEL SZABO

Currently an associate professor in mathematics education

at Pennsylvania State University, Cecil Trueblood is particularlyinterested in the teaching of mathematics in the elementary school.Michael Szabo, also at Pennsylvania State, is an associate professorof science education. His educational interests center on instructionaldevelopment, individualized instruction. and complex problem-solving.

Although much has been written on thevalues of mathematical games in the ele-mentary grades and many game books havebeen published, little has been written thatwould help classroom teachers design, pro-duce, and evaluate games for use in theirclassroom. The focus of this article is topresent a set of seven criteria that were de-veloped in a summer workshop for inser-vice elementary teachers who decided thatthey wanted to be able to produce metricgames and related activities that would fitinto their "metrication" program.

The teachers in the workshop began byasking a practical question: Why shouldbe interested in producing my own metricgames? They concluded that the gameformat provided them with specific activi-ties for pupils who did not respond to themore typical patterns of instruction. Theyfelt that in the game format they couldprovide activities of a higher cognitivelevel for pupils who had difficulty respond-ing to material requiring advanced readingskills.

The teachers then asked a second ques-tion: Does the literature on the use ofmathematics games contain any evidencethat would encourage busy classroomteachers to use planning time to develop

45

their own games? The available profes-sional opinion supported the followingconclusions :

1. Games can be used with modestsuccess with verbally unskilled and emo-tionally disturbed students, and studentsfor whom English is a second language.

2. Games have helped some teachersdeal with students who present disciplineproblems because they are bored with theregular classroom routine.

3. Games seem to fit well into class-rooms where the laboratory or learning-center approach is used. This seems re-lated to the feature that games can beoperated independent of direct teacher con-trol thus freeing the teacher to observe andprovide individual pupils with assistanceon the same or related content.

Plan for development

If for any of the reasons just cited youare interested in designing and evaluatingseveral of your own metric games, howshould you begin? Simply use the followingchecklist as a step-by-step guide to helpyou generate the materials needed to createyour game. Use the exemplar that followsthe checklist as a source for more detailed

46 A METRIC HANDBOOK FOR

suggestions. Each item in the checklist hasbeen keyed to the exemplar to facilitatecross referencing.

CHECKLIST GUIDE

. Write down what you want yourstudents to learn from playing yourgame. (Establish specific outcomes)Develop the materials required toplay the game. (Make simple ma-terials)Develop the rules and proceduresneeded to tell each player how toparticipate in the game. (Writesimple rules and procedures)Decide how you want students toobtain knowledge of results. (Pro-vide immediate feedback)Create some way for chance to en-ter into the playing of the game.(Build in some suspense)Pick out the features that can beeasily changed to vary the focus orrules of the game. (Create the ma-terials to allow variation)Find out what the students think ofthe game and decide whether theylearned what you intended them tolearn. (Evaluate the game)

The exemplar

Establish specific outcome'

By carefully choosing objectives that in-volve both mathematics and science pro-cessessuch as observing, measuring, andclassifyingthe teachers created a gamethat involves players in the integrated ac-tivities. This approach reinforces the phi-losophy that science and mathematics canhe taught together when the activities aremutually beneficial. That is, in many in-stances integrated activities can be used toconserve instructional time and to promotethe transfer of process skills from onesubject area to the other. The exemplar'sobjectives are labeled to show their rela-tionship to science and mathematical pro-cesses.

TEACHERS

1. Given a set of common objects, thestudents estimate the objects' weight cor-rect to the nearest kilogram. (Observa-tion and estimation)

2. Given an equal-arm balance, thestudents weigh and record the weights ofcommon objects correct to the nearestcentigram. (Measurement))

3. Given an object's estimated and ob-served weight correct to the nearest centi-gram, the student computes the amountover or under his estimate. (Computationand number relationships)

Make simple materials

The following materials were constructedor assembled to help students attain theobjectives previously stated in an interest-ing and challenging manner.

1. Sets of 3-by-5 cards with tasks givenon the front and correct answers and pointsto be scored on the hack. (See fig. 1.)

2. A cardboard track (see fig. 2) madefrom oak tag. Shuffle the E's (estimatecards), O's (observed cards), and the D's(difference cards) and place them on thegameboard in the places indicated.

3. An equal-arm balance that can weighobjects up to 7 kilograms.

4. A pair of dice and one differentcolored button per player.

5. A set of common objects that weighless than 7 kilograms and more than 1

kilogram.

6. Student record card. (See fig. 3.)

Write simple rules and procedures

The rules and procedures are crucial tomaking a game self-instructional. In thefollowing set of directions notice how a stu-dent leader and an answer card deck serveto ease the answer processing needed tokeep the game moving smoothly from oneplayer to another. It is essential to keep therules simple and straightforward so thatplay moves quickly from one student tothe other.

L

TEACHING THE METRIC SYSTEM: ACTIVITIES

The eslirnoted \NNW of

the brick. is _kg

Front of card

The observed

weight Is kg

Front of card

I .t1 the 'blanks1he .:iif f erence

.nn brick' r;an chser E_stimofed,y-Ight kg .Thserled

,)f br "k.

Front of card

E Card

la Card

D Card

Fig. 1

. Number of players, two to six.

2. The student leader or teacher aidebegins by rolling the dice.

The highest roll goes first. All players startwith their buttons in the "Start Here" block.The first player rolls one die and moves hisbutton the number of spaces indicated onthe die. If he lands on a space containingan E, 0, or D he must choose the top cardin the appropriate deck located in the cen-ter of the playing board or track and per-form the task indicated. (In the exampleshown in figure 1 this would be Card E3.)

2 points1kg.

Back of card

2 points

kg

Back of card

.4 points

Difference _kg.

Back of card

47

The player then records the card number,his answer, and the points awarded by thestudent leader on his record card. The stu-dent leader checks each player's answerand awards the appropriate number ofpoints by reading the back side on thetask cH-d. He then places that card on thebottom of the appropriate deck and playmoves to the right of the first player. Theplayer who reaches "Home" square withthe highest number of points is the winner.At the end of the play each player turnsin his score card to the student leader whogives them to the teacher.

48 A METRIC HANDBOOK FOR TEACHERS

HOME; Start _

Here, )

E

D

Subtract '

2 points

SkipForward2 spaces

You loseI turn

Fig. 2

Provide immediate feedback

By placing the answer on the back ofthe task card and appointing a studentleader, the teacher who developed thisgame built into the game an importantcharacteristic, immediate knowledge of theresults of each player's performance. Inmost cases this feedback feature can bebuilt into a gameby using the back oftask cards, by creating an answer deck, orby using a student leader whose level ofperformance would permit him to judgethe adequacy of other students' perform-ance in a reliable manner. Feedback is oneof the key features of an instructional gamebecause it has motivational as well as in-structional impact.

Have students record diagnostic infor-mation. The student record card is an im-portant feature of the game. The cardshelp the teacher to judge when the diffi-culty of the task card should be alteredand which players should play together ina game, and to designate student leadersfor succeeding games. The card also pro-vides the player with a record that showshis scores and motivates him to improve.

Go backI Space

L_._ Add

1

2 points

This evaluative feature can he builtinto most games by using an individualrecord card, by having the student leaderpile cards yielding right answers in onepile and cards with wrong answers in an-other pile, or by having the student leaderrecord the results of each play on a classrecord sheet.

Build in some suspense

Experience has shown that games en-joyed by students contain some element ofrisk or chance. In this particular game aplayer gets a task card based upon theroll of the die. He also has the possibilityof being skipped forward or skipped backspaces, or of losing his turn. Skippinghack builds in the possibility of getting ad-ditional opportunities to score points; thisfeature helps low-scoring students catch up.Skipping forward cuts the number of op-portunities a high-scoring player has toaccumulate points. The possibility of add-ing or subtracting points also helps createsome suspense. These suspense-creatingfeatures help make the game what the stu-dents call "a fun game."

TEACHING THE METRIC SYSTEM: ACTIVITIES 49

Student's name

Card number

E3

E2L_

Answer given

2 kg.

Date

Number of points_

2

I kg.

Fig. 3

Create the materials to allow variation

A game that has the potential for varia-tion with minor modifications of the rulesor materials has at least two advantages.First, it allows a new game to be createdwithout a large time investment on the partof the teacher. Second, it keeps the gamefrom becoming stale because the studentsknow all the answers. For instance, theexemplar game can be quickly changed bymaking new task cards that require thatstudents estimate and measure the areaof common surfaces found in the classroomsuch as a desk or table tops. By combin-ing the two decks mixed practice couldbe provided.

Evaluate the game

Try the game and variations with a smallgroup of students and observe their ac-tions. Use the first-round record cards asa pretest. Keep the succeeding record cardsfor each student in correct order. By com-paring the last-round record cards with thefirst-round record cards for a specific stu-dent, you can keep track of the progressa particular student is making. Filing thecards by student names will provide alongitudinal record of a student's progressfor a given skill as well as diagnostic infor-mation for future instruction.

Finally, decide whether the students en-joy the game. The best way is to use aself-report form containing several singlequestions like the following, which can beanswered in an interview or in writing:

1. Would you recommend the game tosomeone else in the class? Yes . No

2. Which face indicates how you feltwhen you were playing the game?

3. What part of the game did you likebest?

4. How would you improve the game?

Concluding remarks

The procedure just illustrated can begeneralized to other topics in science andmathematics. The following list providessome suggested topics.

1. Classifying objects measured in metricunits by weight and shape

2. Measuring volume and weight withmetric instruments

3. Measuring length and area with metricinstruments

4. Classifying objects measured in metricunits by size and shape

5. Comparing the weight of a liquid to itsvolume

6. Comparing the weight of a liquid withthe weight of an equal volume of water

7. Predicting what will happen to a blockon an inclined plane

8. Comparing the weights of differentmetals of equal volumeWhy don't you try and create some

games for each of these topics? Then sharethe results with your colleagues. Additionalexamples developed by the authors areavailable in "Metric Games and BulletinBoards" in The Instructor Handbook SeriesNo. 319 (Dansville, New York, 1973).

Reprinted front the Nthy 1572 and April 1973 Ant rttit.tic TY N(111.12. Copyright 1972, 1973, National Council of Teacher!:or N1,1dieniati,s.

Activities that contribute to the student's personalunderstanding of key concepts in mathematics.

Prepared by George Immerzeel and Don %S'iederanders, Malcolm Price

Laboratory School, University of Northern Iowa, Cedar Falls, Iowa.

Each /DLAS presents activities that are ap-propriate for use with snalentS at the VariousleuelS in the eleMentary school. After you harechosen the activities that are most appropriatefor your .1 tridents, remove the activity Sheetsand r.,produce the copies you need. After asheet has been used, add your own commentsand file the materials for future use.

IDE.4.9. for this month relates to units ofm casut The focus is on the metric systemActivities at the lower levels use the familiarnumber line to relate basic units of measurewithin the metric system. tipper level activitiesuse the number line to relate familiar Englishunits and ba.ic metric units,

For Teachers

Objective: Experience in relating basic units of linear measure using a metricnumber line

Levels: I, 1, or 3

Directions for teachers,

I. Remove the activity sheet. Reproduce a copy for each student.2. Have a student measure with a meter stick the width or height of a

familiar object in the front of the room. Have everyone put his pencilon his metric number line to show the measure reported by the student.

3. Be sure your students understand the directions for each part beforethey proceed.

Comments: Metric measures are likely to be more important to yourstudents' lives than English measures. It is the school's responsibility toprovide learning situations in which the student relates personally to metricunits of measure. If your students have not had experiences in actualmeasurement with centimeter scales prior to this time, such experiencesshould precede the use of this activity sheet.

50

E

Put

an

X o

n th

e m

etric

line

for

each

of

thes

e m

easu

res

.

Nam

e

Cen

timet

ers

O10

2030

4050

6070

8090

100

110

120

130

140

150

160

170

180

190

200

210

O1

2

Met

ers

10 c

entim

eter

s30

cen

timet

ers

35 c

entim

eter

sI

met

erI m

eter

10

cent

imet

ers

ET

3L11

'-=

4--19

cen

timet

ers

28 c

entim

eter

s70

cen

timet

ers

_r

Nam

e ea

ch o

f the

se p

oint

s on

the

met

ric li

ne.

Cen

timet

ers

AB

CD

EO

1020

3040

I).

60 I

7080

9010

011

012

013

014

015

0

1i

XX

'X

iI

X16

017

180

190

200

210

II

'1

X'

OI

2

Met

ers

Ace

ntim

eter

sB

cent

imet

ers

Cce

ntim

eter

sD

cent

imet

ers

Ece

ntim

eter

sE

met

ers

52 A METRIC HANDBOOK FOR TEACHERS

I A For Teachers

Objectives: Experience in relating basic metric units of weight using anumber line model

Levels: 2 or 3

Directions for teachers:

1. Remove the activity sheet and reproduce a copy for each student.2. Hold up some familiar objects such as a chalkboard eraser labeled 52

grams and a book labeled 525 grams. Have each student use his pencilto show each weight on his metric number line.

3. Have students do the first set of exercises on the activity sheet. Note thatthe abbreviation for grams is used on the drawings.

4. Ask if any of the weights shown could have other names (1000 grams1 kilogram and 2000 grams = 2 kilograms).

5. Have students do the remaining exercises. The second set of exerciseshelps to focus on the two names for special points on the metric line.

Comments: These activities are not meant to replace experiences thatbuild the student's basic referent for units of metric weight. Prior to usingthis activity sheet, the student should have personal experience in weighingfamiliar objects using metric scales.

Los Angeles 3,352 km

London 6336 km

Boston 1560 km'

* You are here.

Moscow 7,968 km

New Orleans 1,486 km

-------.-- Paris 6,624 km

Seattle 3,301 km

Washington,D.C. 1,099 km

Tokyo 10080 km

Where are you ?

Put

an

X o

n th

e lin

e fo

r ea

ch m

etric

wei

ght.

Nam

e

Gra

ms

010

0 20

0 30

0 40

0 50

0 60

0 70

0 80

0 90

0 10

0011

00 1

200

1300

140

0 15

00 1

600

1700

180

0 19

00 2

000

2100

0 Kito

gram

s

300g

.

700

g

1000

g.

1100

g.

1

1500

g

2000

g.

2100

g.

IQ)

50a

1I 2 65

0g.

Nam

e ea

ch o

f the

se p

oint

s.

Gra

ms

AB

CD

010

0 24

0 30

0 40

0 50

0 64

)0 7

00 8

0019

0010

6011

00 1

200

1300

140

0 15

001

1600

170

0 18

00 1

900

2600

210

0X

'

20 K

ilogr

ams

1

Agr

ams

Bgr

ams

Cgr

ams

Dki

logr

ams

Egr

ams

gram

sD

gram

sF

kilo

gram

s

54 A METRIC HANDBOOK FOR TEACHERS

For Teachers

Objective: Experience with the metric system of weight

Grade level: 1, 2, or 3

Directions for teachers:

Remove the student activity sheet and reproduce a copy for each student.If feasible. e ich student should he given a sugar cube as a referent. Underless ideal circumstances, vou may exhibit one or more sugar cubes to he surethat each student has at least a visual image of the object that weighs "2grams.-

Directions for students:

1. Note that each stack is made of sugar cubes and that each sugar cubeweighs 2 grams.

2. Decide how much each stack weighs.

Comments: For sanitary and health reasons, it is impractical to have eachtudent personally build and feel the weight of each stack of sugar cubes.

This experience may easily he converted to a hands-on laboratory experienceby using one-inch cubes that weigh approximately ten grams each. In class-rooms where the students have had previous experiences with three-dimen-sional geometry, a work table with a box of wooden cubes will suffice as anaid for those students who can't visualize the stacks as pictured. Expectstudents to attack these problems in a variety of was Some will count bytwos; others will find the number of cubes and multiply that number by two.Some students may solve G by simply multiplying the answer to 1) by five.

ti.viver.v

. 6 8. 8 C. 16 D. 12 E. 12 E. 20 .1". 60

Name _

Sugar Cube

How much does each stack weigh?

A

C

grams

grams

2 grams

B grams

D grams

E gramsF

grams

grails

56 A METRIC HANDBOOK FOR TEACHERS

rE For Teachers

Objective: Experience with ,,v'ight and length using the metric system

Grade level: 3 Or 4

Directions for teachers:

Reproduce a copy of the activity sheet for each student. You may wish toexhibit or even pass around several new pieces of chalk. Students may note thatmost pieces of chalk are not identical and correctly conclude that the S centi-meters and 10 grams arc approximations. Once the students have the basicreferent in mind they should work independently on this activity.

Comments: The total length as an end-to-end chain of individual pieces isan important concept. It should not be expected that this concept is intuitivefor all students.

A nswers

1. 20 grams, 16 centimeters 2. 5 Ensams, 4 centimeters

3. 15 grams, 12 centimeters 4. 25 grams. 20 centimeters

Name

'1 CHALKLength: 8 centimeters

} Weight: 10 grams

Estimate the total weight and the total length of thepieces of chalk shown for each exercise.

Weight- gramsLength- centimeterF

Weight. gramsLength. centimeters

WeightLength.

gramscentimeters

We grams Length. centimeters

58 A METRIC HANDBOOK FOR TEACHERS

IA

For Teachers

Objective: Experience with equivalent English and metric measures

Levels: 4, 5, 6, 7, or 8

Directions for teachers:

1. Remove the activity sheets that you feel are appropriate for your students.Make copies for each student.

2. Use one or more of these activity sheets in a measurement sequence.Note that the names of some units are abbreviated in the drawings onthese sheets.

3. If students disagree regarding certain answers, they should be encouragedto perform a physical measurement to determine the correct answer.

Comments: These activities are not meant to replace the laboratoryexperiences that build the student's basic referent for measurement and thevarious units involved. Their use should follow laboratory experiences thatinclude actual measuring of length, weight, and volume using instrumentsthat measure in metric units. Contemporary science programs include thenecessary measuring instruments calibrated in metric units. If the instrumentsare not available in your school, contact your science supervisor.

" The fever is gone. Your temperature

is right at 37°"

SN ame

Metric

0 100g 200g 300g 400g 500g 600g 700g 800g 900g 1 kg 1100g

II I f ri ji t I TI I I -T-IT-1-r- I I 1 I I

0 4oz. 8oz. 12oz. I lb. 4oz. 8oz. 12oz. 21bs 4oz. 8oz.

English

Put an X on the measurement line for each measure.1

8 ounces 9 ounces -2- pound

1 pound 2 ounces 1-,-1 pound 2.2 pounds

40 grams 100 grams 500 grams

10 grams 1000 grams 1050 grams

Estimate the weight in grams of each object.

I poundBUTTER

BACONpound

grams

grams

CANDY

10 oz

//

BABY FOOD

grams

grams

Name

500 I 1500 2 2500 3 3500 40 ml liter ml liters ml liters ml liters

1ii

1i ri 1

1

IIi

1f1

1

1

1 1 I 70 1 I I 3 2 3 I

cup pint quart pints quarts quarts gallon

Put an X on the measurement line for each measure.

I cup 3 cups

IT liters 2-if quarts

2.8 liters 495 milliliters

Use the measurement line to estimate the metric measure

for each English measure.

CUP/

2 pints

8 cups

3550 milliliters

Use the measurement line to estimate the English measure

for each metric measure.

MILK

9 litersquarts

quarts

TEACHING THE METRIC SYSTEM: ACTIVITIES 61

For Teachers

Objective: Experience in using conversion tables: English system to metricsystem and metric system to English system

Grade level: 5 or 6

Directions for teachers:

1. For each student, reproduce the activity sheet and a copy of the conver-sion tables printed at the end of this section.

Have all students study the first two conversion tables. Be sure that theycan "read" them and note their relation to each other.

3. Encourage students to work independently.

4. Observe students to be sure they learn to read the tables accurately.

Comments: The tables give all measures to the nearest hundredth. Therounding causes some apparent inconsistencies in the tables; for example,6 centimeters is 3 times 2 centimeters, but the corresponding table entry-2.36 inchesis not precisely 3 times .79 inches. You may wish to discussthis situation with some of your students.

Answers

I. a. I inch b. 8 inches c. 2 inches d. 8 centimeterse. 4 inches f. 8 inches

a. 1 mile b. 2 kilometers c. 6 miles d. 50 kilometerse. 20 kilometers f. 100 kilometers

3. a. 1 kilogram b. 3 kilograms c. 5 kilograms d. 50 pounds

4. a. 25.4 centimeters b. 76.2 centimeters c. 68.58 centimeters

Name

Use the conversion tables.

1. Draw a ring around the longer measure.

a. 1 centimeter or 1 inch b. 8 certimeters or 8 inchesc. 5 centimeters or 2 inches d. 8 centimeters or 3 inchese. 4 inches or 10 centimeters f. 20 centimeters or 8 inches

2. Draw a ring around the larger distance.

a. 1 kilometer or 1 mile b. 2 kilometers or 1 mile

c. 6 miles or 10 kilometers d. 50 kilometers or 30 milese. 12 miles or 20 kilometers f. 60 miles or 100 kilometers

3. Draw a ring around the heavier weight.b.

10 5pounds kilogramsj

5pounds

3kilograms

4. Complete each statement to make it true.

a. A stick that is 10 inches long is centimeterslong.

b. A line segment that is 30 inches long iscentimeters long.

c. A line segment that is 27 inches long iscentimeters long.

TEACHING THE METRIC SYSTEM: ACTIVITIES 63

A

SFor Teachers

Objective: Experience in using conversion tables: English system to metricsystem

Grade levels: 6, 7, or S

Directions for teachers:

Reproduce for each student a copy of the Personal Data Sheet and ofthe conversion tables on the following page.

Place these pages on the activity table along with a tape measure, sev-eral rulers. and a bathroom scale.

3. Encourage students to fill out their Personal Data Sheet as an individualproject. Allow a week for this "extra.-

COMMCWS: Sonic students may be sensitive about making "public'' someof their personal data. You may avoid unpleasantness by allowing studentsto skip any data they consider too personal. Sorry, it is impossible to provideanswers for the Personal Data Sheet.

A

Use your conversion tables to help you complete this sheet.

PERSONAL DATA SHEET

Name: Date.

Age years months

I. Height: feet inches

centimeters

meters

2. Weight: pounds

kilograms

grams

3. Waist: inches centimeters

4. Chest: inches centimeters

5. Span: inches; centimeters

6. Reach. inches; centimeters

7. Pace: inches; centimeters

8. Length of shoe inches; centimeters

CONVERSION TABLES

/ Inches Converter I Centimeters

s 6-

Inches I 2 3 4 5 6 7 8 9Centimeters 2.54 5.08 7.62 10.16 12.70 15.24 17.78 20.3222.86

Centimeters

Centimeters

Inches 39

Converter Inches

2

.79

3 41.18 1.57

5 7 8 91.97 236 2.76 3.15 3.54

I Miles Converter

Miles I 2 3 4 5 6 7 8Kilometers 1.61 3.22 4.83 6.44 8.0`,T.; 9.66 11.27 12.87 14.48

> LKilometers Converter Miles

Kilometers I 2 3 4 5 6 7 8 9Miles .62 1.24 1.86 2.49 3.11 1 3.73 4.35 4.97 5.59

--3-1 Pounds Converter "Kilograms

Pounds 1 2 3 4 5 6 7 8 9Kilograms 45 .91 1.36 1.81 2.27 2.72 3.18 3.63 4.08

TA- [Kilograms

Kilograms I 2 3 4 5 6 7 8

Pounds 220 4.41 6.61 8.82 11.02 13.23 15.4317.64 19.84

Machina the Metric System: Guidelines

Rria hitCd 1.,M1 tha \pril 197 i:( I: a, iit R. Copyright 197 National coon,:ii of leak:hers of Mathematics.

Schools arc g ing metric

1' IZ 1) .1 . I-1 ELGREN

.4 retired rescard? thrntLcr, Ere,/ //cit:ren /Ia.+ !wen On officer in1/0' 1, trir .1,,,t ;WWII, (liar prepwev and ,lictribturs

inctric. educational (ads. lin hay in 1/1(' (1,1fTlif,11

nictru. sy.%tent to rcplocc the .vet,' .snots now Inthe I1115.1

Or.2-et the lentith Of King Fdgar's foot.the length from the nose to the tip of thefinger. the lem,th of three barley corns laidend to end. the amount of land that can heplowed by a yok,. of 0 \en in one day. For-get, if Voll 11;1 not already ,0, thrmother of square feet in an acre, the dif-ference between a dr quart and a liquidquart. the number of pecks in a bushel, and;ill the rest of the system of measure, thatarc learned with difficulty and forgottenwith the greatest of ease.

"lhe legal system of measure in the

United States is actually the metric system.It was adopted by an :lc t of Congress in1866. Children should hose been educatedin this language of measure following thatimportant step in the improvement of oursystems of measure. Chades Sumner. thesenator from Nlassachusetts who sponsoredthe 'Metric Bill of I S(16 that made the met-ric system legal in the United States, statedat that time, -They who hayc alreadypassed a certain period of life may notadopt it. but the rising generation will em-brace it and es cr afterward number itamong the choicest possessions of an ad-vanced cis

l he sciences saw the advanta,:es of themetric stem. adopted it. and have usedit almost exclusively. It was not, however,accepted for general use. and schools ap-proached the use of the metric system in away that gave it little encouragement. the

69

following, are some of the poorly conceivedpractices:

I. Nletric measure was not studied as asystem by itself.

2. People were not taught to THINKMI: I Ric.

3. Textbooks often contained only a

single unit on the system, and problemswere merely conversions from one systemto the other.

.4. The unit on the metric system wasfrequently at the end of the textbook. Asa result, it was seldom taught. Teachershad little knowldege of the system, and itwas omitted because of lack of time.

Now a new day has dawned. Followingthe three-\ ear metric study by the Na-tional Bureau of Standards and the Secre-tary of Commerce, Maurice It Stans. whowas Secretary of Commerce in 1971 whenthe study was completed. made the follow-ing reconunendations to the Congress ofthe United States:

That the United States change to the

International Metric System deliberatelyand carefully:

That this be done through a coordinatednational program;

That the Congress establish a target dateof ten years ahead;

That there he a firm government com-mitment to this goal;

7() A NH' TI:1C HANDBOOK FOR TEACHERS

eu I Ihit nriorik he gken to educatingeu.eit \nit:nu:an sc-hoolchild and the pul-u-lh.- laret: to :111111% in metric terms.

I Iii' need for our school, to ytor ulh: Congru's, H ;RA. I here is gooilituas.'Ih that the\ :let,Larett LI de uen turues he inning indut I he metritu euno.ersion hill.

ra,,tutl iii .1tieu,.. h\-

,. n sone: :tote in the tienate. l heaii :\pectcd vote for the

inct:. anti Presidt.it sufurinu, the hid.

\V hat. s.uult! h: (hunt- lil the held of tudn-C,;1,111, t,i i .11 110

,\ ,tent hV INCH NOthat tc.ichers and ruinds learn to think inuhis Lune:met: tut' measuirt:. I)t, not Irt uo

learn or ',each nuetHe sftum ihron,.:11

prolden),, and do not trt tolearn ttonu.:1-,ion. I.earn the metricsu.stcm ttselt. t iii'CK Nil I

;Ind

art: tuau.cui.

Hilo re teyillooks are chainueLl, getmastic uyoul..)-uuuok: for each leacher rand

each [-mild. Then the s% stein can he learnedtY.th individual effort.

ti:C/W7Cr 1.if ihc Licult,,- tai

is: authority- for the school. Ilecan the 1i-dorm:Hon and makurialsuutuce,,.un. to cliahle the school to (a) NI 1. 1.-

5. Falco:al-ie.,: teacher, to become mem-uulF ;in uurLlani/uution that wilt send them

literature that t: \plains 111i2

pri imFormation on Mr.:CS of

.uitts. ;Ind publishes a newsletter thatkr:LT :der( to metric proeress and

de\ elorments in the teachine (if Hints ofnie..sture and their use.

I each tlic metric su,stem to all pro,-uc-aFt-Ilcus. for the chanee uo the new

u,staus ol measure i, not just a mathe-u

The working units of the metric systemare casv learn. The unit of kll./11 IS themeter (or metre) ; the unit for mass is the

',Ind the Unit for volume, the liter (orlitre ). To I (INK Nil TRW, it IS well to learnthe dunce hasic units in combination withthe prclkes remi, and 1,71u.

For u-ractical purpose,, the whole sutstemcan he suunahiriietl ;is follow-s:

Inoo millimeters !mum - I ineuer urnutau min I centimeter ftilli

m I kilometer (krillII All) mullituram, trim) -- i lur am (I.!)Huai I kiltlo.lain11'0) k]..

1 metric ton iiinuIllulaer, unit == I hoc) ill

on tunule enter leers len1=- 1 eulNt. dec umuter '.1.1n1

I' milliliter and the cubic centimeter havethe same \ohmic. 1 he term kitiallicr t, flitrc,:ommentled----it is equal to a cubic nucter

m I. u,YInch is more easily wit:I-stood andused.

Machinists measure in millimeter,:,t;:idesmen measure in centimeters andmeters: clothing sizes arc giken in centi-meters. And greater lengths are in metersand kilometers. Nlasu, is measured inff:1111,, grams, and kilograms hy the chemist;erams antl kilograms by the shopper; andkilograms and metric tons when largequantities dr;.. invol cd. t \1a,:, is thetit:amity cut matter, whereas weight is a

force. the earth's attraction for a givennnis. Generally. the term masy- is meantwhen tyc use weiL'ilt.) pharma-cists. ehemiuhs, and bacteriologists usethe term, milliliter. centimeter, and!ih:r. Consumers will make purchases ofgasoline and other liquids in largequ;Int;tic,, will he sold hr the cubic meter,

hrough c' crythav Use.; the metric systemCan he learned in a short time.

TEACHING "FHE N1ETRK SYSTI NI: GUIDELINES 71

In conclusion, I repeat the recommenda-tion made by the former Secretary of Com-merce, \Ir. .\laurice II. Scans, that earlypriority he given to educating every :\meri-can Nt.: hool,:hitd and the public at large tothink in metric terms.

Fur the price of 53, the folIowLig ma_tcrials can be obtained from the I\-letricAsociation, 2004 A41 Street, Waukegan,IL 60085, a nonprofit organization in-

terested in the dissemination of metric edu-cational materials and information:One 20 cm plastic rulerTwo 1.5 in plastic, flexible measuring tapesTwo copies of Metric Units of Aled.snrc, a book-

let

One copy of Afetric Stepp/eine/it to Science and:Slinlienzutics, a workbook for use by theteacher and the pupil

One Go sti:rauc bumper stickerOne price list of metric educational aidsA copy of the last newsletterAn annual membership in the ,\letri. ..1ssocia-

hon that includes a subscription to thequarterly newsletter

ct limn th, \ I 1,7 j \11i111,11 III I Hi 14 (.1,P,1 /VW( 1474, N.iiinn.ti 'min,

Metric: not r,f, but howNCTM METRIC IMPLEMENTATION COMMITTEE

The metric system will soon become themajor way by which we measure

the height of a person,the mass (weight) of a hamhurger,the area of a carpet,the room temperature at which wecomfortable.

A metrication bill is likely to he passed inthe current session of the United StatesCongress the Senate passed a metricationhill in a previous session. Business andindustry have already taken strides towardcomplete metrication in an effort to enhancetheir position in the world market. -lheautomobile industry, for ex,,mple, is pro-ducing engines and even complete auto-mobiles to metric specifications.

The metric system will he the majorsystem used by students now in schoolthroughout most of their adult life. Soschools are now beginning to teach metri-cation to all pupils. As mathematicseducators, we have a responsibility forproviding leadership and direction in metri-cation so that these young people will hecompetent in day-to-day life with measure-ment in the metric system. We need tothink carefully about the implications foreach level of schooling:

What can be learned and understoodthoroughly at the various grade levels?

What approach should be taken with stu-dents who have some knowledge of boththe American' and metric systems?

What are the needs of' students and of ourcolleagues in courses outside nv he-!nudes in home economics, science,industrial arts, or geography?

are

I. The term Amerwan. rather than Br'irrsh, io toedbecame the Britoh have adopted the metric sy,termthe system toed in the United State, k no longerthe firitoh Sy,tern.

7-)

What responsibility does the mathematicseducation community have for the educa-tion of parents, and for aiding the mediaor any others who present metrication tothe public?

Fhe slogan of the National Council ..)f

leachers of Mathematics is THINKMETRIC, which emphasizes thinking in themetric system. What does it mean to thinkand to function effectively in any measure-ment system?

General guidelines for teachingmeasurement

It is imperative that we use metricationas a means to putting new vigor in theteaching of measurement. Too often thestudy of measurement has consisted only ofwritten exercises on worksheets or pagesfrom textbooks, w;th major emphasis onconversions between units and on operationswith so-called "denominate numbers." Thelack of attention to activities that encouragethinking and estimating in a measurementsystem has meant that measurement hasbeen viewed as dry and dull by pupils andthat our instruction is not adequate for thestudents' real needs. What, then, shouldguide us in planning a sequence of activitiesthat will put new life in the study of measure-meat and assure that major ideas aretaught and learned?

1. Choose art appropriate unit cif measureand tree it to measure (.4 variety (V' objects.The most important component of thinkingin any measurement system is -a thoroughunderstanding of the unit used most often.In the American system, the initial majorunit probably is the foot; in the metricsystem, it is likely the meter or centimeter.Repeated experiences with the basic unitshould lead to estimation of lengths with nomore than about ten percent error.

TFACIIINd THE NIETRIC' SYSTF11: GUIDELINFS 73

To highlight the nature of the measuringprocess, and to emphasize the arbitrarynature of standard units, it is advisable tobegin with a nonstandard unit. For length,straws, pencils, or paste sticks are readilyavailable and easily used. From such exper-ience a standard unit, such as a meter, canhe made from string or cardboard and usedby pupils with a double goal learningabout measuring length and learning themeter as a unit.

The unit chosen should he appropriate tothe size of the object measured. You wouldnot use a kilometer to measure the lengthof a room, nor would you use a millimeter:you would likely use a meter.

2. Use multiples of the basic unit, as theneed arises for a larger unit, and subdivisionsof the basic unit, whcre smaller units art'needed. By referring larger and smallerunits hack to the basic unit, which is wellknown, estimation of larger and smallerunits is easier. Furthermore the constructionof a measuring system becomes moreevident.

Limitation of the number of larger andsmaller units taught makes the learninggoals more realistic and more manageable.The goal of thorough learning of a fewlarger and smaller units is much preferredto mere acquaintance with many differentunits.

3. Limit expectations of mastery of co/I-D:TWO/1S Siit'll/I 1.1 Measuring .9SteM toCOMO10111 y used units adjacent in .size.

In the American system, feet are usuallyrelated to miles, and inches are related tofeet: but not inches to miles. In metric,centimeters are usually related to meters,millimeters to centimeters: but very little isdone in relating millimeters to meters.

4. Use the approximate nature of themeasuring process in the physical world as atheme. Helping children to give measure-ments using language such as "more than2 meters but less than 3 meters" and "alittle more than a kilogram" providesexperience with approximation. It also

points to the needs for smaller units andfractional numbers.

5. Use measurement as motivation forfractions, for decimals, and for arithmetic.There is no source of applications ofarithmetic quite comparable to those thatarise from measurement. An active measure-ment program provides a needed stimulusfor learning many topics in arithmetic thatare considered important to teach.

6. Use the actual units as often as possible:avoid the scaled-down versions often found intextbooks or works/''as, It is folly to tryto teach the meter by drawing a scale versionof it. Students have great trouble in respond-ing to measurement questions and estimatesusing such "distorted" representations. Thetime for sealed drawings is much later,after the initial units are well learned andwell understood.

No matter whether it is the American orthe metric system, a sense of active involve-me,: is essential. New life will emerge frommeasurement it' simple relationships areemphasized, more estimation is done, andthe process is directly taught especially atan early age. The flavor of "hands-on"experience is much more important thananything else. Creative and interestingactivities can he developed and can he usedin the full range of management schemesfrom whole-class to independent work.

Specific guides for teaching themetric system

With an active point or view aboutmeasurement in general, what guides shouldhe used specifically for teaching metric'?

/a the elementary school, teach metricand American systems as dual systems.This means that the major emphasis is onteaching the metric system in itself, withrelatively little attention to the relation ofunits between the two systems: we want toencourage thinking within a system.

The need for American units in everydayaffairs is likely to persist for some time afterthe adoption of metric units, Even though

74 A METRIC HANDBooK Fox

Betty Crocker is in the process of changingmeasures in recipes to metric, many oldbut good cookbooks will still be around!Wile:, we do become fully metric- four toeight years from now we shall he able tophase out the teaching of the Americansystem.

2. Begin with the metric .system. Themajor adult usage by our present-day pupilswill he metric. By teaching it first, we showits importance and we provide the neededframework for its eventual emergence asthe only system. Further, we avoid thetendency to overstress conversion.

3. for children in the middle and uppergrades, begin measurement with the meter asthe basic unit. For children at the primarylecel, who have trouble handling so large aunit, begin with the centimeter, or with a unitof 10 centimeters. The 10-centimeter unitserves the needs of young children and isneither too large nor too small for themto handle. The name /0 centimeters canhelp in counting by tens.

For mass (weight) the basic unit is thekilogram: and for volume, the liter. Theseunits should be learned initially. beforestudying multiples and subdivisions of them.

4. Teach only the common/y used muttiptesand subdivisions, and their correspondingprefixes. Such an approach minimizes thenumber of prefixes and units a pupil is

required to learn. The units shown in tableare sufficient through grade 6. Note thatthe prefixes deka and hecto are not includedbecause of their rare use. Dec/ is used moreoften than deka or 'recto, but it is less usedthan centi or milli: hence it is omitted.

A few other prefixes are needed in moreadvanced science courses: for example.milligram for weight in medicine. Such

Table 1

Length 1%,,eight) ( apacity

kilometer (km)meter 1m)centimeter (cm)millimeter (11)111)

kilogram (kg)gram (g) liter (

nulhhtcr (ml)

TEACHERS

specialized uses should be left to the sciencecourses and taught there.

Language usage should he natural andunforced. Teachers should try to use thelanguage and symbols correctly, however.Pupils will not use periods behind symbolsfor the units if the teacher uses no periods.

5. For pupils who already know bothmetric and America)? .systems, and forparents, approximate concersions can bemade. Activities involving approximate con-versions might he deliberately planned forpupils in the upper grades. Students shouldhe taught approximate equivalents thatallow them to estimate in either system.given a measure in the other.

The following list is suggested:

I kilogram is a little more than 2 poundsI meter is a little more than I vlard1 kilometer is a little more than 1 2 mile1 liter is a little more than / quart

There is, however, almost no need for anypaper-pencil conversions between metricand American. it is important that theemphasis be on understanding the twosystems, and that the arithmetic of con-versions he minimal.

Grade placement melnic topics.

The question arises directly, what shouldhe the placement of metric topics? Onesuggestion on placement is the following:

Meter and centimeter can he introduced atthe primary level and reinforced at allsubsequent levels..1-he initial work shouldbe done to emphasize estimation withthe basic unit.

Tenth of a centimeter the millimeter conhe the topic for middle and upper grades.Decimal notation will he helpful.

Weiv,ht, which is more difficult to estimateand so requires more maturity on thepart of students, may he introduced inupper primary, first using the kilogram.The gram can appear in the middlegrades.

Capacity or volume, using the liter as thefirst unit, can he introduced in the upper

TEACHING THE METRIC SYSTEM: GUIDELINES 75

primary, with the milliliteralso called acubic centimeterappearing in the middlegrades.

Celsius temperature can be introduced atany level.

Much of the current material on today'smarket does not deal with the reallysignificant concepts of the metric system andmeasurement. Rather, it tends to he a seriesof paper-pencil activities using the gamutof metric notions, without regard for theprinciples of estimation and measurement,or for the components of the metric systemthat our students will lind necessary. Forexample, present-day metric materials oftenask students to perform complicated anduseless conversions within the metric systemitself changing mm to m, km to din, ormm to dm. Just as useless are exercisesrequiring addition of unlike units such asdm and m. Rarely are units mixed in ametric problem. Instead of using 3 dm 5 cm,the measure would he given as 35 cm.Instead of 3 m 22 cm, we would say 322 cmor 3.22 m. Poor instructional materialsmake the metric system seem complicatedfor children when it is relatively easy. Suchmaterials should he screened and notpurchased.

Metrication in the community

Schools should be expected to helpparents understand the metric system.Parents will want informatioq on what theirchildren are being taught so they can providereinforcement at home. School people ingeneral, and mathematics educators in

particular, should take the initiative inpresenting metrication to the community.Let us not make the mistake of waiting forparents to demand it. It is our responsibility.Creative teachers can prepare single-concepthandouts to send home periodically tothe parents. Perhaps a metric newslettercan be developed with teacher and studentinput. Libraries could prepare displays ofmetric materials. For some school districtsthere is the possibility of developing parentworkshops as an activity of the parent-

teacher organization, Metric education is

our responsibility and we have the equaltasks of working with our students andwith our community.

Conclusions

Our goal of metrication in the schoolprogram is realistic. The amount of newknowledge required is actually quite small.Success for the mathematics educationcommunity in making the change willdepend on how well we follow guidelinessuch as those given here. It will depend onthe degree to which we provide appropriateexperiences for maximum development ofstudent understanding.

In summary, these are the guidelines:Teach students to THINK METRIC.Concentrate on those units necessary from

the utilitarian standpoint.Develop meaning and feeling for units

through experiences centering aroundestimating. and checking of the esti-mates.

Minimize conversions. Do not immersestudents in the morass of computingconversions between the metric andAmerican systems nor even within themetric system itself.

Use metric units at every opportunity. Thisincludes use in other subject-matterfields. Elementary teachers use themthroughout the day. Secondary teachersteach metric exclusively and provideinservice materials for your colleagues inother subject areas.

NI FUR ICAT ION IS UPON US. IT ISNOT IF, BUT I- IOW,

.VCTA/ Metric ImPlementationCOMMittee

130vd Henry. C hairmanStuart A. ChoateDonald Firl

rlt,111 the Nta, I,/71 At< moat ur Ft wk. Cop,right i977, National ( leacher, ,,INIathelnaacs.

Teaching the metric systemas part of compulsoryconversion in the United States

VINCENT J. HAWKINS

nhithernatics lea, her at Toll (;(ite 11101 . hot'/ inlVurtrit A. Rhode Wand, l'in, ent I.N. al.so .scrutnt; (11 re.Seil 1 ( le

coordinat,It It the pall (;ate .tltric I'rojec 1.

Ihere is no doubt that the metric s,stemis rapidly becoming the standard systemof measures for the entire world. Overninety percent of the countries; have al-ready adopted the system. Eer. England,from whom we inherited our customarysystem of weights and measures, is morethan halfway through its own ten-year pro-gram of conversion.

The main fact emerging from the recentstudy of the National I3ur, .1u of Standard:is that conversion is inevitablethe onlyquestions are when and ;?,w. The recom-mended when is the creel of ten year,thus providing time to plan for and mai, ethe change. The holt. can be answeredtwofold:

I. Constant exposure to the metric sys-tem through advertisement, conversation,relabeling of all commercial goods, alongwith a continual phaseout of the customarysystem.

2. Proper educational programs of in-struction on four different levels from pre-school through high school.

Since the British are still in the processof conversion, a complete report on theeducational aspects of British metricationis not available. We do know that from thebeginning the British counted quite heavilyon the education system to make metrica-

76

lion n smooth, gradual process. Englishstudents who are now entering the ele-

mentary L,rades are learning to think in

metric terms as naturally as their parentsthought in terms of inches and pounds.Students in the higher grades are success-fully breaking the habit of thinking in theold terms. The book publishers and edu-cational equipment suppliers are well aheadin production to conform to the metric sys-tem. Teachers are being trained in specialcourses to teach the system effectively.

The proper education of students, usinga master plan that involves all grades, is

the most important factor for a successfulcons ersion. An educational survey, con-ducted as part of the U.S. Metric Study,showed that changing textbooks and equip-ment would cost about S I billion. If thesechanges were made for no other purposethan for conversion, then the expense couldbe tabbed wholly to metrication. The factis, however, that most textbooks arc re-placed after a few years of use. The hulkof the expense of conversion can be ab-sorbed with regular revisions that providea planned exposure to the metric system.

Most educators agree that learning themetric system is quite easy. The simplicityof metric tables leads to the assertion thatthe metric system can be learned in just onehour, and continual work at applications

TEACHING THE METRIC sys-rEm: GlimELINEs

would make the system completely secondnature at the end of one year. Young peo-ple are more receptive to the system thantheir elders are. In fact, partly due to thepsychological effect of a totally new systembased on ten, the slow learner learns themetric system more readily than he learnsthe customary system. In the English sys-tem, there are more terms to learn: inch,toot. yard, mile, acre, ounce, pound, ton,pint, quart, gallon, peck, bushel, dor:ett, andgazes. in the metric system, a child needs tolearn only eight terms: meter, liter, and,i;ratn; the prefixes milli-, centi-, and kilo-,and hectare and metric ton.

Once metrication has been given thegreen light, the metric system can becomean integral part of the mathematics cur-riculum for each of the grades. The ap-proach to the metric system will need to beadapted to the various grade levels.

Preschool

Up to age five, children should he ex-posed to the metric system as a standardsystem of measurement, even though theirparents will still he using the English sys-tem for everyday, practical purposes. Thefamily should help children realize that onesystem of measures is on the way in, andthe other is on the way out, Childrenshould he encouraged to use metric units.Educational programs would help the pre-schooler realize that what he is learning isextremely useful in everyday life. Educa-tional toys and planned television programscould be beneficial.

Kindergarten

At this age, children are exposed to theidea of sizelarge versus small, tall versusshort, and heavy versus light. Studentsshould learn to associate the basic unitswith familiar objects: the meter as some-thing that is slightly taller, or shorter, thanthey are: a decimeter as long as, say, asharpened pencil; and a liter of water asabout the same amount as a quart ofwater, and weighing about the same as alarge book like a dictionary.

77

First grade

Addition and subtraction can he taughtusing the meter stick as a number line. Theintroduction of the fraction 1/z should heaccompanied by the introduction of themillimeter as a part of the centimeter.Students should be taught that is equiv-alent to and 5", . They should ex-amine two different containers of I:II-litercapacity, but of ditlerent shape, and findthat both can be used to till a I-liter con-tainer.

Second grade

Students at this level learn about mak-ing change, inequalities, and large numbers.The teacher can show the relationship be-tween terms--how cent is related to centi;how ma/ is related to milli. The conceptsof more than and less than can he illus-trated: thirteen centimeters is more thanone decimeter. One dime is less than $0.13.Students should learn that there are 100centimeters in I meter, 100 millimeters in

1 decimeter, 100 decimeters in I deka-meter, and so on. Groupings of 100 toform 1000, and groupings of 10 to form1000 can be studied with the help of themetric tools.

Normally at this stage, fractions such as' 2, and "I are studied. In the

study of metric measurement, these arepassed over. Students will learn the tenthsinstead--1, , " i , and so on.

Third grade

Sums up to 100 can be associated withthe meter stick. For example: How manydecimeters and how many centimetersshould he added to reach the 95 on themeter stick? The introduction of the Celsiusthermometer, with 0 degrees freezing and100 degrees boiling, provides further oppor-tunities for additions up to 100. Conceptsof larger numbers can be introduced withthe number of millimeters in a meter. Thedegree of error can he introduced with theidea of the nearest half centimeter. Thisties in with inequalities on the meter stick.

78 N1E.FRIC HANDBOOK FOR

With the introduction of area and volume,students learn the terms square and cube,and they should be taught estimation. Forexample: How many squares one meterlong on each side do you think will coverthis floor? Or, How many liters of waterdo you think it will take to fill this tank?Foreign road maps can be used for theintroduction of the kilometer and thehectare. By this time in their academiccareer, students should be well aware ofthe importance that ten has in everydaylife, as well as in all mathematics.

Fourth grade

Averages of metric measures can hefound. It is important that equivalence belearned thoroughly. Students should knowthat 97 on the meter stick means "7,meter, or 9 , decimeters, or 97 centi-meters, or 970 millimeters, and what mustbe added to each of these to ohtain theequivalent of one meter. The process offinding the least common denominator forthe purposes of measurement is practicallytrivial. Weights are taught at this level, andthe gram should he learned in associationwith such things as a small piece of chalk.The kilogram can he illustrated with a

heavy hook and, possibly, in associationwith capacitythe relation of a kilogramto a liter of water. Decimals can he taughtentirely by the metric system. Commonfractions can also be related to the metricsystem. The example 9 Ill 4- 7 dm 33 cmcan also be expressed as 71 ""i.This example, of course, can be changedentirely to meters, decimeters, or centi-meters. There is no confusion with the leastcommon denominator as there would bewith the customary system. The relation-ship between the shifting of the decimalpoint and the corresponding fraction shouldhe evident.

Fifth grade

For further practice at this level, studentswill he exposed to more difficult material.The terminology of the metric systemshould he implemented as much as pos-

TEACHERS

sible. Problems in multiplication and di-vision in measurement can be studied atthis time. A typical example in multi-plication might he

(6.5)(1 in 7 dm 2 em) (6.5)(1.72 m)

Of course, finding the decimal equivalentof a measure must he learned. Manipula-tion of measures becomes easier for stu-dents because there is less mechanicalwork.

Sixth grade

Time, rate, and distance problems withmetric units are not usually taught until aphysics course is taken. However, with thenational changeover to metric, this is

brought down to the lower grades, whichgives more time for other things in physics.'Hie use of the metric system can facilitatethe more complicated inequalities with frac-tions, especially in area, volume, and per-imeter. The liter should be studied morecarefully, with applications in both capacityand weighta tub that is 1 meter by I

meter by I meter will take 1000 liters to fillit, and each liter of water weighs I kilogram.An experiment with several students using aseesaw is a way to show comparisons. Theintroduction of percents, with further workin the degree of error, can he taught usingcentinieters.

The metric study in the sixth gradeshould also include a film documentary onthe history of the system and its worldwideuse, with examples in commerce and in-dustry.

Junior high school

Students should know what a squarecentimeter looks like and that 100 squarecentimeters covers an area equal to I squaredecimeter (100 cm' =- I dm"), and so on.They should he somewhat familiar with thenotation cm'. Fractional parts of squareunits should be taught. For example. 1/2ern' may he a rectangle that is I centimeterin length and 5 millimeters in width, with

TFACIIING THE METRIC SYSTENI: C;VIDELINES 79

an area equal to SO square millimeters.After such simple ideas are learned, stu-dents should have no difficulty in findingthe area of any simple geometrical figureusing fractions or decimals.

In the first year of junior high school,the student should have plenty of practicein applying his knowledge of linear meas-ure to the formulas. Volume and its rela-tionship to weight are studied in the sec-ond year. Students should he shown modelsof cubes, cylinders, and the like for help invisual perception. The use of metric unitsin the formulas should nu, he a problem:but, again, notation needs to he explained.The notation for cubic centimeters is cm".A volume of 1...2 en' is equivalent to arectangular solid 1 centimeter by I centi-meter by 5 millimeters. As in area, workwith fractions and decimals in volume isimportant, and not difficult, as long as thestudents know all the components of linearmeasure.

In working with volume and capacity,the basic thing to know is that 1 liter con-tains 1000 cubic centimeters (I / == 1000cm.l. All expansions of capacity intometers or cubic meters, or reductions ofcapacity into millimeters or cubic milli-meters. can he done easily. More workwith inequalities can be provided. For ex-ample:

Complete 1r 0 /t <, or = :.001 liter . 1 cubic centimeter

The laboratory approach would be an oh-vious aid in this type of instruction.

The third year of the junior high shouldbe given to more difficult applications ofall previous work with the metric system.The only new topic that needs to he cov-ered is scientific notation.

Senior high school

The metric system is taught in the tradi-tional way in high school science classes.Work with the metric system can he con-tinued in courses like geometry and tri-gonometry, with applications to problemsinvolving measurement.

There are two important points to makeclear:

I. The metric system should he taught asa primary language.

2. Conversion manipulation should not beused at all.

The model program outlined here is

designed for the graduated exposure ofstudents to the metric system. The exclu-sive use of the metric system will reducethe need for common fractions and, thus,the time given to teaching fractions. Esti-mates vary, but mathematics teachers saythat in the elementary schools fifteen totwenty-five percent of class time is spentteaching the details of adding, subtracting,multiplying, and dividing common frac-tions. They believe that much of this is un-necessary. As Lee Edson pointed out inAmerican Education, if the metric system,with its simpler decimal relationships, weretaught, teachers could quickly and easilygive their pupils the basic principles offractions and then continue to other aspectsof mathematics.

There is no doubt that the schools arethe key to complete metrication. Fewadults who have lived with the inch-poundsystem all their lives will ever completelylearn to THINK METRIC, but children whoare introduced to the system when theystart school, and even before, will always-MINK METRIC.

During the period of metrication, notall school children will he exposed to themetric system in the same manner. Thereare four categories of exposure:

I. Bane, for the preschooler and gradesone and two.

2. Lower intermediate, for grades three,four, and five.

3. Upper intermediate, for grades six,seven, eight, and nine.

4. Advanced. for applications in the tarsciences, such as chemistry and

physics.

Table 1 is a chart of how the categoriesmight be distributed for the successive

80 A METRIC' HANDBOOK FOR TEACH FRS

Table 1

Programmed conversion chart

1 3

Kindergarten 13 13 B

First grade 13 13 13

Second grade 13 13 B

Third grade 50' ; B50' ;L

25'; B75'.;11.

1.

Fouth crack 50''; 13 3(1' ; B 25' ; B50'", 1. 70' ; 1. 75'; L

Filth grade 50' ;13 40' ;13 13

50' ; L 60'; 1., 75' ; L

Sixth grade and plus .10' .; 13 25' B 75`.; L60'.; L 75' ;1. 25' ;U

B = BackL Lamer intermediateL: L:pper intermediate

years of conversion. For example, if a

student entered the seventh grade duringthe initial year of conversion, what wouldhe study'? For the first two years he wouldhe instructed in the basic and lower-inter-mediate phases. When he had the appro-priate competence, the upper- intermediatephase would follow. Since all of this is pre-sumably new to the student, it is importantthat no mention of the English system.for comparison or conversion, be made.With much applied use, the student willsee the simple relationships for himselfand make the necessary mental conver-sions.

To help students in their learning, themetric system should always he in evi-dencedisplays in the classroom can in-clude such things as foreign road maps;16-mm and 35-mm strips of film; culturedpearls; military shell casings; wall charts.like those distributed by the Department ofCommerce; snow skis; and of course, acomplete set of the standards of the unitsof measurement of the metric system.

There is no doubt that the change to themetric system is an enormous undertaking

Year of metrication

4 5 6 7

B B 13 13

B B 13 13

B B B 13

50';L50' ; U

25'; 1.75';U

I. 1

U U

8 9 10

B B B

13 13 B

B B

1. 1. I,

1.

U U Ll

But neither is there any doubt about thebenefits that will result. The physical as-pects of convertingchanges in industryand the likecan be described in dollarsand cents, but mental conversion cannothe measured. There will always be theunconvinced people who feel that thepresent system is perfect and that perfec-tion should not be tampered with. It is

generally true that nearly all of us proceedon the assumption that whatever is, is

right. This is probably the major reasonwhy America has been afraid of metrica-tion. Many people, in the United Statesand around the world, want this problemovercome, and the way to do it is througheducation.

References

Edson, Lee. "Metrication," American Education(April 1972) : 10-14.

Halsey, Frederick. The Metric Fallacy. Wash-ington, D.C. : American Institute of Weightsand Measures, 1920.

Manchester, Harland. "Here Comes the Meter,-Reader's Digest 100 (April 1972): 19.

Metric curriculum: scope,sequence, and guidelines

M A IZ I I, YN N. SU D A N1

The metric system is merely a system ofmeasurement that replaces another systemof measurement. However, metricationprovides the opportunity to: ( I) reevalu-ate the measurement component of themathematics curriculum: (2) reconsiderwhat we know from experience and fromresearch about how children learn mea-surement concepts; and (3 ) restructurethe curriculum and modify instructionalpractices. The scope and sequence sugges-tions and the guidelines that follow arenot exhaustive; they are intended to callattention to sonic of the imnortant factorsto keep in mind when planning a program.

Scope and sequence suggestions

Primary lets!

I. Develop the. basic prerequisite skillsand understandings ahout measurementby having the studenta. Match, sort, and compare objects

long:short, heavy/light, large/small, andso 00.

b. direct comparisons of two ob-jects by placing them next to each otherto determine which is longer or shorter,heavier or lighter, larger or smaller, andso On.

c. Compare three objects, developingthe idea of transitivity ( that is, if A is

Adapted from materials amt copy-righted by the Agency for Instruction:it Tele-vkion, 1974. Used by permission.

81

shorter than 13 and B is shorter than C, thenA is shorter than C).

Place several objects in order, fromlongest to shortest, heaviest to lightest, andso on.

e. Make direct comparisons by using athird, huger unit to describe the compari-sons. For instance, give the child threesticks, Have him tell the length of the firstand second sticks in terms of the length ofthe third stick.

f. Combine lengths, masses, and vol-umes, using physical objects. For instance,put the water from four glasses into onecontainer, or put two desks together.

g. Transform objects for comparison,applying the idea of conservation (that nolength, mass, or volume is lost in theprocess). For instance, pour sand that isin two differently shaped containers intotwo similar containers.

h. Compare by iterationplacing ob-jects end to end a number of times, pour-ing over and over, and so on. This relatesmeasurement to a process of counting.

1. Use metric measure!: in "play" activi-ties. Metric measures and terms shouldhe used in everyday experiences, althoughthe metric system itself is not under dis-cussion. Metric terms should be used, forinstance, when recording temperatures ona daily calendar or marking heights ofpupils on a wall hanging.

2. Extend the concept of measurement byusing nonstandard (arbitrary) units.a. Use varying units. For instance, give

each student a different length of string;

82 A ME IRIC HANDBOOK FOR

have the student measure an object andreport his measures in terms of that length.Develop the reason for using a commonmeasurecommunication with others.

b. Have the class choose ;111 appropriate(common) unit and use it to make indirectcomparisons. Measure a variety of objects:wing J ohn's foot as the unit, have eachchild make :1 copy (model ) of the lengthof John's foot and use it to measure theroom desks, and so on. 17.ven when acommon unit is used, their measures ill

probably not all agree; discuss the approxmateness of their measurements.

c. Measure between limits, reportingthe measurements as "between 2 and 3units," for example.

d. As the need arises for a larger unit,use multiples of the basic unit; subdividethe basic unit when smaller units will facili-tate more accurate measurement. (Sincewe have a decimal system of numerationbased on powers of 10, it follows that forease of calculation the subdivision of theunits should also be based on 10. Measure-ment with the metric system can thus beintegrated with other topics in the curric-ulum.)

. Develop the idea that a coinstandard unit is needed for communicationwith others outside the one classroom.

3. Having established the background fordeveloping a decimal standard systemof measurement. gradually introduce thevarious standard units of the metricsystem and the instruments used tomeasure in these units. (This listing isgeneral: many specifies must be add,(1.)

a. The meter and centimeter shouldprobably be introduced first. The childshould have practice in measuring to thenearest centimeter with the ruler and meterstick. He needs to he taught how to holdthe ruler to make careful measurementsand how to draw lines that reflect carefulmeasurement. ( lhe child needs similar in-struction on how to use other measurin,2instruments.)

b. After some practice in measuring.

TEACHERS

the pupil should learn to L,imate metriclengths. Resin with gross comparisons ("Isa meter about the length of the schoolbuilding or about the length of the book-case'r ), then develop finer ones (''Abouthow many meters long is the room?-).

c. Discuss the need to use appropriatemeasurement unitscentimeters to mea-sure the width of a book, meters to measurethe length of a room, kilometers tc, mea-sure the distance between two cities.

d Begin to develop the relationship ofthe metric system to the numeration system.For instance, explore counting on a metricruler and on the meter stick; note theI -; 0- 1 00 correspondence.

a. Provide activities in weighing with abalance, first using the kilogram, since it iseasier to handle than the gram weight.(Weight is a more difficult concept ac-cording to research and is more difficultto estimate. )

f. Introduce the liter as a unit for mea-suring volume.

Develop time concepts related to thehour Lind minute.

It. Use the Celsius thermometer in

(:\ eryday situations, having the child readand record temperature.

Intermediate level

I. Develop the relationships between theprefixes. stressing the relationship to thedecimal numeration system. Use decimalnotation. (The most ofH.ni used prefixesare milli, eenti. dcci. and kilo. )

2. Introduce the symbols for the metricunits as the unit is introduced: m. din, cm,mm. km. g. kg. I. nil.

3. Teach the relationships among mea-sures. For instance, for length, developsuch relationships as

mni100 mm

1000 mmIll cm

1011 c m

m

cutdmni

dinni

km

(emphasis shouldbe placed onthese four)

TEACAING THE METRIC SN'STENI: GUIDELINES 83

Later teach such relationships as

num = 0.1 cm == 0.001 me (Lni m

0.001 kin

4. Measure to the nearest millimeter; tothe nearest milliliter; to the nearest gram.In the elementary school, the distinction

between mass and weight can he noted, butthe term "weight'' will probably he used.Mass is sometimes thought of as theamount of material is an object. Weightis the measure of gravitational force on amass and varies with the location of themass (object). the weight of an astronauton the moon is less than his weight on theearth because the force of gravity is lesson the moon; he may he weightless in aspace station. His mass, however, is thesame in all three places. We have been soaccustomed to using the term "weight- in-correctly that it will still probably be usedin cases where the correct term is "mass.-

5. Develop understanding of rectangu-lar solids---1 liter , 1 cubic decimeter(dm') --, 1000 cubic centimeters (crui.').

6. Concert from one metric measure toanother. stressing the relationship to thenumeration system (10-100-1000). De-velop the ability to convert mentally.

7. Introduce addition and subtractionwith common measures. Compare with re-grouping in addition and subtraction al-gorithms. Later use multiplication anddivision with measures.

S. Develop angle measures ( which arethe same in both metric and customarysystems) .

9, Work with metric units in problemson perimeter, area, circumference ;Ind areaof circles, volume, time, temperature, iindSO on.

1 U. Develop understanding of the rela-ti(mships among units for length. volume,and mass.

11. Develop the id..a of accuracy andprecision of measurements and of signili-cant digits.

12. Extend time concepts and tempera-ture ideas.

13. Discuss the history of measurement,presenting selected aspects to indicate howvaried systems of measurement developed(including but not limited to the customarysystem of measurement),

Seer Mchlry (and ali10 1Crel

I. Units for such quantities as force,pressure, work, power, and electricityshould he presented in science and voca-tional education courses as the need arises,

2. Generally, the problems at the juniorand senior high school levels will not hemuch different from those at the adultlevel. The very things that will be taughtto the elementary school child will alsohave to be presented to the older studentand to the adult. More extensive develop-ment of the metric system will be neededin some classesfor example, in scierv_ethan in others. Metric units can he pre-!.cnted with limited reference to or com-p, rison with a few customary units. Theemphasis should he on making actual mea-surements with metric instruments. No

problems should he presented that involveconversions from customary to metricunits or vice versa. The decimal nature ofthe system should he stressed in realisticproblem settings; comparisons to the mon-etary system may he particularly helpful.The workshop approach, in which thestudent actually makes all types of mea-surements with metric instruments, is highlyfeasible and desirable.

Guidelines for teaching measurementwith the metric system

1. [he focus of instruction should he onmeasurement, with the meiric system evolv-ing and taking its role as the standardsystem of measurement.

2. Before children can understand themetric system or any other system of mea-surement, they must have experiences inmeasuring. They must understand %%Ilia

measurement is. Some prerequisite skillsand understandings are essential before anystandard measures arc used.

84 A :METRIC HANDBOOK FOR TEACHERS

3. After prerequisite knowledge andskills ,tre attained, nonstandard measuresshould be used to develop the concept ofwhy a standard system of measurement isneeded, as well as to extend the conceptsof what measuring means and of howthings can he measured in various ways.'(Only when the child understands howarbitrary the choice of a unit actually is

will he realize the importance of stan-Jarditation in meitsurenhint and appreciatethat the history of mcasurm,i1 is essen-tially a struggle for standard' i.ition.)

4. The logic of usinti ri;..asurementsystem hosed on ten---to cr.rrr.h nui 'withour numeration system----is that devel-oped, and the metric system is introduced.

5. Begin with linear measures, becausethe metric units of length are the basicunits from: which the units of mass andvolume are derived. For children in theintermediate grades, begin with the meteras the basic unit; for smaller children whoplay hose difficulty handling the meterstick (and who do not yet know the num-bers to I iii sufficiently). begin with thecentimeter as the basic unit.

(t. In the elementary school. ';.each themetric siistem as the system of measure-ment; later, the eustomai'\ system may hediscussed as one of the other system, ofmeasurement. Schools inav itti,c to teachsome of the customary measures-with teaching the metric system for

e. since the country as %%1101: 'A ill

offer examples of both far .,ear, to come.the mi_itric and customary sterns

is dual or alternatii.c ,,sterns -! -the cu.;-tornan, si.stcm happen, to he the one thechildren' parents used.

7, Avoid coni.crsion exercises, concen-trating on use the metric system. Theindii.idual need- to learn metric measure-ment itself and thus learn to think inthat language of measurement. (ihildrenwho haiie not learned arty system of

have little difficulty learningand the metric ,. -gent.

1.lmit comersiiiih within the metiiicstem to commonk used tiniN tuljacent

in size. Present-day metric material; oftenask students to perform extensive con-versions, such as changing kilometers todecimeters Just as ni;eless are exercisesrequiring addition of unlike units; rarelywill this he needed in actual metric situa-tions. Instead of stating 4 decimeters 3centimeters, the measure will he given as43 centimeters: instead of 5 meters 16

centimeters. we will say 516 centimetersor 5.16 meters. Children will need to un-derstand the relationship hetween mea-sures, but they should encouraged touse the standard form.

9. l'se actual units and measuring in-struments: avoid completely the use ofsealed-down sersions sometimes found incurrent text materials.

I)ei.elop the understanding that theappropriate instrument should he used for(Thiel-Lint measurement purposes: the meterstick for length, the balance for weight(mass). the container for volume, theclock for time, the thermometer for tem-perature.

11. The units should he introduced atrite point at which they are to he used.concentrate on those units necessary fronta utilitarian standpoint at all age levels,including adult. Do not teach the metric-unit tables per se.

2. rs:imations should he emphasized,such a,. 'About how many nuatehbookcitisirs long is [lb: desk?" or ":About howmafti. grams of sugar do you put in a cupof coffee.' Verify estimates with non-standard measures and later with metricmeasures. Deselop the meaning of anda feeling for the size of units through ex-neriences cAtering on estimating andchecking those estimates.

13. Stress the idea that measurement isapproximate. Schools have childrenmany illustrations of -exact- measures:measarenrint is not as precise as we havemade it seam. Precision is partially de-pendent on the unit of measurement11,2.

I -1. With pupils who already ki nw thesstern, is well as adults.

NG THE NI E"FR1C SYS]. N1: (AIM:LINES

appaiximatc conversions may he needed.Relate metric measures to common ob-jects anti to body. measures.

The meter is a little longer than a yard.The liter is a little larger than a quart.Fhe gram is about the weight of an ordi-

nary paper clip.File kilogram is a little heavier than 2

pounds.

Body temperature is about 37 C.

15. Use metric mits at every opportu-nity, including other subject matter fields.

lo. The prefixes should be introducedas they arc :-..eeded. Association and pre-sentation of the complete set of prefixesshould be done late in the developmentand then it should he presented only toserve the function of notimg the orderlinessof the metric system and its relationshipto our numeration system. The prefixes

dcci, centi, and :nilli arc the onlyones that will need to he stressed in theelementary' school

17. -leach only the commonly used

85/E-6

multiples and subdivisions and their cor-respw:ding prefixes and symbols; for in-stance,

cm i()1) cm == 1 III

nun lUmm= 1 enlkm 1 NO in I km12, kg 111011 g = 1 kv,

1 , nl1 1000 ill 1 I

15. Stress the importance of correctsymbol usage, which is the same in alllanguages.

19. Special emphasis should be given tosymbols for area and volatile units thatcontain superscripts. Additional emphasison exponential and scientific notation willhe needed in the elementary school.

20. Discourage the use of commonfractions with metric units except whenneeded to develop specific quantitativeconcepts; when a fractional term is used,write it in decimal form, that is, "one-half-is written as ".5- or "0.5.-

Looking at the illeasurement P10Cnif

Ten basic stepsfor successful metric measurement

.10N I.. HIGCTINS

A. adults we have been measuring forso long that we now tend to take the mea-suring process for granted. Today's pushfor conversion to the metric system pro-vides us with new opportunities to redesignnot only the system of measurement thatwe teach in our schools but also the waywe teach measurement itself.

Research over the past thirty years sug-L,ests that measurement is a much moreinvolved process than we have commonlyassumed. Measurement require: the (level-OpnICtit of several prerequisite skills anduncierst.inclii,:s that ;ire often overlooked.The following, list suggests some of theseprerequisites. along with a possible se-quence in which they might be developed.The list is intended as a summary. Theissues and critical aspects of each step canhe found in the other article; in this hand-hook. We have (milled this list to treatlength, area, volume, and mass ( weight)simultaneously. Researchers have foundthat many of these understandings must belet-e/Hped by children, instead of merelybeing 1,4 I en by textbooks or teachers, andthat a child's development does not pro-ceed simultaneously for length, area, vol-ume, and mass (weight ). Neveriheless, theprerequisites are similar for all and can heeasily discussed together even though theymay be separated in the curriculum. Com-pare this list of ideas and suggested activi-ties with the textbooks used in your school.Are you skipping any imporOnt steps andprerequisites?

89

I. CH/Hpuo direcdy. Extend the ideasof "more" or "less" to -longer" and-shorter,- "heavier- and "lighter,- and"higger- and -smaller" by placing objectsdirectly next to each other. Be sure toextend the activities to include not onlylength but also area, volume, and mass(weight ) as well. For length, compare pen-cils. pieces of string, chalk, and so on.For area, compare sheets of paper orS111111;11- paper cutouts by placing one overthe other. (keep the shapes being com-pared geometrically similar at this time.)For volume, compare boxes and cans byplacing one inside the other (filling activi-ties should come later). Compare the mass(weight) ) of objects by lifting and byplacing on simple two-pan balances.

2. Cm/paro three objects. Three oh-jects B, and C' may be compared bycomparing A with B and B with C, Developthe property of transitivity for length. area,volume, and mass (weight) : if A is lessthan 1? seal B i; loss than C, then A is lessthan C. which can be verilicd by a thirddirect comparison. Note also cases wheretransitivity does not apply. For example,if A is less than B and C is less than B, wedo not know how A and C compare with-out a direct comparison.

Apply the transitivity principle to Com-pare two widely separated objects. Hiveone child hold a short pencil on one sideof the room, and another child hold a pieceof :balk on the other sick of the room.

91) A Nil.. I RI(' MA NI-MONK FOR I' \CIILRS

Vhieh look; lancer '.. Choose a third objectof intermediate length, perhaps ;1 dowelor stick. ;ind use the transitive property bycomparing lint the pencil to the dowel ;111(1the eh,dk to the dime].

3. ritiott kittaJr)// «itiooiN it! ,Jttifttr,

children ha\ e learned and indstered the

transitit, it ). propiartii.t, they c.in place se\er :tl object, in order of inereisint or de.creasing ,wit H. repeated use or frmisitkio..Fins serial ordering dependshilt k 1101 the steps in Inca-;man, that

1. C,Jtinpartt iNtlire(76. Basle to the

ine,tsuretnew proc,.., is the usia of a thirdlarger unit that can K iisi:(1 to help corn-p.tre. \\je net around the inapplicability rfthe tr,Insitke properiti Itty cojistriLttillijjmarkint.:) the si/c tit one object tin till.,

lamer milt :111,1 using the con,trueted ;tieto compare to the second object. Foresample. ahoose dime' long.ar than eitherthe pencil or the chalk IIl the previousevimpfe. CompAre the dittAel to the penciland mark the pencil's length on the dowel.hhan comj,tire the tuarked length on thedowel to the length of the chalk,

ctimpare the areal of pAo hr

tracing one or them till tl 1:1P2e ,Meet of

paper (the la. jter unit I. !lien carry theI arked sheet of paper to the tither Irl-allgle tor comparistill

ljt.tto ohmic-. cat. he comparec indirectlyby using it volume of either sand or vvgoeras an intermediate. A, before, the olurneof sand or %ater ;P.:Waffle must ha largerthan the volume of either of the tv.t) con-tainer, being compared Fill one of thecontainers with sand oi t.vater. discarding

.\I;tsse, cvcights l nisi he compared hething sand or tA.iter to -,..(tmtruct- ti massequal to the first. file con.tructe(1 massmat; then he compared to the nt.'s, of the

5'. I (;).c?o,

Develop the 'Rica that basic prop-

erties can he :aided. Comp;Irc the lengthof a pencil to the length of two pieces ofchalk hv placing the chalk pieces end toend.

('ompare the area of four small squaresto that of a large square by fitting thesindller squares together like a Kurile totrill a net.% tiren. C01111,:tle the volume of,e1C1,11 l'10e10, tti til:It 01 larger 1)0 1,V1,1C1-011g the hiOck. COInp:Ill: the WCIL!.11t

of sta% er,il paper clips to that of a safetypin by placing all the paper clips on onen;In 01. a ;Hid the sale t_\ pin on theother.

friot,iornt ttitjotic ft)). (.(0)titari\,),).Often it is helpful to transform objectsto make them appear more nearly alikeor purpose, it i..,scnti:o to

process is the fact that no length, arca,\ttl itiiii', of Ilia must H. lost in the trans-formation process--this is the basic ideaor

Iol'll? fl i,1,,2(111 pith toothpicki. Com-pire the length (perimeter) of the polygonto the length of a pencil by placing thetoie,hpicks end to end to form a straight

Compare the area of a triangle to thearea of a rectangle in cutting the triangleinto piaecs that c,tn he fitted together toform a new shape that is appro\lmatelyrectangle.

(Ionip.ire a hall of modeling clay with aHoek of modeling clay by pinching thecorners off the block and adding them back:it other Tots. Fse this same transforma-tion to compare both volume and massciveight

7. (',,,,,pair /ty iter(iiimt, l wo objectscan now be compared by using a thirdintermediate object that is smaller thaneither of the object, being compared. Asmall length i, used to construct the lengthotj a pencil by placing copies of the smalllenj:th end to end until the additive lengthi; equ,d to that of the pencil. The number'it times the small length is used or re-peated i; counted. and this number be-comes the measure of Tile length of the

LooKING At' THE NIEASCRENIENT PROCESS 91

pencil. If we want to compare the lengthof the pencil to the length of a piece ofchalk, we can get a number for the measureof the length of the chalk in the samemanner and compare lengths by comparingnumber.

This process is what most people thinkof when they think ()I' measure. There aretwo important things to note iibout it. Firstof all, the iteration process reduces mea-surement to a process of counting, but

counting is the last step of the 111C:ISUrC-Incnt process. The steps that we havesummarized earlier form the necessary pre -requisites for the successful C01111)10tI011 ofthe counting proce,s. Secondly, the inter-mediate -small length- forms the unit ofmeasurement. When %%,..f count. we rousthe counting equal or congruent units. Thisis why the repetitive or iteration process isimport:un--it guarantees that we countc(.ngruent units. The unit we choose mayhe arbitrary. as long as we tiSe the sameunit for till HIC;I:NlIrCIIICIIIS (20111-

P:1ft'.When measuring areas by iteration we

may he arbitrary not outs in the choice ofthe size of the unit but in choosing theshape of the unit as well. .lhe problem isto choose a shape for a unit that can beused to construct many differently shapedregions by repetition. The most practicalshape to choose is a square, %%llicit can heused to construct hoth square and rectangu-lar regions. -To measure regions of othershapes, we first transform them into rec-tangles and then measure the resultant rec-tangle with the unit squares.

Children should develop the iterativeprocess for measuring areas by coveringsquare and rectangular regions with smallsquare tiles and counting the number oftiles used as the measure. Later. the useof graph paper or ti transparent grid mayhelp speed this process. \\Then this has beenmastered, then parallelograms, trapezoids.and triangles may he measured by trans-forming them into equivalent rectangles.

The iterative process for mettsurim.z vol-urn,' follows the same pattern as the process

for area. Small cubes are chosen for unitvolumes, and these may he used to con-struct rectangular solids. Irregular volumesfire transformed into rectangular volumeshefore measuring.

To measure mass (weight), a small unitmass must first he chosen. Identical copiesof that unit are added to the balance panuntil the mass heing me:isured is matched.The number of unit masse:; used is thenc(mnted. and the resulting, number becomesthe measure of the mass. For classroomusc, paper clips make practical small-unit111:1,,2S.

S. Afeavnre between limits. Unless theunit has been chosen with respect to theparticular object being measured, it is gen-erally not possible to construct congruentlengths, areas. or volumes by :I simplerepetitive process. When measuring thelength of ti pencil. for example, a childmay find that five units form a lengthshorter than the pencil, while six unitsform a length longer than the pencil. Chil-dren should then report the measure ofthe length :is between 5 and 6 units, orS in (, (where in is the metvurc

An arithmetic of measure limits can hedeveloped. If the measure of the length ofa piece of chalk is between 3 and 4 units,then the measure of the length of the penciland the chalk end to end is between 5 t 3

and 6 f 4. In a similar manner. the lengthof two pieces of chalk is between 2 f/ 3

and 2 ' 4.

(), ..tibait.i\ion of writs. \\lien childrencheck the arithmetic of measure hunts bymeasuring the combined lengths directly,they usually bind that the interval betweenthe limits of the actual measure is less thanthe interval between the limits given by thearithmetic. thils, although the arithmeticgives the measure limits for the length oftwo pieces of chalk its 6 and S. actual mea-surement of the length of two pieces ofchalk laid end to end may give limits of7 and S. This happens, of course, whenthe length of the original piece of chalk isIMICh closer to 4 than to 3.

A METRIC HANDBOOK 1,01Z 'FFACHNIZS

The difficulty that is raised here usually'suggests to children that the interval be-tween measurement limits should be madeis small as possible. There are two ways

to accomplish this--either choose a smallerunit of measure, or subdivide the unit ofmeasure into smaller pieces. If one choosesa smaller unit, the interval between themeasure becomes NinitlIcr, but the countingprocess becomes more difficult, since thereare score eopi.'s of the unit to count. BYsubdividing .1 Lirger unit, \ye make bothaspects easier. Counting is easier, since wehave fewer larger units to count. The in-tervals between measure limits is smaller--after the last whole unit, we C:111 switch tothe smaller subdivided unit.

Children should subdivide basic units inms manner they wish and practice mea-

suring silk the new subdivided units oflength, arca, volume, and mass (weight ).

W. Calculate with measurements. Thebest way to subdivide measurement unitsdepends on In1W they ;ire to be used. Wecan often shorten the counting process bydoing simple calculations on basic mea-surements. For example. suppose we W'Lltltto measure the area of a rectangle. We cancover it with unit squares and then count.Or we can count the number of squares

;don,: the length of the rectangle and thenumber of squares along the width of therectangle and multiply these numbers to-gether to obtain the total count. In similm-

multiplying the length, width, andheight of a rectangular solid is a shortcutfor counting stacks of unit cubes,

lihe algorithms for calculations dependOn the place-value system used in our nota-tions. To keep our calculations simple. thesubdivision of units should parallel ourplace-value system. Since we have a deci-mal system of notation based on powers ofID. it follows that for ease of calculation,the subdivision of the unit should also hebased on powers of 10. This ii e,riu

what the metric system Marc! this isthe point at which the metric .system be-comes ail rantat,,cous.

Other subdivisions may be useful in

other applications. It is often useful to

consider halves or thirds of units. particu-larly in estimations. but no other subdivi-sion procedure is as useful for computationis the decimal di\ isions used in the metricsystem. As teachers of young children, wemust look ahead for the system that shill

ultimately he most useful to them and usethat sYstem from the time standard unitsare first introduced. That is why it is im-portant to think metrie--now!

FZ,Ti'lil!,1 rn,n1 the 1`171 ( 1')71, (oolKil of NI.Ithem.ltik.,.

I hinking about measurement

LESLIE P. S'1F.FFE

ii., ,INociarc rn. Inalhemenicx an.,n at Inc1 'nil c nay .. -t the /Lc, Licor:,,in. 1.,'%11,' Si,

rrwarch wan

thin in!? In niallicinaIiiv. II, r;

cif 01 a Li,//re'rel/CC re.scarch

thc ;1'p,'. 1N Iiti article v

erc,a1 1,10't; tin,/in Nrich

It is essential that teachers of mathemat-ics have g,lod grasp of the thinking basicto the mathematical content they present totheir student;. \\Tien confronted with thegeneral area of thinkinil. nlathenlaiics edu-cate I includinL, teachers) have tradition-ally Lurned to psychology for help. Somemathematics educators arc currently turn-

ttmard theories of cognitive develop-ment in their quest to understand thinkingbasic to mathenudics. .1 he search is richin content and rewarding in terms of im-plications for mathematics teaching. Theimplications ,ire not necessarily in termsof pedagogical polcmics but lie in the in-sight that teachers can gain into the think-ing of the Yount! child and the origin, ofthat thinkino.

Piager, theory of cognitive de% elopmentis a theon, intelligence. Onc of its; fc:,-

93

tures is that it describes structures ofthought and changes such structures lln-ticP.D.) a child changes in chronologicalage. Nlajor stages of de% clopment of them=structures lia%e been identified hr Piaget.Crucial ages at which changes in structuresoccur ha \,e been isolated it :ippri,\illi-,,L.7,y

eighteen month, of age. so.en vears of agc.and mei\ c rears of age. Stage, may heidentitied in terms of the Jge inicrwals.from birth to ciohteen month, of

eighteen Months to seen \ ears of aoc.seven years of ;,,,e to tv,elve \ ears of age.

id twelve years onwards. Two stages arcparticularl of concern in this paper----thestage from eighteen months to seven \ ears.called the stage (If preoperational repre-sentation., and the Stage front \c,..11 vcars

ty,clve ears. called the stage of eon-L.Tete operations. In the case of individual

94 A Nil: TRIC HANDBOOK I OR l'EACHERS

children. exceptions in when stages occurhave been noticed. but not in the sequenceof occurrence.

In the theory, overt action is yie\vcdsouiee from hich intelligence originates.As the child grows older, the ()yell actionsare internalifed t they occur in the mind 1and become operations or thought, or,for short, operations. Such operations ofthought Yie\Yed ha\ ing structure.Concrete operations ,Wed

to denote Chit children em think in a logi-cally cohLrent manner about objects thate\ist ami about actions that are possible.

(;ill perform coneiete operations,however, in the imme(.1i.ite tib\C'nCi' ofobjects. Sonic CiffiCletc opera-tions are classification, ordering.. counting,and the rundament,i1 operations of the logicof classes In WI1t fi,111

the operation or ordering is discussed.:\ child can perform an overt act of

comparing Py 0 .trine's sal'. and Bi to:tsccittin if i is ;ts 10112, as B, .1 is

shorter than B. or if .-1 is longer than B.In order to e parthdiv, the

conditions that have to hold for such overt:lets to represent operations of thought. itis neeessary to consider a child placed in it

situation where he is asked to order a

bundle of strings from longest to shortest:!..tun-,' the are of diilerenl

lenifths I. ko avoid the possibility of thechild performiug the ordering by lust look-ing at the strings. the strings haYe to beelse 011")h.411 iii length so that ;my twostrings :ire not obYmusly ltif differentlengths. I hat is, to find the relation thatholds. ;I child actually has to compare thestrings by placing one end point from eachstring adjacent to :mother and comparinethe ti) remaining end points '.s it Ii thestrings drip\ n taut (see lie.

I ), To performthe ordering, task. the child must take r,(,,i)strings. sit .-I and B, .ntd math. compare

1,

:1 B C

A longer than B

B longer than C

Iig. 2

thein. suppose he finds that is longerthan B. Ile then must take tt third string,ind compare it with either ,-1 or B. lin:Tinethat he compares it 55 ith B ;Ind finds thatB is longer than C (see lie. 2). At thispoint. if the overt action represents opera-tions. there is no necessity for the child tooyerth. compare anti C. lie is able, inhis mind, to compare C on the basisof the fit .mises that . is longer than B and1? is longer than C. That is, the child isable to compost: the 1\ \o relations "Alonger than H- and "B longer than C" andinfer that is longer than C. which is onecentral aspect of the operation of ordering.In mathematics. it is what is knomi as theprinciple of transitivity. Although not un-equivocal. some research (Smedslund 1963)shows that the average :lee at which chil-dren acquire transitivity of length relationshe, somewhere between ',even and eightyears,

Imagine now that when the child com-pared string ( ith string B he found.instead of B longer than C, B shorter thanC (see tig. 3 I. In this case. if the (-wenactions represent operations, there is noneeessity for the child to make :my further

ert comparisons to ascertain that B isshorter inan or that ( is longer than B.lie is able. in his mind, to construct a eon-seise relation. \\ lien the child thinks, "Bis shorter than it is a relation that hitsa directionality from B to C. The converserelation. (' longer than B.- has

B

A longer than B

B shorter than C

LOOKING IT -1111 MEASURF \I ENT PROCESS 95

tionalit% from ( to P. In other words. thechild starts wit!, /3, procecds to C (

shorter than C. then hack to It (( longerthan It ). such an e\pression of re\ ersl-hility is essential in the structure of theoperations.

In the case of Him: 3, the child cannotestablish. in thought. ;my relationhetcen .1 and ( the comparison ofit and and al and B. It is essential Cr ohint to iwertk compare :tit1/41 coinplete scriation l ordering ) of al. it andC. ..\ssume that the child Comic! .1 longerHan C. so the order of the strings is :k Ili

4. I he child mu -t now take .inoiherstring. say /). Imagine hi: rind, that /3 is

longer than I) with one overt comparison.The concrete operational child %%kink! nothave to mertly compare al with /) nor Cwith 1) to infer the correct relations, Fore \ample. to infer that :1 k longer than /).he eiiii!cf take at feast two different route,

estahlish the relation. First, because heknows :hat al is longer than /3 and It is

ioniIer than /), he knows that .1 is longer/): or, s:cond, hecaise he know, that

.-1 I, ',Inger than ( and ( is longer thani). he :an infer that al is loner than /).most :Ilicient strategy on the part of thechild would he to rea-on that B is shorterthan hoth .1 and C. /) is shorter them B.

I) is shorter than Huth .md C. Chil-dren in the stage of preoperational repre-sentation do not exemplify such flexibilitof thinkin,i.

()tiler strucHral aharactcristics of con-crete-orerational thimght exist. but titerare Netter hoilagfi) oil in other cows0.s.The ,hike characteristics of transiti%

and re.ersihilit% discussed in the context ofseriatiorkot-strings tas:: ;ire quite general

utd arrIv to other oru.'r relations: that is.

A C BI r.

A longer than r

C longer than B

"less than" and "more than" for numbers.One point should he made explicit. That is,it is not necessarily true that ;ill childrenwho ;ire eight Years of age are in the stageof concrete operations, nor is it necessarilytrue that all children who are six Years ofage :ire not in the stage of concrete opera-tions. It is important to note that children:ire not. in the present curriculum. taughtdirecfk to think in the ways characterized.Tikk do s» without formal instruction,

\ow, consider a child who is placed in asituation where he is asked to categorizea collection of "linear" ohjects ( sticks.strings. pipe cleaners. etc, ) into subcollec-tions, where the subollections ;ire formed1-k using the relation ";is, long ;is." That is,;ink ohiect in a gken suhcollection is as

kin, is ;my other ohject in that suhcollec-lion. ;Ind ;my two ()Neck, taken from differ-ent subcollections are of different lengths.Some structural aspects of concrete opera-tions noted earlier :ire involved, as well asothers. The child can start by selecting anobject. say al, and finding ,inothk:r, say /3,o that is .1, low, is /3. The child may

now find ;mother object. sir C. is long asal. If these overt actions represent opera-tions. there is no ne.:cssitv for the child toovertly compare It and C. Ile is able. inthought. to compare al and ( on the prem-ises that :1 is is long is It and al is its loneas C. For the concrete-operational child.knowing that .-1 is as long as /3 also impliesthat the child knows that it is ;is loni, as .4reversihilik ). and knowing that it is as

long as al and al is as long (" implies thatthe child is able to infer that it is as long ;isC t transitk its ). Given that the ,.`-..Id iews

1. B. and (' as heing as long ;is one:mother, he can freely us.: any one of thethree as rcpre-cnIiilive of the others. or. inother words. t, . principle of substitu-tion. Nloreiwer, the child can compose rela-tion, as follows, If he finds an object I)such that I) is not as loig (', he knowsthat /) is not as Joni, as either ,4 or R without overt comparisons because al is as longas ( and It is is Ion), is (".

1 he above principles of thought at the

96 A METRIC HANDBOOK FOR TEACHERS

A131 132 113 114 135 116 137 138

sIng.e of Conecele operations ',U.0linear objects into subcot_

lections on the basis or is long :H.- !lowsuch thinkin, Is insoked IIl measurementtasks

NI,....isurement is a process whereby aHtemitcr is assigned to some ohjeet. Con-sider. for example. a line segment .1 as anohjeet to he measured taut piece ofstring is ph\ representation of a linesegment ) and another segment B usuall\considered as a unit of mensuremeem se-lected to measure the st.sgment mea-sure B is laid alongside and iteratedas many times is necessary. Consider thespecial case in figure 5, where .I is rightunits in length. the child must conceiveof }Icing partitioned into eight sub-parts, determined by the iteration of B,each of which is as long Is 13; that k, B, is

lorp, B. R. is as long is B, so onOhviouslv, the child //trot conceive, in

thought. that 13, is as long sincethere is no possibility of directly comparingR. and B. "pile ,:.:E1C holds for any two ofthe subpart.. 'Um). the child must coneci\ ethat B. I-. shorter than B, B (R, R.means the sei21112Ilt formed B).13, R. than 13 13 B

and q) on, .,11(.1 at the same time he mustconceive tha; B, R. is longer than 13,.

B, 132 is longer than B, andso on. Or, in other words, a stick one unitlong is shorter than :1 stick two units lone:a stick two units long is shorter than a stickthree units long: and at the same time. astick two units long is longer than a stickone unit long: a stick three units long is

longer than n stick two units long: and soon.

Too often measur,.ment is begun ill theclementary school 1)y tiikin, a foot ruler.marked off into units of an inch or frac-tional parts thereof. and :milking it toobjects to he mensured. obtaining nnswerssuch a the hook is eleven inches long-

,,,suminr a hook was to He measured I.Children certainty like to measure objects.However, if they have Junkie their ownmensuring instruments, two children maywell find "different- answers for how longthe hook is. 'Flit: resolution of such a con-flict may lead to II quite powerful notion-a unit of measure for each child that is aslong the unit of measure for ;in\ otherchild. In the resolution of such conflicts.other structural characteristics of concreteoperations become immediately apparent.For exnmple. consider a simple situationsuch as that depicted in figure 6, Segment.-1 ( for example. the length of the bookmeasured by t%\o rulers, ruler I and ruler 2.With ruler 1, segment. .4i is found to hetwelve unit. long. and with ruler 2, eightunits 101"q!,. HOW can the same segment havetwo "different- lengtlis? It is essential atthis point that a child be able to simulta-neously conceive that the unit of ruler I is

shorter than the unit of ruler 2 and thatthere are more sub.egments of ruler I thanthere are of ruler For a child in the staLteOf preoperational representation. Pinget'stheory could predict that he is not able toconceive simultaneously of both relations.A child at the stage of concrete operations,however. could establish both relations and

Fig. 6

Segment A

Ruler I

Ruler 2

1.00KING AT THE MEASUREMENT PROCESS 97

Coordinate thc:Ii.:11% it .1,0111(1 seen]that such I1 Child would a more pow-erful conciTtuall/ing Jibilitr than the pre-operational child as regards construction ofstJindard units of measurement.

To illustrate mote lucidly the point co-cernino the construction of standard unitof measurement. consider the three situa-tions depicted in liguie 7. In each situation.a child is iisked if he can make a lengthcomparison of the two polrgonal paths.polygonal path mar he thought or as a set01 COI1I1Ceted at the end points.)1 he piths are not !nor dile so that the childcannot make a direct, orert comparisonbetrreen them. First he must ascertainrrhether the subsegments of each path :ireof the same length. The child could do thishr selecting stick ( a unit ) long as onesuhscment and then comparing it \sifteach such subsegment in that pith. as :d-read\ discussed relative to Figure 5. Ilemust. hir necessity. he able to usc Iran-sitir Ur and re\ ersibility in order to con-ceive of. path I of situation I :is beingpartitioned into subsegmcnts all of thesane length. Note that in path 2 of situa-tion 3 one slIhsek.1.111,211t is shorter than theunit and one is longer, so that no ct)in-p:ffi,on is possible, on a logical basis, ofpath I and path 2. Mc:ours:2. mire sophis-ticated means :ire :ivailable to m:ike thecomparison.

In situation 1. all stili..L,Ancrit!-: of pathI are of the ,same length and Jill subseo-mews of path 2 :ire the same length. It is

Situation I

Path 2

necessary for a child to compare one sub-segment of path I with one subsegment ofpath 2 through the use of an external stick(thus employing transitivity and reversi-bility) to ascertain that they are of thesame length. Then by establishing that onesubsegment of path I is as long as one sub-segment of path 2. the principle of substitu-tion must be employed to ascertain thatany subsegment of path I IS as long as anysuhsegment of path 2. Now, in order tocompare the lengths of the paths. the childmust either construct a one-to-One mappingbetween the subsegments of each path orelse count them, In each event, the con-clusion. path 2 is longer than path I, is

based on the premises that there are moresubsegments in path 2 than in path I andthat each subsegment of path 2 is as longits each subsegment of path I.

Although in situation I it is not neces-sary for the child to establish and co-ordinate both relations in order to reach acorrect conclusion, in situation 2 it is. Insituation 2 each subsegment of path I is

as long as any other subsegment of pathI but longer than any subsegment of path2, all of which are en the same length, Thetwo premises on which a child must baseany logical conclusion are: there are moresuhsegments in path 2 than in path 1,

turd each subsegment of path 1 is longerthan each suhscgment in path 2. Consider-

just these two premises. the child mustperceive that it is not possible to comparethe lengths of path I and path 2. In figure

Situation 2

o

0-

Fig. 7

-.4

Situation 3

C

98 A NIETRIC HANDBOOK FOR TEACHERS

6, a similar situation exists. There, how-ever, the polygonal paths forming the rulersare se,t,quetrrs. that are proximal to eachother and to sygment .4, so that visualcompat'isons are possible. \o conclusionsabout the relatke lengths of the two rulerscould be ascertained from the two premisesthat the unit of ruler 2 is longer than theunit of ruler l and there are more subscg-limits of ruler I than of ruler 2. Those tworelations considered simultaneously wereused to C.17)1(11.11 why two rulers, for whichit was t,,it'en that they were of the somelength, could have different numbers aslengths. In situations 1, 2. and 3 of figure7, the task is to compare the lengths ofpath I and path 2 ( which are not movable )H. !LI knowledge of the lengths of thesubsegments and how many such subseg-mems form the paths. In situation I sucha comparison k possible. In situations 2and 3 such a comparison is not possi-

ble. In situation 3, the reason a compari-son is not possible is different fromthat in situation 2. In path 2 of situation3, if a child selects a stick as long tis

subsegment ( say .-f I. he will find that thereore exactly two subsegments (C and /)not as long as the stick I C is longer than.4 and I) is shorter than ,-1 ). If the childis at the stage of concrete operations. heshould know that even though there areIs IlLtfly subsegments of path I as thereare of path 2. he cannot compare the

lengths of the two paths. It is essential thata child have the ability, before such situa-tions are presented. to ascertain whether the

1 2

subsegments of any path are of the somelength. It would also seem necessary thatthe abilities outlined above concerning sit-uations I. 2. and 3 of figure 7 be presentbefore the number line is used as a modelfor operations with whole numbers.

Considering the above discussion, for achild to conceive a treed for the construc-tion of a standard unit of measurement itseems necessary that he be placed intosituations that are resolvable but that in-voke conflict ( fig. 6) or into situations thatare not resolvable ( situations 2 and 3 offig. 71. Such situations must be carefullyselected so that the thinking necessary forresolution ( ilonrCtiOhnion ) is availableto the child.

Once a class of children have constructedrulers with :a standard unit of measurement

an inch ). they should use the rulersto measure objects. In such usage, conceptsof inner and outer measure become impor-tant in forming approximations to thelength of objects that are not a wholenumber of units long. In figure 8, the innermeasure of segment . is six units and theouter measure is seven units. The lengthof the segment is between six and sevenunits. The inner measure is the greatestnumber of units that are completely in-c ludcd in the segment. and the outer mea-sure is the least number of units needed tocompletely include the segment. If thechildren agree to make two units out ofeach one in their ruler, then a closer ap-proximation could be made to the length ofsegment .4 I see figure 9 I. In figure 9. see-

5Fig. S

7

1 2 3 4 5 6 7 8 9 10 11 12 13 14

9

Segment

Segment

1.1101ING ,\ '1111 NI EASURENIENT 99

rnctit .1 1, 1),:tccil 13 .ittLi 14 unit', O, in t)itt:11 Atii.111,ort- \Vc,,ley Put,-11,1m,,. 0., ptto).t,,:rtit, of tit,: unit. :Hid 11

7 unit,. It k t11:1t lhC llllltl fccl (.in,l,cr:!. N.. ;ohl S. ()pner. Tifc,,ryno conflict in \ Iv. Hew' /) 11111,,tlieL

.0,(1cd ( 41k, CIIIICe-11.111, Iill irtiCr lik2t.:C0,1%

pro\ n1).11101). hi !Ili Of hi,. ,,f bucl!"...,.n.r. Lon-

tionc S ;ink] 0. Jul:\ Rinrlf.... R. I \`.%ofilt1 IL,1,t includc unit, ill /', sc,.f., /6.0,r! ( tft'IlLi. 1,11

112(,:l:', ( 10'1 11/11101iiI1C;11 111,',111,1.C111L'Ill. MCI lit2 Unit, Of 111;.:,1- \.Y.: s,.1101 ut Ititicittion,,111.t..1:1,...itt, otra 1(164,

:111d ::0111, CH1011', h10111 \I. I . Stelfc. I..lit'cmcnt to noll1c1. ( /,.con;i \ clopck.1 conccrt k1! C:(11.1.11 /1,11, of 0

contLfr,...ncc on te,,....th.:11 in;itlic-Such, llo cr. not ,if:it I c..1,1ftn,

ficr. .,1,11.:ct; (if filinhif1:1 Lniver,ity, 1970.tion;t1 Iti nic-,:\ 11;..cn in N. c\I of lc:h.:Nei, ul NI0then-Kitic,

,11,1 1 (In I ii.Lt illycNti.2.:ttion foundprinCiplc tr;111,iti\ it \ \ \ sine,fItm,f, I. .1)e% clornuent (. kni,:tcic I

it \. of I in ( (1/1/,/ if,tit ion. ;I, \L'cll19r,',

Ctniccivc t \\t\\ F.n....!cfi. /',/;:,n 1, Ir-wcre .111 found:111(11;11. 1-.noiwi1 ,itutttiott,

'li't,'. DO Chilled ll. Ole \\1,,,..011,111 1L',C.11.,:l1;mil c\;ifiiplc 11:1\ C 1)ccn IlleilidCd ll hint 1 ;Mt.', 1)C..eitpIllerli .. CI Cr lof Ct,211111,:: I.C,1[11-

1C;IC[ICI. V.Ith ;l littic 11112.Clillit% C:ill :1`.,C!.-;!, 111:1.. .\\ .IILINC I(Ontl the N.:tion;11 ( ,inel. 1,,,-

such ;n1)ccts, of thinkill,2. in her t:l.t,,rot)nt. s....mt,,,1 .ind colic:2.2 Icle \ ',ion, Illoonlifiton.Inuf.r.m.f.

Bibliography. .tn(I I. R. \1,1

\ 1-11:.,cn. II., \I. ILirtufl. ;mil .1. Stochl.

\ I of c,in'an ( f..

Reptionitl riont thii I chru in, 1971 :\i I I /I \II I IC Ti \t III it. Copyright r 1973, Nation.t1 Council Feitchttri; of Niathent.ities.

Teaching about "about"

HAROLD C. TRIMBLE

profes.vor of educatirm Ohio State University,Harold Trimble describes himself as ''a would-be teacher,.

iontla' alwayA: as Other duties permit.-

o many adults mathematics is a mys-tery, an occult science beyond their powersof reason. To test this bald statement, talkto boys and girls of high school age andto adults. Listen to what they say. Ob-serve what they do in simple problemsituations that involve numbers. Noticehow they try to "remember what to do"or begin to perform random calculations.Notice how few people wade into a prob-lem confident of their own good sense.

Now consider a child who studies al-gorithms for addition, subtraction, multi-plication, and division in grades threethrough six. Much of his time is spent

learning to find "right" answers. Perl,apshe concludes that mathematics is the placein the world where exact answers are avail-able and required. At this point in his

mathematical development. a child mayencounter what seems to him to he a con-tradiction. Elc has become accustomed tousing numbers to record measurements.And now the teacher tells him that allmeasurements are approximations; that, inthe arithmetic of measurement, there areno exact answers, and, even worse, cor-rect calculations do not necessarily produce"correct" answers,

This is, of course, only one of the criti-cal moments in which a person may learnto believe that mathematics is u mystery.Perhaps it is an effective one for manychildren. At least it would appear to pre-sent a fine opportunity for a child to losehis intellectual self-confidence.

There are, of course, two ways to tryto correct such a misunderstanding. Youcan provide corrective experiences justprior to the critical moment. Or, you cantake a longer look to discover how themisunderstanding was built into the cur-riculum over the years preceding the criti-cal moment.

Let's look first for a short-range solu-tion. As a teacher you may not be in aposition to rewrite the program of study.Often the best you can do is patch upmistakes that previous teachers have made.

Have you seen the advertisement in

which one razor blade corrodes morequickly than another one? The edge of arazor blade appears to the eye as a linesegment. What could be more perfectthan the edge of a blade used only a fewtimes? Yet, under a microscope the edgebecomes a mountain range! A teacher canuse pictures like these to dramatize the

100

LOOKING Al' THE MEASUREMENT PROCESS 101

contrast of the perfection of mathematicswith the imperfection of reality. Then hemay be able to get children to see howcalculations, correct in the mathematicalworld, fail to reflect the rough edges ofthe physical world.

My own lack of confidence in talkingto students, even in using pictures todramatize the contrast of the mathematicalmodel with the everyday reality, is basedon experience. My efforts along these lineshave failed. Because of this, I am con-vinced that boys and girls need to use acorrect mathematical model for a whilebefore they can appreciate and understandthe incorrect, but good-enough-for-practi-cal-purposes, model now presented in

most school mathematics programs. So,to my mind, even the short-range solutionrequires more than showmanship.

I propose, then, providing children withexperiences in which they contrast themathematical idea of measure (a real 111.1111-her, length of a line segment, weight, or thelike) with the physical idea of measure-ment (the process in the real world). Achild readily accepts a thing as what it is.It has something that grownups call a

measure. This may he a length, a weight,a time, or another idea-level property. Thetrouble is that real lengths don't fit rulersperfectly. The measure is an idea. It is whatit is. The problem is to express the meas-ure using numbers on a scale. This is thebasic problem of meaurement. Carpentersare content to call a measure 121/2. Ma-chinists might call it 12.49. They meanabout, and about means good-enough-for-the-purposes-at-hand. That is why wallshave floor moldingsto heal the differencebetween the lengths of wall boards and theheights of rooms. But about is not a goodmathematical word. Sometimes we agreethat "about 121/2" means 121/2 V.; ; but

a better way to say this is to speak of ameasure in, about which we know that12'4 < in < 123/4. At least I like it betterbecause it keeps things straight. There is atheoretic-il h:ng called a measure, in. Butthe process of measurement is a process of

comparison that inevitably leads to makingan approximation. So the best we can knowabout in is that it belongs to an interval.For example, we may know that

121 < Sri <

or we may know that

12.485 < on < 12.495.

What we know will depend on such prac-tical questions as the type of measuringtools we use.

In present-day programs in arithmetic,children begin the study of measurementby measuring things that conic out as wholenumbers, such as 2 or 3 or 5. This seemswrong to me. It would he much better, Ithink, to begin with thinks between 2 and3, or between 5 and 6. If early experi-ences with measurement have implantedthe idea that things really do have mea-sures like 2 or 3 or 5, children espe-cially need to work with the between ideaas they begin to perform calculations inthe arithmetic of measurement.

A good problem to get fifth or sixthgraders started is to ask How far abovethe first floor of the school is the secondfloor of the school? There are, of course,a lot of ways to measure this. Fine. Supposeone child, or committee of children, dropsa weighted string down the stairwell anduses a yardstick to measure the length ofthe string. Suppose another child. or com-mittee. measures the rise for one step,multiplies by the number of steps, andcomes up with an answer. The answers aredifferent. Who is right?

In a context like this one, it will he easyto convince fifth graders that a stairstephas a riser of some definite height, but it isnot possible to say exactly what this heightis. The best one can do is to make a state-ment like: The height is h, where

< h <

For sixteen steps like this one, the height is, where

16 X 71 < H < 16X7y.

102 A METRIC HANDBOOK FOR TEACHERS

Notice that the calculation required isdoubled when you work with intervals.Notice that 16 = 11(1 and !673 = 118, so that an interval ' s inchlone ( 7'4 to 73ii ) becomes an inters l 2

inches long (116 to 118 ) when vou multi-ply it by 16. I believe that it is in contextssuch as this one that the absolute errorsand relative errors of the arithmetic ofmeasurement can he made sensible tochildren.

What I am proposing, then, is involvingthe children in a few problems in whichthey perform simple measurements: lead-ing them to notice that different methodsor different observers may get different re-sults: having them express the measuresusing intervals; helping them perform cal-culations with intervals for a time beforetrying to teach the more conventional ap-proach to the arithmetic of measurementthe rounding of answers.

The next example may be more suitablefor junior-high age children. When a boyruns up a flight to stairs, he does work. Ifhe climbs h feet and weighs w pounds, hedoes ;nu foot pounds of work. If this takest seconds, his power is expressed as

I! .toot pounds per second.

Since 1 horse power is 550 foot pounds persecond, we can write

h tt= hone power.

You can have children mea .ore their horsepowers using this formu.a. They mustmeasure h in feet and, for each child, win pounds, and r in seconds. To get a usablemeasure of t, a stop watch will he needed.Again, it will he easy to get argumentsgoing about whether or not Joe is reallymore powerful than Sam. Again, intervalswill he needed to allow for the inaccuracies

LOOKING AT THE MEASUREMENT PROCESS

of measurement. For example, the childrenmight agree that

< h< 113, (ilemeen II ft. 8 in.and I I ft, 9 in.),

II I < w < 112', (Weight of one boyto the nearest pound),

and 2( < r < (Time for the same boyallcming an errorof second tither

Notice that

(11) X (111.1, ) x )' < ' <(550) X (2) /

(550) X (2DThere are opportunities for computationalpractice. The more students who compete.the more practice. Important mathematicalideas are also involved. For example, thesmaller the divisor, the greater the quotient.

Between the two examples given, thereare, of course, many of intermediate dif-ficulty. The idea is not to teach physics, or

103

surveying. It is, rather, to work with inter-vals in order to develop some appreciationof the effect on the answer of approximatedata. With such a background, it may bepossible to get children to understand andappreciate the arithmetic of measurementas a shorter way to find answers "goodenough for practical purposes." Workingwith approximations vnd rounding answersis not really mathematics, it is a practicalwar to shorten calculations with intervals.

You may want to think about the longer-range solution. If you buy the idea that thecorrect mathematical model for calcula-tions with measurements is calculationswith intervals, you may wish to:

1. Plan the first experiences with measure-ment to fit the realities of the situation.Have children measureand record the

1 04//e6--- A METRIC HANDBOOK FOR TEACHERS

answer in the form 3 < in < 4, ratherthan say "the length is 3."

2. Use the students natural desire forgreater precision to motivate the use ofsmaller units of measure, and, later, ofcommon fractions and decimal frac-tions.

3. Lead up very gradually to the idea of areal number as a limit defined by a setof nested intervals. For example, thenumber 17 has been a source of greatconfusion. However, children accus-tomed to recording a measure as an in-terval should not find a set of intervalslike the ones that follow surprising.

3 < < 4

3.1 < r < 3.2

3.14 < < 3.15

The number r is the measure definedby the ratio of the circumference to thediameter of a circle. It can he approxi-mated by intervals obtained by mea-surement and, later in the study ofmathematics, by intervals obtained bytheoretical methods.

4. Use the arithmetic of intervals as hack -ground for the idea of a vector space.

In this instance, notice that when youmeasure with an agreed-upon unit ( hun-dreths of inches, for example) and re-cord a, h, . . . in terms of that unit,then for measures mi and m2, definedby a<mr < h and c < < d,a) m, = m2 if any only if a = c and

b = d, andh ) tn, + nr2 is defined by a < mt

+ m2 < h + d.c) For an imeger n, nmi is defined by

na < <So this arithmetic of intervals is remark-ably like vector arithmetic. It has theadditional, rather unique, property thatd) mim2is defined by ac < < bd.

Whether vou seek a short-range or along-range solution for the problem ofmaking sense out of the arithmetic of meas-urement, I have argued that it will help toview a measure as an unknown element ofan interval. I have suggested asking boysand girls to perform calculations with inter-vals. I have pointed out several advantagesof such calculations. But, mainly, I claimthat intervals provide a mathematical modelfor the arithmetic of measurement thatcontrasts with the rather sloppy, this-works-even-though-it-isn't-really-right ru-brics with which we currently mystify chil-dren.

Metrication, Ildreasure, and litlathanatics

Metrication, Measure, andMathematics

ALAN R. OSBORNE

The United States has achieved an en-viable industrial technology. The inter-changeability of parts, standardization ofassembly-line procedures, planned obsoles-cence, commonality of marketing practices,and the efficiency of management that char-acterizes our industry have solidified a

remarkable number of traditions as shownin Daniel Boorstin's The Americans: TheDemocratic Experience (1973). The per-vasive orientation to mass production anduniform consumerism has established mea-surement as a critical component of Amer-ican industry. The measurement units andprocedures are an essential tradition of themass-production, assembly-line techniquesthat characterize our technological, indus-trial society.

But this measure tradition has createdan extensive problem for trade and, ulti-mately, our position of leadership in theindustrial and commercial world. The restof the world is metric. The economy ofthe United States is dependent on tradefor our continued health; it affects eachof us. The sellers of goods manufacturedin the United States are finding marketsclosed. Just a few years ago the benefitsof expanded world trade did not outweighthe bother and the expense of convertingto the metric system, Now it is necessaryfor the United States to shift from the tra-ditional, comfortable English system to themetric system in order to enjoy the benefitsof continued and increased commerce inmanufactured goods with the remainder ofthe world. This is recognized and acceptedby leaders in government and industry

107

( U.S. Congress Senate Bill 100), and thecountry is in the process of making theshift to the metric system.

As conversion is instituted, the countryand individuals face new problems andadded expense. The comfortable intuitionsand effective estimations that are part ofevery adult's way of life will not apply. Thewrenches and other equipment in the work-man's tool box will have to be replaced.In the same way, industry will need toretool with considerable inconvenience andexpense. Measures in the comfortable Eng-lish system will become anachronismsuseful perhaps as literary referents of sym-bolic valuebut this is an adult problemand not the problem of a child who willfinish school and move into a metric world.The child should not he hound by the in-tuitions and traditions of heritage thatadults possess.

The task of this paper is to considerimplications of the conversion to the metricsystem for the schools. As such, it will notargue the efficacy of the conversion; thisis well documented and discussed else-where (in publications such as De Simone'sA Metric America: A Decision Whose TimeHas Come, 1971). The advantages arcclear. It behooves the schools not to makea big fuss about conversion but rather totreat it simply as the down-to-earth act ofreasonable, intelligent people. Otherwise.we run the profound risk of helpine, chil-dren anticipate nonexistent difficulties inhandling concepts and processes of metricmeasurement within the metric system. Thisis not to say that there are no difficulties

108 A METRIC HANDBOOK FOR TEACHERS

in the teaching and learning of measure-ment. Rather, the difficulties are not at-tributable to the conversion to and use ofthe metric system. Indeed, measure con-cepts do provide major difficulty for teach-ers and for children. The fact that mostbeginning physics courses at the collegelevel devote extensive amounts of time tothe teaching of measurement is testimonyto difficulty in establishing measure withchildren. This is to say that considerableattention need he given to considering theproblems of teaching measurement in theschools. Given the fact that conversion tothe metric system will entail modifyingcurricular content, we submit that the timeis opportune for us to address basic prob-lems of improving the teaching of measure-ment. If the content and measurementactivities and curricular design must be interms of a different system of measure-ment, should we not also modify the natureof the pedagogical design to treat difficul-ties in learning measurement?

The move to the metric system providesthree different but not unrelated opportuni-ties to modify the curricular and pedagogi-cal design of teaching measure. Each ofthese will be discussed here in terms ofsuggesting important characteristics of mea-surement that need to he established withchildren and highlighting some problemsand unanswered questions about learningmeasurement. First, the nature of measure-ment in the mathematical sense will becontrasted with the nature of measurementin the scientific sense. Second, the psychol-ogy of measure learning will be discussed.Finally, the problems and procedures ofestablishing a metric intuition will he con-sidered.

The nature of measure

A student teacher in an eighth-grade-mathematics class faced the task of plan-ning lessons about measurement for hisclass. He looked at the pages of the text,which contained such words as "relativeerror,- and "accuracy." Turning to hissupervising teacher and displaying dismay,

he stated, "I thought I was going to teachmathematics. This looks like physics. I

chose not to major in physics because Iliked mathematics best. Now what is thisbusiness of teaching science? Shouldn't thisbe the responsibility of the science teachers?It certainly is in the science texts and cur-riculum.- The supervising teacher hesitatedand said, "I have often thought the same.I teach measurement because it provides alot of opportunity to provide children prac-tice with fractionsconverting betweenEnglish units is a good, relevant contextfor drill in multiplication and division. I

also no'ice that my children have troublewith mLasurement, and they will need theseskills later on in life."

There is an element of truth in theremarks of both the student teacher andhis supervisor. Measurement is a scientificprocess and skill. Children do need theseprocesses and skills. Premetric curriculacan be rationalized in terms of a relevantcontext for drill in handling fractions inconversion between units, but analysis ofmeasurement with a broader perspectiveindicates a much larger potential than ineither of these points of view. First, mea-surement does provide one of the bestenvironments for establishing the conceptof a mathematical model, one of the pre-miere and most productive concepts ofmodern mathematics. Second, the processesof measurement are firmly based on theconcept of function. A functional approachopens vistas of power in using the measureprocesses.

What is measure?

Teachers of mathematics are often con-fused about the goals and objectives ofteaching measure concepts and skills. Theconfusion stems partly from not possessinga clear conception of the distinctions be-tween measure in science and measure inmathematics. Lacking a clear perceptionof the differences and distinctions, teachersfind it difficult to design productive in-structional materials and to plan effectiveteaching strategies. The difficulties for

METRICATION, MEASURE, AND MATHEMATICS

learners stem from the ambiguities arisingfrom the common words and parallel proc-esses used by scientists and mathematiciansin describing and talking of different con-cepts and goals.

The scientist has a different coal in

using measure than the mathematician. Heis attempting to describe reality ,:arf..rollyand precisely in order to predict futureeventshe is building it model of reality.This may be illustrated as follows:

The model is a system of concepts andpropositions reprsenting reality.. Conflictswithin the system predict how reality willbehave. The soundness and usefulness ofthe model as a description of reality is

determined by the correctness of the pre-dictions.

"I'recEction of an event\Sithin reality

(Reality) >Occurrence or nonoccurrenceof the event

If the predicted event takes place, then themodel "fits- reality and has a degree ofsoundness and usefulness. If it does notoccur, then the model does not "lit."

The building of models depends onmany inductive and observational proc-esses. The matching of the model to realityis strengthened if a mathematical matchingcan be achieved. This means providing afoundation of measurementa quantifica-tion of quantities and phenomenafor abase of comparison for the m(ii.r21.- OneeritiLl process in assuring it good fit of themodel into reality is the scienti.fs mea-surement process. It necessarily dependson and involves observation and is. con-sequently, subject to error.

109

Suppose, for example, a scientist desiresto build a predictive model for the expan-sion of a metal rod, ti 13, as a function ofchange in temperature. Verifying the pre-dictive rule (a part of the model) dependson accurately ascertaining the length of therod under two different conditions of tem-perature. (See fig. 1.) The scientist mustplace a ruler such that a point of the rulerfalls on point A. This requires judgmentand may introduu error into the system.The scientist then observes where point Bfalls on the scale. If he observes that 13falls between S cm and 9 cm, he mustdecide which is closer to point B. Againjudgment. and, necessarily, error is intro-duced into the system, and this processmust be repeated for the other temperaturecondition. The scientist may improve hisjudgment by using a more finely graduated

or by introducing some other observa-tional techniques that refine his judgment.He then applies the rules of the mathe-matical system to the model and establisheshis predictive rule. The reasoning may in-volve mathematical observations, such asmultiplication and addition, that have thepotential for magnifying the effect of errorsin the observational process.

Error in scientific process or in the useof models can be made in another mannerperhaps the internal structure of themodel has flaws; the reasoning may not bevalid or may lead to inconsistencies: in-appropriate rules may introduce systematicerrors leading to useless rules that do notpredict the event accurately when thescientist verifies the rule. Correctness withinthe reasoning of the model is the concernof the mathematician. He operates withinthe model by providing the syntacticalrules that govern the appropriateness ofoperations and the relations between op-erations within the system. Indeed, for themathematician the model assumes a realityof its own, and he may not even carewhether there is a corresponding reality.He is concerned only with internal con-sistency, validity of the reasoning, andwhether solutions exist for problems within

I I 0 A ME ERIC' HANDBOOK FOR TEACHERS

A B

0 I 2 3 4 5

Fiji. I

the model. 'Hie scientist is concerned withwhether or not the system will fit intoreality. Carl Allendoerfer's article for par-ents---The Nature of Mathematics"(19(,5 1 provides a straightforward ;tnaly-sis or the types of thinking that are usedin building mathematical models.

\lost people do not operate as eitherjust the mathematician or just the scientistour world of mathematics and science isnot so idealized. We do not typically con-cern ourselves with building an accurate"tit" of a model into reality with mathe-matics. Rather, we take previously devel-oped. intact, systems or models and usethem. The measure concepts and opera-tions within Euclidean plane geometry arean example of such an intact system. Con-sequently. we are concerned with bothestablishing measurement as a series ofobservational processes and using measurewithin th intact model.

The child needs to consider measurefrom the standpoint of both the intactsystem and the observational processes. Heneeds to deal with, accept, and incorporatethe intact system so that he may workwithin the world of future mathematical!earnings. He needs the observational proc-esses in order to learn future scientific con-cepts and because this is the sort of mea-sure that he will find most practical in hisadult life as "the man on the street.- I3utthese processes and intact systems alsoprovide the child with some of the skillsand understandings necessary for copingwith measure today as well as in futureadult life. The child needs measure toquantify hi.: world, to make mathematicalas well as verbal descriptions of his en-vironment. NIcasure and measurement con-cepts give the child the means of describingthe work: ..1iout him.

6 7 6 6 to I

Length

The example of a scientist determiningthe length of a rod was fraught with obser-vational difficulties. These could not beevaded and required judgment on the partof the scientist. Necessarily. error and in-exactitude were involved. Length in themodel system or in mathematics is not sub-ject to observational difficulties: it is exact.Corresponding to a pair of points A and13 is a Nin:zie real number that is called thedistance from . to B. That is to say, dis-tance is a function that maps segmentsa line into the set of real numbers.

The distance function may he defined orcharacterized several different ways de-pending on the particular axiomatic struc-ture used to describe geometry. The clas-sical axiornatization of Euclid. the structurefor the usual Cartesian coordinate system.and the intermediate coordinatization ex-emplified by SNISG's grade-ten geometrycourse each provide variations of the de-fining characteristics of the distance func-tion and the restrictions to which it is

subject. Usually, the axiomatic structureof geometry in elementary and junior highschool mathematics texts is implicit ratherthan explicit. Rather than being concernedwith the niceties of the restrictions imposedon the function by the axiomatic structure.authors emphasize the characteristics ofthe distance function that hold for each ofthe implicit axiomatic structures. Thesecommon characteristics are precisely the"big- ideas that a student needs for futureencounters with more formal treatment oflength. They provide the organization ofhis cognitive structure that assures readi-ness and receptivity for new concepts.

Let us examine the distance functionwith the intent of identifying some of these

NIETRIC'ATION, NIE AMAZE. AND NIATHENtATICS 111

important characteristics. In the manner ofthe writers of elementary and junior highschool texts, we will not provide an ex-plicit description of the axiomatic struc-ture of geometry. Rather, the distanceconcepts spelled out below hold in anyEuclidean context. The intent is not toproduce a minimal, independent, descrip-tive set of statements but to identify a

sufficiently comprehensive set of primitivelength concepts to allow the student thepower to deal with most situa:ions.

We define the distance function d in

terms of a set :11 of segments on a line.The function d maps elements of .\! intothe set of nonnegative real numbers. \\ewrite:

1(a). d: or

1(h). d( A, B) C- R. where A and B arethe endpoints of the line segment.

In the more leisurely paced developmentof school mathematics, the luxury of aless complex notation is typical. Distanceis labelled simply as mAB or .-I B. This de-velopment uses the functional notation toemphasize the functional character of dis-tance and because the it A. B) notationallows a more precise rendering of theprimitive ideas that, taken together, con-stitute distance. In the notation d(A, B),the d names the functional rule, the argu-ment A, B indicates the segment endpointsin order (we are considering the distancefrom A to 8) and the entire symbold(A, B) is the real number signifying thedistance.

The distance function needs sonic otherproperties if it is to be sufficiently powerfulto treat the usual problems encountered inschool mathematics. In order to strengthenthe function, some rather obvious char-acteristics are neededthe sort that are soobvious to the experienced adult that it is

easy to overlook establishing them in theelementary school classroom. but the dis-tance function is not complete withow.

these primitive concepts. We state themfirst in the functional notation:

2. d(A, B) = d(B-1). The distancefrom a first point to a second is the sameas the distance from the second to the first.

3. If d(A, B) = 0, then A = B, andconversely. the distance between twopoints is zero, then the points named byA and B are not different but the same.The "and conversely- indicates that thedistance from a point to itself is zero.

4. I f B is between A and C, then d(.-t, B)d( B, C) = (1(.4, This assures that

if two segments are laid end to end on theline, then the sum of the length of thesegments is the number naming the lengthof the union of the segments.

5. If B is between A and C, then awhole number p can be found such thatpld(A, C). This may betranslated to say that if segment AB is apart of segment AC, then it may be copiedenough times to go beyond C on the line(see fig. 2":. A mathematician would statethat this makes our geometric space Archi-med i a n

_L-4=2-i- 1 -1

A Bp copies

2

6. If segment .48 77: segment CD, then((A, B) = d(C, D), and conversely. Thissays that segments that arc congruent havethe same length, and if two segments arethe same length then they are congruent.

When taken altogether, these properties (1through 6) constitute the idea of distanceon the line. This complex of ideas providesthe intuitive insights each student shouldacquire in matriculating through school.But are these characteristics best acquiredby providing experiences with them sepa-rately. or should the young child encounterthem en masse? Many children's early en-counters with measure are in terms of corn-

1 1 2 A METRIC HANDBOOK FOR TEACHERS

putational formulae. These formulae andexperiences subsume and depend implicitlyon the six primitive properties stated above.Consequently, the ideas are muddled to-gether rather than providing the child op-erational control and understanding ofeach property. Because they are so obvious.to the adult, it is easy to overlook the prob-lem of the child who has difficulty with oneor more of these ideas.

The distance function is frequently"strengthened- by imposing a coordinatesysiem on the line to turn it into a num-ber line. :\ function is used to map the realnumbers onto the number line in a one-to-one fashion. If is such a coordinatizingfunction and and B points of a line!, thatdL-1. B) 11( . This assuresthat the number line is dense, ordered, andArchiniedian. ( By dense we mean thatbetween any two points a third can befound.)

The ruler-placement postulate of severalgeometry texts currently used in Americanschools describes implicitly a coordinatiz-ing function. The label "ruler placement''is suggestive of the close parallel betweenthe physical reality of the use of rulersand actions with rulers but is actually a

description of the ideal world of mathe-matics within the model of reality. Use ofthe descriptive "ruler placement'' label maywell obscure the distinction between physi-cal reality and the world of mathematics.

Coordinate systems are intrinsic to mostadult experiences with measure. Typically,coordinates or scales provide the base forcomparison of size. Most conversions fromone measure system to another are based

0 32

-32 = (0-32)_

-1778

on the scales possessing properties I

through 6. Whether within the metric sys-tem or between the metric and Englishsystems, the shift from one scale to anotherinvolves the same type of transformations.Most commonly. the transformation is sim-ply a dilation, a stretching or shrinking,caused by multiplying each number of theoriginal scale by a positive real number.Converting a meter scale to a kilometerscale is accomplished by multiplying eachcoordinate on the meter scale by 1/1000.Note the transformation simply relabelspoints. The shrinking is only apparent be-cause of the relabeling. The point is thatthe mathematical principles are the samewhether the conversion is within one sys-tem of measure or whether it involves boththe English and metric systems. Many stu-dents have become so bogged down in thecomputation associated with conversionthat they have missed the basic idea of thetransformation.

The conversion of one temperature scaleto another illustrates one more transforma-tion that is sometimes employed in conver-sion, namely, moving the zero point byshifting or translation. Both translation anddilation are used in this conversion. Sup-pose a line is labeled to serve as a modelfor Fahrenheit temperature. This is a co-ordinatization of a line. By subtracting 32front each coordinate, we can shift thecoordinate system. Multiplication of eachcoordinate by 5/9 shrinks the scale suchthat we have established a new coordinatesystem, one which serves as a model forCelsius temperature (see fig. 3).

All that has been accomplished by these

212 Fahrenheit

shifted by subtracting 32

0= (32-32) 180= (212-32)

4 shrunk by multiplying by 5/9 Celsius0=5/9 (0) 100

1

Flu. 3

METRICATION, MEASURE, AND MATHEMATICS 113

transformations of stretching or shrinking(dilation) and shifting (translation) is con-version from the Fahrenheit scale to theCelsius scale. Since each transformation isone-to-one and onto, the process could hereversed to derive the Fahrenheit scalefrom the Celsius scale.

The transformations of dilation andtranslation provide the mathematical meansof converting between the measure scales.Dilation, the multiplication by a positivereal number, is messy in the English sys-tem, with fractions such as 1/12, 1/36,and 1/5280, and with whole numbers lead-ing to comparable computational drudgery.A primary payoff of conversion to themetric system is that dilation is always interms of multiplying by a power of ten.

Conversion from one measure scale toanother is an exceedingly useful propertyof a measure system. Dilation and trans-lation do not alter the basic properties ofthe underlying coordinate system Lit:. is

the scale. The ratios of distance, tweenpoints labeled in one system arc . sameas the ratios of the distances between thepoints in the other system. That is, theratios are invariant under the transforma-tions of dilation and translation, The meas-ure systems most commonly used by scien-tists and by the man on the street possessthis property of invariance. Such measurecoordinate systems are frequently calledTali() .wales.

The distance function was establishedabove in a peculiar manner so that it wouldhe exactly analogous to children's firstclassroom experiences with measure. Thepeculiarity is that the domain of the func-tion is restricted to a single line; that is,any measuring must be done on a singleline. Children's preliminary work in theearly grades with number lines and withrulers is typically so restricted. Perhapsagain because it is so obvious to adults, itis easy to overlook the difficulties in mov-ing from measure in the one-dimensionalspace of a single line to the more advanta-geous situation of measure within two- andthree- dimensional space. Rather than care-

fully developing the mathematics of a moregeneral length function, four specific prob-lems of moving from the distance func-tionwith limited domain and restrictedapplicability in the world of mathematicalmodelsto the more general concept oflength will he highlighted.

Problem I. How can lengths of segmentson two different lines he compared? Infigure 4, is iitC7 greater than, equal to, orless than mAB?

Fig. 4

The nature of the problem requires thata relationship be found that will connectthe distance functions for line k and line 1.The critical characteristic of segments thatneeds to be established is that all congru-ent segments have the same measurewhether they are on a single line or not.We note that property 6 of the distancefunction is the parallel of this problem butis restricted to a single line. Indeed, as anidea, the single-line case is contained inthe new property applying to more thanone line. The particulars of the mathe-matics of establishing that the congruencerelation partitions the space into equiva-lence classes of line segments will not headdressed. Rather, the point of this dis-cussion is that the analogue of the mathe-matical problem is not intuitively obviousto all children as Piaget's seriation inter-vews show. This problem of mathematicsreeds to he addressed as a learning andteaching problem if children are to under-stand the concepts of length.

Problem 2. How does one find the lengthof a broken line segment'? That is, given afigure like figure 5, how can one assign anumber that will give the length?

114

A

A METRIC HANDBOOK 1:0I2 TEACHERS

13

1)-

Fig. 5

Ultimately the solution of this problemprovides the capability of computing theperimeter of polygonal figures. Bs using thesolution to the first problem and allowingourselves to add the distances for eachsegment, we can comput.: the length of thebroken segment. Careful treatment of thisproblem ..could require a mapping from theset of all broken segments to a single line onwhich we could apply the distance function d(see fig. 6). We would need to define ourmapping neatly for each segment so that themap of 13. the endpoint of segment .,113, andthe map of 13, the endpoint of segment BC.would fall on the same point a of line /, sothat the images of the segments overlap inno o; her points and so that the length of animage segment is the smile as the length of

A

the original segment. This is similar toassuming a nonstretchahks string is on thebroken segment ABCDE and is moved tothe line / on which we can apply the dis-tance function. We should restrict thisma pping to apply to only a finite number ofsegments constituting the broken segment.'leachers need to provide children with avariety of activities transforming brokensegments onto a coordinatiied line. Thismin as simple as constructing a brokensegment with strung-together soda strawsto he picked up and juxtaposed against anumber line or the reverse, which is picking

up a bendable number line and laying it on abroken segment. The experiences shouldaddress helping children acquire a feeling forthe conservation of length under the trans-formation, but it is much easier to focus onthe outcome, namely, that we can find thelength of a broken segment by summing thedistances of the segments making tip thebroken segment.

Pro/i/cm 3. How does one determine theminimal distance between two points? Infigure 7, is 4,1, C') < B) -1- d(13, C) forany point /3 Er. I?

The solution of this problem is generallyaccomplished by assuming an additionalcharacteristic for d, namely, for any threepoints of the space /1, R, and C. c/(/1, C;

/3) r d(B, C ). Note that thisstatement encompasses the single-line case.

Probicm d. the final problem is morecomplex. We would like to be able to deter-mine the length of curves in space. Givena curve r. as in figure 5, how can we deter-mine its length?

r

8

Solution of this problem is necessary ifwe are to be able to find the circumferenceof circles, the length of a portion of aparabola, and other such problems in

which the figure is not a broken line. Solu-tion of this problem requires a more refined

MET'RIC'ATION, MEASURE, AND MATHEMATICS 115

treatment of broken line segments in orderto generate the set of all broken line seg-ments that approximate the curve so thatlimit or bounding processes may be ap-plied to the set. Children can gain someinsight into the limit processes involved bycomparing the measure of a curve formedby a jumping rope placed on the floor withapproximations made by first using five-centimeter sticks and then using decimetersticks. A careful mathematical solution tothe measure of curve length may be foundin A. L. 131aker's Mathematical Concepts.of Elementary Measurement (1970).

This concludes the mathematical de-scription of distance and length. The de-scription depends on determining a char-acteristic function, the distance functiond, and then exploring carefully the prop-erties desired for this function. Primaryamong these properties were those relatingthe joining segments and addition, congru-ence and having the same measure, andfinally comparison.

The distance function and its definingproperties are ubiquitous in that theypermeate the world of this mathematicalmodel and their analogues are those usedand observed in the world of physical real-ity. They are obvious; but therein lies thedifficulty in working with the young child.As adults and teachers, we possess a globalgestalt of the distance function ,ind itsproperties. Since the analogues of theseproperties are present within the world ofphysical reality, it is easy not to give themsingly the attention they need if they areto be acquired by children. The implica-tion for teachers in planning for instruc-tion is that activities for each property needto he sought and included within the child'spreliminary experiences before stressingthe computational formulae for length anddistance.

The characteristic that has no analoguein the world of physical reality is that thereis a single, unique number called the dis-tance between two points in this idealworld of the mathematical model. How-ever, the physical world of observation

with the inexactitude of ruler placementand reading offers no such succor to thelearner.

Area

"lhe idea of area is mathematically simi-lar to the idea of length. There is a char-acteristic function for area just as forlength. It possesses defining propertiessome of which arc similar to the definingproperties for the distance function. Ana-logues of most of these properties are evi-dent in the parallel world of physical real-ity. These primitive, defining properties onwhich a full-blown area function is builtare often slighted by a too rapidly pacedinstructional sequence to the familiar form-ulae for figures such as squares, triangles,and rectangles. The primitive subconceptsconstitute the important intuitive founda-tion for the children's acquiring the com-putational formulae.

The domain of the area function is lim-ited initially to the most simple case,namely, the set of all polygonal regions.A polygonal region, such as the one dis-played in figure 9, is a closed, broken linesegment together with its interior. Thefigure may be cut up or partitioned into afinite number of trianguk.r regions by con-necting appropriate vertices with straightline segments. We shall label the set of allpolygonal regions I? and all elements ofthe set r.

Fig. 9

The area function A associates a singlepositive real number with each polygonalregion r. We write:

1(a). R . or

UN. A(r) r= 9i .

I I6 A METRIC HANDBOOK FOR TEACHERS

Among the several primitive, definingproperties for the area function that aresimilar to the distance-function propertiesis the one concerning the areas of con-gruent regions. For the distance function,we require that congruent segments be as-sociated with the same real number. Anal-ogously, the area function maps congruentregions to the same real number. We write:

2. If r, and r and r, r2, thenV) (/'.2).

For the distance function, if c/(A, B) =d(C, D), it could be concluded that inCD, but the area function does not behaveso nicely; equal areas are not necessarily theresult of the area function applied to con-gruent regions. A rectangle that is 2 units by8 units has the same area as a triangle with abase of 4 units and an altitude of 8 units.The area-measure characteristic functionpossesses unique defining properties settingit apart from other characterizing functions.

The area function, like the distancefunction, needs a mechanism for associatinga number with combinations of elementsfrom the domain. The join of two non-overlapping polygonal regions should havethe same area lumber as the sum of thearea numbers ( f the two regions. That is:

3(a). If r,, E R and r, and r, shareonly points if their boundaries, then

-L AO 2).

Piaget's observations of small children sug-gest that thei acquiring a feel for the sub-tractive vers'on of this property is a keyelement in heir forming an operationalfoundation f a- area. This subtractive ver-sion is, narrh Iv:

3 ( b ). A ( A (r, ) = A (r., ).

This proper y of joining regions and theprevious property concerning congruenceprovide a necessary, logical foundation forexamining area in terms of tiling, or cover-ing a large region with uniformly sizedpieces and counting them to ascribe a num-ber as area to the entire region.

The three primitive properties of thearea function that have been specified donot provide a means of associating a num-ber with a region. Instead, they providerules that this function must obey whengiven a rule for association. Typically, twodifferent rules for assigning numbers toregions are used and intermingled in texts.One is the unit approach, which is a

necessary component of the tiling, or cov-ering, approach discussed previously. It

simply identifies a standard unit with aparticular region. This is usually a squarewith a side of length one unit.

4(a). If a square. sh has side of lengthI, then A (s,) = 1.

This provides a means of assigning areasto all polygonal regions if suitable theoremsrelating diffetcm types of regions are de-veloped. We need, for instance, to relatethe area of a square with side different thanone to the area of the unit square. Theareas of other polygonal regions, such asrectangles, triangles, and trapezoids, needto he related to the square. The unit lengthof the side provides a means of tying areato the real numbers, yielding several niceproperties, such as order.

The tiling, or covering, approach basedon the unit measure ultimately forces thelearner to cope with problems of incom-mensurability. The fundamental idea ofcovering is simple and intuitive. This is notso for the ideas involving incommensur-ability: the concepts involved requireconsiderable mathematical maturity andsophistication. Rather than face these so-phisticated concepts directly, most textselect to step around the problem by in-corporating an equivalent approach thatprovides a more direct connection withthe real numbers. We state that for a poly-gonal region that is a rectangle:

4(b). If b is the length of the base of arectangle r and ii is the altitude, then/1(r) = 1)11.

This provides an immediate association ofall rectangular figures with real numbers

METRICATION, ME ,....)RE, AND MATHEMATICS

without our having to impost: theprocesses necessary in develorng version4(a) carefully. It should be rioted that4(a) ;Ind 4 (b) are mathematically equiv..-lent: one can he deduced from the odic:.

Three comments are in order. First, wetypically concentrate our instruction on thenumber that we a!is.ciate with each poly-gonal region rather than the functionor association itself. Classroom talk aboutthe area of a rectangle of sides six units andfive units identities the area as thirty areaunits. This labeling of the result of thecomputation, or the range value, as thearea function ignores the mapping. It is

appropriate to label the resultthirty areaunitsas the area, but the function and itsprimitive, defining characteristic are a por-tion of the foundational intuition that eachchild needs.

Second, the careful mathematician wouldwant to prove that if a polygonal region iscut up or partitioned in two different ways,then the same real number naming thearea results. For example, if the pentagonalregion in figure ID were to be cut up firstas shown by the dashed lines and secondas shown by the solid lines, the mathe-matician would deem it necessary that thearea function applied to the three triangles

I)

front the dashed-line partitioning give theSallie number as the area function appliedto the five triangles front the solid-linepartitioning. This desire for uniqueness on

117

the Dart of a mathematician appears strangeto many individuals because it scents soobvious based on our experience withmeasurement in the real or scientific world.It takes just so much paper to wrap arectangular box, no matter how we CIA itout (see lig. I I ). This is another case inwhich an adult assumption leads to ignor-ing the need to provide a child with foun-dational, intuitive experiences.

Fig. I I

Third, the choice of an area unit, be itexplicit as in 4(a) or implicit as in 4(b),is arbitrary. We could have selected a unittriangle, a unit hexagon, or even a circle,although we would ultimately have to dealwith the same problems for polygonal fig-ures. The choices of 4(a) and 4(b) havean advantage in that they provide almostimmediate access to solution of severalfundamental problems concerning the areafunction and because they parallel so di-rectly the instructional sequence of mosttexts, It should be noted in this contextthat Euclid's treatment of area did not in-volve a characterizing measuring functionthat depended on the real numbers. It alsodid not provide the same direct tic to themeasurement of distance on a line.

The domain of the distance 'functionwas extended to encompass length in theplane and the length of curved lines. Inan analogous fashion, the area functionrequires an extension of clonmin in orderto treat areas of closed figures that are notpolygons. We need to be able to apply ourfunction to circles, ellipses, cardioids, and

1 1 8 A M ETRIC IIANDROOK FOR TEACHERS

other such figures, and, as in the case ofthe length function, we need to also ex-tend the area function to treat areas notcontained in a single plane. As in the caseof the length of curved lines, the problemis solved by applying limit processes ap-propriately.

In nniiry, area measure is a functioncharacterized by mapping closed figures tothe positive real numbers in such a waythat congruent regions have equal areas.A joining of a finite number of nonover-lapping regions produces an area equal toadding the areas of the regions, and a rulestating specifically how the real numbernaming the area is produced. As in the

ease of the length function, the primitive,defining characteristics are a necessary partof the foundational intuition that childrenneed. These defining characteristics are soobvious to experienced adults that they arefrequently overlooked in designing instruc-tional sequences. The p-imitive, definingcharacteristics are paralleled in the real

world that the mathematics models.The properties that were encountered in

area and in length are nicely parallel. Ineach case there is a defining function. Eachfunction has several characteristics in com-mon (see table I ).

Volume

fhe mathematical model for the measureof volume shares many characteristics of thedistance and area functions, Volume measureis a function C that assigns to measurablesets in three-space a positive real number.

Table 1

1)1%1a/it e

Children need experience with primi-tive subconcepts of volume measureparalleling those appearing in th,2 t hieabove, namely the concepts of congruence,additivity, unit, and comparison. Ratherthan explicating these concepts in terms ofthe characteristic function V, it suff.ces tonote (I) that children need experiencesdirected toward the attainment of theseprimitive subconcepts and (2) ,hat theproblems children encounter closely parallelthe difficulties with the analogous sub-concepts in other measure systems. Itshould he noted that children acquire thesesubconcepts for different measure systems atdifferent points in time mos( children donot attain additivitv (com.ervation) forvolume at the same time they attain addi-tivity for distance.

The experiences that have been foundto facilitate children's attainment and ac-quisition of these primitive subconceptsdemonstrate a close comparability to thoseneeded for the analogous concepts in othermeasure systems. The use of blocks tobuild volumes in a fashion similar to theuse of area units as coverings for polygonalregions can strengthen children's intuitionsfor the unit, additivitv, and congruencesubconcepts.

Volume does present some unique prob-lems corresponding to one subconceptwith no exact analogue in length or areameasure, The problem is the measure ofirregular volumes. Nice parallelapipcdspresent relatively minor difficulties mathe-matically. The characteristic volume func-

it-err

I. additivity The join of two nonoverlapping ,CgMent,ha-, the ,a me measure as the our or themeasures of the individual segments.

2. taut The eoordinatiiing of the line providesthe unit needed to assign a measure to asegment.

3. comparison Segment .-I segment 13 mean,< dint from the triangle inequality.

1. congruence Congruent ,contents have the samemeitsure.

The join of two nonoverlapping regionshas the same measure as the sum of themeasures of the individual regions.A unit is needed to assign a measureto a region. It is Usually derived from arelated distance function.

region Y means< Et.

Congruent regions have the sameneasure.

ETRICATION, MEASURE, AND MATHEMATICS 1 1 9

tion assumes its uniqueness through theprinciple of Cavalieri. Cavalieri's prin-ciple may be exemplified as follows: Sup-pose you have a deck of playing cards as

pictured in figure 12(a). The deck is

-pusher as shown in figure 12( b).Cavalieri's principle asserts, informally,that although the conformation of the solidhas been changed through parallel dis-placements of the points in planes, the vol-ume is unchanged.

Fig. 12

This primitive concept is necessary forthe full development of volume measureand for the deduction of computationalformulae for many shapes, but note thecomplexity of the statement, Fortunately,the concept is more simple for children toapprehend than to state. It is a Piagetianconservation concept. Children need expe-rience in deformation of solids embodyingthe Cavalieri principle. For instance, takea Slinky (one of those coils of spring steelthat children like to run downstairs) andfill it with peas. Remove and count thepeas. Deform the cylinder of the Slinkyby pushing, and see whether it will holdthe same number of peas (he sure theSlinky does not stretch in length). TheCavalieri principle provides a unique con-servation principle that children need toacquire. Ultimately, the Cavalieri principleis realized in terms of being able to find thevolume by multiplying the area of any crosssection parallel to the base of a cylinderby the cylinder's height. Similar to areameasure in that the volume measure de-pends on measures of lesser-dimensionedspace, the Cavalieri principle provides thetie to the area and length functions.

The treatment of volume suggested bythe foregoing differs substantially from

what happens in many elementary schoolclassrooms today. It suggests considerableexperience in examining rectangular solidsformed by stacking blocks and determin-ing their volume. It suggests deformingthese volumes in prescribed ways to em-phasize the subconcepts. Most early in-struction in volume, however, coniesthrough experiences with liquid measure.Experience with liquid measure is impor-tant, but children need experience withboth volume of solids and volume ofliquids. At some point, the child will needto relate liters to cubic decimeters. Theconcept of liquid-volume measure appearsto be easier for the child to acquire. Thejunior high school child frequently findshimself being introduced to volume ofsolids with little intuitive experience exceptfor the presumption that volume of solids''behaves'' quite like area. The child is fre-quently left to his own devices to bringliquid-volume measure and solid-volumemeasure into a common system.

Angle measure in the plane

.\ final example of a measure functionin the sense of the mathematician's niceworld of exactitude as opposed to theworld of practical measurement is that ofangle measure. The measure function forangles is considered because history andtraditions confuse the definition of themeasure function, as well as the fact thatangle measure is more complex.

The length function and the area functionare intuitively more obvious than anglemeasure for several different reasons. First,it should he noted that we can talk naturallywith precision about length and area. Theword length denotes an attribute of a linesegment. The word area denotes an attri-bute of a region that is different than theregion itself, but in the case of anglemeasure, the word angle is used for the angleitself and for the measure of an angle. Oftenyou will encounter statements such as ''Theangle is twenty-five degrees.'' We can speakof the area of a square and know we arereferring to the measure function. However,

I 20 A METRIC HANDBOOK FOR TEACHERS

not having a special word for the measureattribute of angles, we need to use a phraselike (nigh. measure in order to remove theambiguity of confusing an angle with itsattribute. Some texts attempt to remove thisambiguity by writing ABC for the angleand in 4: ABC for the measure of r ABC.This helps, of course, but does not removethe problem of ambiguity when teachers andlearners are operating in an oral mode.

Second, the characteristics of the angle-measure function depend on the nature ofthe definition of the domain and its ele-ments. It would be nice if we could simplysay the domain is the set of all angles inthe plane, but the history of mathematicsindicates that there are at least a half-dozen alternate ways of defining angle.Angle has been defined by saying it is therotation of a ray around its endpoint, bysaying it is the wedge consisting of all thepoints of two rays with a common end-point and all of the points of the rays inbetween, and by indicating that if two rayswith common endpoint lie entirely withina half-plane, the angle is the set of pointsof the rays. Many of the alternative defini-tions are to he found in currently usedtexts. Corresponding to the choice ofdefinition of elements of the domain is a

set of intuitions that children need to ac-quire in order to possess a "feel- for theangle-measure function.

Finally, the nature of the range of theangle-measure map has a potential forproviding confusion. The student of tri-gonometry., in coping with Dcmoivre'stheorem, may want to distinguish betweenangles with measures of 75'', 435", and800- . From our experience we know anglesof measure 75" and 285" are remarkablysimilar. (Is this a problem of the natureof the domain or of the range of the func-tion?) A long, historic tradition of usingdegrees as the unit of the range set exists,when for many problems it is more nat-ural to use radians and provide a moredirect tie to the real numbers.

Rather than carefully describing thecharacteristic function for angle measure

and becoming hogged down in the prob-lems discussed earlier, the following sec-tion highlights some of the desirable prop-erties for the function. The function shouldmap into the set of real numbers suchthat

I. Every angle shall have a measure.

2. Congruent angles have the same rangevalue.

3. If' two angles are adjacent (that is, theangles have a common vertex and side butare nonoverlapping), then the sum of theirmeasures should be the same as the mea-sure of the angle formed by their "outer''rays.

4. Given the measures of the anglesformed by three rays from a common end-point (see fig. 13), it can be determinedwhich ray falls between the other two. Thisis to say, given the measures of AOC,

BOC, and 4.40B, selection of one of4- 4-AO, BO and CO as falling between the othertwo should he apparent.

AC

Fig. 13

5. Given the measures of two angles, itcan be determined which angle is larger.

6. A unit measure for angles needs to hedetermined.

Most of these properties have an analoguein the length function and area function.

The primary difference in the measure-of-angles function comes from the fact thatwe generally prefer that the range bemodulo a real number, If degrees arc therange unit, then the angle measure A is ex-pressed in terms of A', a real number lessthan 3(4).

A'(mod 360)

METRICATION, MEASURE, AND MATHEMATICS

This is to say, A' A n,360 whereintegers and 0 < A' 360. In terms of

radians, A --- A'(mod 2rr). It should henoted that a ratio-scale transformationconverts the degree-unit scale to the radian-measure scale.

Each of the properties discussed abovewarrants the attention of the teacher or theinstructional-matrials designer in order tobuild the intuitive foundation from whichtie learner operates. The building of theintuitive foundation is more complex in thecase of the angle function than in the case ofthe length function or the area functionbecause the attributes of the domain set andthe range set are affected by the choicesestablished in respect to our traditions andhistory.

Angle measure is also complicated by thefact that alternative natural means ofaccurately describing angles and theirmeasures exist. H. is quite reasonable todescribe angle measure in terms of ratios.For example, consider L A BC in figure 14.Some young children appear inclined to

C,

lk

Fig. 14

find a point x a specified distance from 13and then to determine a length k on a lineperpendicular to A13 through .v. Thisprocedure could be used to measure allangles, but it poses some inconveniences.This method. similar to a carpenter's use ofhis square to measure angles, is a naturalalternative to measure in the degree sense.

Mass

Understanding of mass is more com-plicated for children than are di' under-standings associated with length, area, vol-ume, and angularity. The crux of the mat-ter is perception. To hold two objects in

121

your hands and say which has the greatermass is difficult. ht determining lengthcomparisons, the child can simply placeone object against another and look. Ap-paratus is required to provide the childwith the perceptual refinement to gen-erate his mathematical model for mass.The child's perception seems a step furtherremoved from the model for mass that heis building than in the case of these othermeasures. Two words, equilibrium andcomparison, serve to categorize sonic ofthe perceptual difficulties involved.

Equilibrium is a word describing thesystem of balance necessary to the mani-pulation involved in finding the mass ofan object. Given a quantity of flour on onepan of a beam balance, as shown in figure15, the learner must established a balance

Fig.

by adding standard masses to the otherpan. Determination of the equilibrium in-volves perceptions of the objects or theirattributes that appear indirectly related tothe original quantity of flour. For one thing,the mass added to the nonflour pan is per-ceptually distant from the flourhow canthe mass added to the nonflour pan be con-cerned with an attribute of the flour? Also,the determination of the equilibrium statehinges on the marker on the scale at thetop of the picture, but this marker is not adirect measure of the massa number tobe part of the mathematical model--but anindication or system of whether a state ofbalance is achieved. This indirectness ofrelation of the underlying perceptual baseto the mathematical model is unavoidable.It introduces a complexity into the learn-ing processes. Developmental psychologistshave shown that children's acquisition of

122 A METRIC HANDBOOK FOR TEACHERS

equilibrium ideas comes quite late in theirconceptual maturity.

The second term, comparison, is anotherconcept that is required of children whenbalances are used to provide the funda-mental perceptual data for building themeasure concepts for mass. These ideashinge on the child having a usable sense ofthe transitive property, "a < b and h <c < c," the heart of the comparativeprocesses. The child should clearly en-counter the transitive property in measurecontexts other than mass; this would givethe child a background on which the teachercould build in preparation for transfer tothe context of mass. Nevertheless, becausethe comparative aspect is so fundamental tomass and is so late in reaching fruition in themind of the child, it demands specialattention by the teacher. Confounded by theperceptual difficulties of a balance system, itprovides the child with significant problems.

The mathematical model for mass gen-erated from perceptual data by the childis very similar to the models for length,area, and volume. Again, the importantsubconcepts desired to characterize themass-measure function arc congruence, ad-ditivity, unit, and comparison. Acquiringthe comparison idea appeals to he at theheart of children's difficulty. In passing.one should note that weight measure pre-sents the sante order of learning difficultiesas mass measure. Mass is a more generalconcept than weight. since weight varieswith variation in the strengths of gravityfields. However, since it is hardly prac-tical for children to compare the weightsof a given mass in different gravity fields,there is probably little to be lost by usingmass and weight interchangeably duringthe early years of school. The nu, --weightdistinction is grounded in scientific prin-ciples rather than mathematical constraints.and its placement in the curriculum is

within the domain of the science educator.

Indirect measures

The importance of mass and weight tothe mathematics curriculum is that it in-

troduces the concept of perceptually in-direct measure. Length, area, volume, andangles are measured with units that arelengths, areas, volumes, and anglestheunit look like the object being measured;but standard masses seldom look like theobjects on the opposite pan of the balance.

Measuring mass is also indirect in thesense that an intermediate device, the bal-ance, is used in the measurement. Unlikerulers or protractors, which simply providecopies of the units used in the measuringprocess, the balance serves to magnify oursenses. Here is the first introduction of ameasuring tool or instrument.

Measuring instruments may also mea-sure in yet another indirect way: they maymeasure effects. In a zero-gravity field,mass may be measured by noting howmuch a spring s stretched when it pullsthe mass with a given rate of acceleration.Temperature is measured indirectly bynoting the effect on the length of mercuryor alcohol in a narrow tube. Electrical cur-rent is measured by the force it effects onparallel wires.

Thus, measurements may be indirect inthree different ways: perceptually differ-ent units may be used, measuring instru-ments may magnify senses, and measuresmay he made by noting effects on otherobjects. The extension of measure to in-direct factors calls for a blending of bothmathematics and science. The propertiesof mathematical measuring frictions muststill be preserved, but properties of scien-tific measurement must also be considered.

The measure functions length, area, vol-ume, angle measure, and mass have beenconsidered from a mathematical point ofview. For each it has been observed thatonly one measure corresponding to a givendomain element exists. Further, some cha r-acteristic primitive subconcepts exist foreach function and are necessary if a learneris to build a functional gestalt. Many ofthese characteristics, such as the equal-measures-for-congruent-domain elements,additivity, and ordering, appear commonto all the functions.

METRICATION, MEASURE, AND MATHEMATICS

Measure in mathematics has a key ele-ment of exactitude that is missing in scien-tific or practical measurement. It is meas-ure in the ideal world of the mathematicalmodel of physical entities. This point ofview concerning a distinction betweenmeasure in mathematics and in the physicalworld is relatively recent. A mathematicianor scientist in the age of Euler would nothave worried about the distinction; indeed,he might not have considered it valid.Freeing the mathematics of measure fromphysical reality awaited a careful formula-tion of limit processes and the arithmeti-zation of analysis, It should be noted, inpassing from this discussion of mathe-matical measure functions to the follow-ing discussion of measurement processesin science, that not all is simple. We havenot attempted to argue the separation butrather have assumed it as a foundation ofour discussion. The philosophical argu-ments are not simple and are beyond thescope of this document. The interestedreader may turn to Eddington (1939),Bridgman (1938), and Churchman andRatoosh (1959) if he is inclined to pursuethese distinctions further. The point of thedistinction is this: the learner needs a dif-ferent sort of intuition when playing in thegarden of exactness than when workingwith the processes of measurement. Some-times these differences are confusing to,and confused by, children.

A concluding point about measure inmathematics is in order before turning tomeasurement in the world of the scientist. Amodern mathematician often considers aspace of objects or entities to he a measureor metric space. A space is defined byMcShane and Botts (1959) to be metric if itis a set s with a function in meeting thefollowing minimal set of conditions where.v, 5:

1 . in( s , ) It

2. (.s 1) m(.v, r') m(y..v) >

3. In( r. I.) + > s. ::1

123

Each of the exampleslength, arca, andangularitymeets these minimal criteria;hut, in addition, because of what our in-tuition and experience indicate is usefuland productive, we 'lave imposed addi-tional characteristics on the characteristicfunction and. inC;:ed, on the entire space.The conditions for a topological space tohe metric are powerful in that they ac-commodate many different characterizingfunctions. Perhaps these three character-istics nei.:1.1 to he stressed above all else,but that alone would not be sufficient toprovide the typical learner with a sense ofcontrol for all the measure systems neededin mathematics. One must cope with thecharacteristics of each metricizing functionin the sense of what gives it a uniqueness.

Measurement in science

The measure concept of mathematics isuseful to the scientist. It is his model forthe prnctical world of observation. Thescientist should understand the conceptsinvolved, for he needs to fit his observa-tions of reality to the mathematical model.In many respects the scientist's world is

more complex. It involves more ambiguity,relies on his observational prowess, andrequires some skills not necessary in theclean world of mathematics. Measurementfor the scientist contains the measure ofthe mathematician and much more. Fol-lowing are examples of measuring scalesin science offered as examples of the differ-ences.

The oeololffst uses relative hardness ofminerals as one tool to help identifyminerals. A mineralogist, Frederick Mohs,suggested in 1822 a hardness scale. Fol-lowing is Mohs' hardness scale:

1. talc (i. feldspar2. gypsum 7. quartz3. calcite 8. topaz4. fluorite 9. corundum5. apatite 10. diamond

To help identify a mineral X, scratch it

with each of the identified elements of the

1 24 A NIEFRiC HANDBOOK FOR TEACHERS

scale. Most steel will not he scratched byapatite but will be scratched by feldspar.The hardness of any mineral can be de-termined this way. There are intermediatestages or; the scale; a fingernail typicallywould relJster about 21'2. To each materialcan be :,signed a scale number by a proc-ess quite similar to the old rock, paper,scissors Lame. flowever, there are prob-lems: some minerals have one hardnessalong the axis of their crystal structure butanother across the axis kyanite) ;therefore, the apparent characteristic func-tion of the metric system simply is not afunction. Further, there is not a nice pro-portionality in the scale. The relative dif-ference between the hardness of diamondand corundum is not the same as betweenquartz and topaz. Indeed, the numbers donot provide the base for operations andcomparisons between points on the scalethat would make for a "nice" measurefunction.

For the geologist Molls' hardness scaleprovides a useful measure system. It is

also a prime example of the problems ofobservation in scientific and practical meas-ure. It does not yield a predictive mathe-matical system. It provides a scale forcategorization and for taxonomic purposes.In terms of our initial description of therelation between the physical world andthe model world of mathematics, the op-erations of scratching and the perceptionof scratching provide the "fit" for :I weaklyordered set that is the model.

(-SIoilef: partially ordered ,e1,1

fittin ,:eratching and ollervine

(Set of ininerIT,1)

The mathematics of the model does notprovide a rich base for prediction. Molls'hardness scale depends almost totally onobservation. It is not a ratio scale, as wasconsidered in conjunction with the scalesused for measuring temperature. Ratios of

distances between relative hardness pointsare not preserved when multiplied by posi-tive numbers.

Molls' scale is an example of anotherbasic category of measure scales, namely,the ordinal scale. It merely assigns a rela-tive order to entities. The large majorityof percentile achievement scales used inschools are also ordinal scales. If one stu-dent scores at the eightieth percentile ona standardized mathematics test and an-other scores at the fortieth percentile, youcannot state that one knows twice as muchas the other; at best you may concludethat one student achieves better than theother. That is, an order has been assignedto their performance on the test.

Nominal scales provide a third categoryof measure function that contrasts sharplywith ratio scales and ordinal scales. Sup-pose a set of scientists was partitioned intothe four categories of mathematicians,physicists, chemists, and psychologists. Ifeach of the four subgroups were countedand the number in each category assignedto the name of the category, a nominalscale would have been created. This typeof measure function is the weakest of thethree types of measure scales in terms ofits inherent mathematical characteristics.It is very useful for many kinds of in-formation-gathering and organizing tasks.

The Richter scale, invented in 1935, isused to measure the magnitude or theamount of energy of an earthquake. TheRichter scale is the range of a function thatassigns a number to the maximum ampli-tude of an earthquake's shock waves. Thelo2arithm of the maximum amplitude is

used to assign the number. Suppose thegraph in figure 16 is the record of a shockwave observed at a seismographic station.The log, (a) 3 = Richter number, ifa is the maximal amplitude recorded inthe course of the earthquake for a givendistance from the earthquake. Richter scalevalues are dependent on the distance tothe epicenter of the earthquake. Empiri-cally derived tables arc used to take intoaccount this distance factor at different

NIETRICATION, MEASURE, AND MATHEMATICS 125

observation posts. Now a is an observa-tion. It depends on standardized measure-ment recording devices. Taking the log ofthe measure is not necessary unless thescientist desires the convenience of beingable to graph on a reasonably scaled andsized piece of paper. Thus, at many pointson the practical side of measuretnent, thearbitrariness in the selection of units is

shaped by the conditions of measurementand the convenience of the measurementdevices. Note that an earthquake of Richtermagnitude S is actually indicating an en-ergy of 10,000 times the energy of anearthquake of magnitude 4. Further. be-cause of the conservative factor of measur-ing only the maximal amplitude rather thanall the cnergy released by the earthquake,it is estimated that an error factor of 10:is introduced. There is, indeed, no exact orprecise measurement of the total energyreleased by an earthquake involved; rather.calculated and refined estimation of a sin-gle symptom of an earthquake's energy isgiven by the Richter scale. A precise andcomplete measurement model for the totalenergy release of an earthquake does notexist, but the Richter scale is a sufficientapproximation to be a useful, predictivemodel.

The Molts' hardness scale and the Richterearthquake magnitude scale are not partof the everyday life of most learners. Theyprovide nice contexts for study of the roleof approximation and the arbitrary natureof assigning scale or measurement valuesto physical phenomena. They are relativelyfree of the confounding effects on learningprovided by prior measuring experienceand linguistic usage. Thus, tl,ey differ fromthe more commonly used base for teaching

- Time

concepts and processes of scientific meas-urement. Length, area, volume, tempera-ture, weight. time, and mass are each usedmuch more extensively to establish con-cepts and processes of scientific measure.The measuring of length and area aretypically used as an instructional base forestablishing key concepts and processes ofscientific measurement with the youngchild.

One characteristic of scientific measure-ment is the idea of arbitrariness. Withinthe measuring context, the measurer mustselect an appropriate unit of measure. Twoconcepts need to be established: first, thatthe measurer has a choice, and second.that some choices are better than others.In the measurement of area, I could electto cover a region with triangles, withsquares, or even with circles. My choicewould depend on what 1 use for my mathe-matical modelwhich would fit hest withthe rules of operation that I desire to useand the type of predictions I want to make.My choice of unit may also depend on thekind of instruments I have available. If I

have a rule scaled in decimeters, then I

probably would not elect to measure interms of centimeters. (This provides a

practical yet compelling reason to replaceEnglish-system measurement tools withmetric devices.) For most measure sys-tems, tradition and obvious convenienceestablish my selection of type of unit. Afterselecting the type of unit, I need to selectthe .rize of unit. I would not elect to use akilometer scale to measure the length ofa room nor a centimeter scale to measurethe distance from Anchorage, Alaska toNew York City. Judgment depends onconvenience and the error factor that is

126 A METRIC HANDBOOK FOR 'TEACHERS

acceptable. The point is this: the selectionof a unit for measuring depends on thecwice of the measurer: it is arbitrary. Inworking within the English system, thematter of arbitrariness appeared to be easyfor the teacher to establish. The same sys-tem- -feet, inches, yards, rods, and miles-seemed so different ( because of the hor-rendous conversion computations) that thechoice factor in sk sting an appropriatelysized unit seemed relatively clear, Withconversion to metric, will it be so easy toestablish arbitrariness and choice of themeasurer as a factor?

Unit selection is not necessary for a

mathematician to have metric space.However, many of the metric functionshave a unit as an important characteristicin determining their nature. The Archi-median characteristic of length and, moregenerally, the tying of length to the realnumbers, implicitly specifies a unit forthe length function. Note, however, thatthe scientific process of measurement con-tains the characteristic of providing theuser a choice of picking size. The selectionof the size of the unit is within the mathe-matical model but does not depend on anyexternal characteristics, such as instru-menN used or what is to be measured.

Another characteristic of the scientific-measurement process is the estimation andapproximation process. This characteristichas several facets. First, and particularlydistinctive, is the fact that measurementscales are based on rational numbers. Themathematician will state that the distancebetween two points is exactly y2 units: butrulers arc designed to measure units andfractional parts of units. Indeed, limitationsthat are imposed by the selection of measur-ing instruments and by the base unit meanthat limit processes cannot be used arbi-trarily to approximate a length that is anirrational number. Necessarily, measuringentails approximation and error, and thelearner must become aware of this funda-mental fact of lift.

Historically, measurement evolved as acounting process. Whole numbers were

used to describe the number of units ofmagnitude necessary to measure an object.Indeed, we could have included measure-ment systems based exclusively on whole-number counting in our discussion ofmeasure. The evolution of measurementconcepts in Greek mathematics can beconstrued as leading from counting to ratioconcepts and ultimately, in the idealizedPlatonic mathematics of Eudoxus andEuclid, to modification of the ratio andproportion to remove the ambiguity ofincoo'..nsurable measures. Hence, in a

re ai !. coping the practicalitiesof measure established the ideal world ofthe mathematics of irrational numbers.

The measurer should also recognize theimplications of error being part of themeasurement process. If a measure in-volves error, then any operations on thatmeasure apply to that error. Learners needto develop an awareness of this, If thelength of an object is observed to be 20cm with an error of 0.05 cm, then if themeasure is multiplied by five, the erroras well as the measure is multiplied by five.

The above discussion of arbitrarinessand approximation has not developed theparticulars of all that is involved. Rather,the point is that among the goals of teach-ing measurebe it in the context of mathe-matics, science, or practical applicationthe teacher needs to help learners realizethat there is more to the process of mea-sure than the ideal world of the mathe-matical model indicates. Error and ap-proximation are part of measurement.

Many of the processes of measuringtaught in school mathematics address prob-lems of improving accuracy or reducingerror of observation in a systematic way. Astudent may understand the measure con-cept of length but may not yet have realizedthe necessity and advantage of carefulplacement of the zero point of a ruler or thezero ray of a protractor. Although thelearning of the metric concepts and thelearning of the observational skills typicallyprogress together, these distinctive observa-tional skills require careful attention in

METRICATION, MEASURE, AND MATHEMATICS 127

their own right. Judging a length in terms ofthe confidence interval derived from theunit of observation is also a portion of thescientific skills and concepts used to controlerror ,)r accuracy. That is, to judge thatpoint B. the endpoint of .771 in figure 17,

falls within one unit around 12 is not re-quired by the mathematics of measure butis required as an integral part of the measur-ing process in science. Although significantdigits, relative error, absolute error, andmany of the processes used to identify andcontrol error have nice mathematics withinthe operations and the background con-cepts, it should he recognized that these arenot part of the foundations of measurement;rather, these processes are dictated by thenecessity for a rationale for processes andskills directed toward improving the ob-servational processes. This is not to say thatthese concepts and skills should not headdressed in the mathematics classroom.Rather, it is to say that children may heconfused in acouiring understanding of themeasure Cum.' and control of theobservationl pr eses and skills if theirencounter vith two concepts are notkept clear ,Aul distinct.

Ruler

II 12 13

Fits. 17

Klopfer's (1971 ) statement of a tax-onomy of educational objectives for theteaching and learning of science recognizesthe discreteness of observational and mea-suring processes. Tying these processes touse of appropriate language, selection ofappropriate instruments, estimation ofmeasurements, and recognition of limitsof accuracy, Klopfer is careful to removemodel characteristics and formulation fromthe observing and measuring processes.Herein lies a problem: for the novitiatedeveloping a measurement system for, let

us say, length by an inquiry or discoveryprocess, it is clear that the length functionis part of Klopfer's separate and distinct"model characteristics and formulation."After the length function and its char-acteristics are acquired, is this a part ofa theoretical model to be tested and, per-haps, reformulated? Clearly not. But, inthe teaching of science, it is not preciselyappropriate to label the length functionas a portion of the measurement instru-ment. Even in talking about processes ofscientific inquiry, some knowledge shouldbe accepted as such. More importantly,the teacher of science needs to think care-fully about the mathematical modellingaspects of his science, recognizing thatnot all models are in the process of for-mulation even in the teaching of inquiryprocesses.

A final word concerning science, mathe-matics, and measurement is in order.The measure functions of the ideal worldof the mathematical model are firmlygrounded in the world of the practicalitiesof measuring. The models are refinementsof what was and is observed in manipulat-ing real objects. The process of relatingactions or operations with objects tooperations in the mathematical realm is

a powerful idea of mathematics andmodel building labelled with the wordhontomorphism. A simple example of ahomomorphism is that of sets with theoperation of union and numbers with theoperation of addition, as shown in figure18, A homomorphism is a function thatrelates operations within one set to op-erations within another set. This is pre-cisely the structure that relates using realobjects in building a model of operations

such as joining line segments end toend for determining length ) to using thecharacteritics of the length function in

building a mathematical model of thelength function. Clearly, children need toacquire concepts, skills, and intuitions rel-ative to the real world of operationswithin the physical and manipulative con-text; this supports and extends the in-

12S A N11TRIC HANDBOOK FOR TF...1cut Rs

t .1

H

.1 a 1?

LUtin!tr)ci

fatictien

ail?)fltadt1:.):1

coor,t,ngc(.1 13)=c(.1)-4-6(1;)-

tuitions ,,ithin the model world of mathe-matics. Hie model \Amid is. to a large

world of symbols ;Ind operationson those smhols, each operation possess-ing an analogue or p;11.:111C1 in the physical11011(1. [he building of homomorphismsbetween the physical world and the con-structed world of the mathematical [nodelis the heart or the ,cicutilic-inquiryprocesses so critical to the developmentof science. Klopferts higher-level ohjcc-ti\es for inquiry teaching and learning arc,indeed. succinctly summari/cd h\ savingthat the ta \onoinv is directed to huddin.2,"morphisms- in a calculated way. Mathe-matics teachers can support and extendthe development of intuitions for 1/4:icrititic

inquiry h\ maintaining a stress on thehomomorphie or model- building aspectsof measure. Ithis is to say, a mathematicsteacher should constantly stress the dif-ference hetween the physical world \\ ith itsoperations iind the model world with its

operations. \ it the same time rellect-ing that the operations arc mirrored in

each Other. ('ltlICCI)H, skills. and intuitionsmust he developed with `,0111c 1.1101*(1111111CS,

in cacti '0,'ent, for learninr in one ,ternI, supportive of lcarning in the other.

Teaching measure: problemsand activities

In the lort.going section, measure wasevunined from the mathematical point ofview. The purpose of this section is to

pcutigo.i...Jciii approaches to mea-sure front the vantage point of instruc-tional ;Ind pschologieal research.

Three remarks are in order before wehegira. 1:11.,,t, the research 1'111(11112S of the

child-de\ elopment psychologists are gen-erallw with the orientation toprimitive subconcepts for each character-istic measure function. The problems inconcept acquisition identified by such re-searchers as Plage!. Inhelder. and Szcntin-ska ( 1960 ) Smedslund I 1963 ) Stefieand ('arew ( 1971 ), Lovell ( 1971 ), Skenip(1971 ). ;Ind Sinclair (1971) lend tolocali/e un the child's failure to incor-porate ;1 suhconcept into his cognitiveschemata for measure.

Second, the word 11111lilion was usedfreely in the preceding section. This sec-tion reflects an orientation toward thechild's constructing perceptual and con-ceptual intuitions based on his interact is

with the physical environment or vt ; . thehomomorphic world of mathematicalstructures :old operations in mciisurement.The components of the homontorphismprovide the base for the intuition. Indeed,intuition in the homomorphic structure ofmeasure is similar to what psychologistslabel ith the word transfer. It differs fromthe traditional views of transfer in that theThorphism- prmides an orienting connec-tion hemeen the structures..lhe functionalconnection is stronger than analogy, CM11-

elements, and other mechanismsidentified is facilitating or establishingtransfer. Transfer is, in a sense, "rigged''or -wired- by the tight, functional relation-ship between the domain and range setsand their respective operations. Intuition.then, is constituted of concepts learned oneach of the "sides- of the homomorphism

the real world of manipulation of physi-cal entities and the world 01 the mathe-matical model.

NI ET I(ICATION, :\ NIATIIF N1ATICS

Third, ideas and concepts Of 111C;ISMC-111CIII ;ire not unique to the particular char-acteristic-function system. There is a

parallelism between systems of measure.In some cases the parallelism is weak:there is little similarity between the struc-ture of Mohs' hardness scale and the mea-sure system for volume. In other cases thesimilarity is strong: in the Finglish system,foot-pounds and 1311'; ''behave'' in thesante way. N'1;1111,' Of the primitive sub-concepts for length have an almost exactanalogue in :trea measure and in volumemeasure. In the traditional sense, theparallel structures indicate transfer to hetin important instructional goal. On theone hand, it appears surprising that trans-fer is seldom an outcome of instruction,given the several measure structures thata child encounters in the elementaryschool. He common elements for themeasure structures----such as additivity,units, congruence, and comparison----wouldappear to facilitate transfer, but childrenand adults often need to relearn these com-mon characteristics of measure systemswhen cncounterin,, a new system of mea-sure. On the other hand, lack of transfermay not he surprising but should he pre-dictable from what is known ahout howchildren learn. .1-he common elements arcsystem ideas----part of an interrelated com-plex that provides the foundation of 11 lca-sure structures. [he child's capability forhandling and usinl, measure structurescharacterizes the mature learner who hasattained the formal reasoning stage of thePiagetian model of cognitive development:perhaps curricular designers and teachersexpect transfer hefore children are ready.Although children encounter each of theideas before attaining the formal reasoningstage and can indeed use the conceptsseparately, can they use them in the struc-tural sense? Perhaps consideration oftransfer as a specific objective of instruc-tion should he delayed until the child at-tains the maturity of Ihe junior high schoolyears and then he addressed with con-certed attention. A potentially signilicant

129

hypothesis concerning the teaching of mea-sure is whether children who are at thePiagetian formal reasoning stage can heled IlloCe easily to transfer concepts fromone measure structure to another if theprimitive subconcepts are emphasized ininstruction for children at the preformaloperations stages of development.

The parallelism between measure func-tions discussed in the preceding para-graph presumes that the functions arelike. .I'he parallelism facilitates transfer

between ratio-scale measure functions.Ratio-scale measure systems possess themost powerful inherent nlitheinatiealstructure and ;ire, consequently. the mostused and pervasive of measure systems.All metric scales are ratio scales. Someuseful metric measure scales are, of course,derived scales in that they are combina-tions of directly measured attributes.Velocity is such a derived scale, since it

is the composition of a distance functionand ;t time function. For derived scales,the ratio characteristic may be obscured:a teacher may therefore want to presentlearners with situations in which all butone of the measured attributes are heldconstant in order to highlight the ratiocharacter of the derived scale.

But not ;III measure functions and scales;ire alike. Previously, we examined ordinalscales, such as Molts' scale, and nominalscales, such as those counting settles de-rived from categorization of sets. "theseprovide the teacher negative instances toincorporate into instruction for the pur-pose of highlighting important character-istics of the ratio scales.

The purpose of the next section is toidentify some trouble spots in instructionconcerning measure ;Ind measurement. Theinstructional problems identified are allconcerned with ratio scales, in recognitionof the pervasive character and importanceof ratio scales. The fact that metric measureis ratio-scale measure attests to the impor-tance of their limitation. Although the dis-cussion of 111;111V of the problems is specificto a single measure context, generally the

130 A METRIC HANDBOOK FOR TEACHERS

discussion may be extended to any ratio-scale context.

Specific instructional problems

Children have difficulty incorporatingthe idea of unit into their cognitiveschemata for measure (Montgomery 1972).Precise information concerning the diffi-culties with the idea of unit is far fromcomprehensive--more basic research con-cerning children's development of the unitconcept is needed. There is some evidencethat children benefit from early, extensiveexperience with unit-domain elements fora given measure function before emphasisis placed on the fact that the measure func-tion maps the domain element to one..=man's article, which appeared in the April1974 A riamietic Teacher, concerns geo-hoards and the area function; it demon-strates an approach emphasizing the unitas providing a base for covering an areabefore stressing the map to the range ele-ment I. Note that this approach can alsobuild to the Archimedian subconcept, andwith the incorporation of counting activi-ties, it can build to a child's acquiring asense that each domain element maps to anumber. Comparable manipulative activi-ties are readily invented for linear andangular measure. The child has experienceswithin the context of the physical worldarid acquires a manipulative base for thehomomorphic operations with the range ofthe function.

The child's progress toward acquiring aconcept of unit has been shown to he im-proved if the instructional strategy encom-passes examination of two measure systemsthat possess unit-domain elements. Mont-gomery has stated that children's under-standing a unit length is enhanced byexperience with area units and vice versa.She has also demonstrated that childrencan acquire this subconcept as early asgrade 2.

The payoff of examining with childrentwo measure systems possessing units sug-gests there might also be an advantage in

comparing measure systems that are notalike in the sense of one not possessing aunit. Research indicates the benefits ofpositive and negative instances for con-cepts being incorporated into instructionalsequences (Shumway 1973), but how manycurricula ask children to compare measure-ment systems, such as Mobs' hardness scale,which lacks a unit clement in the domainspace, to other measure systems, such asarea? Shouldn't learners explore what char-acteristics of a measure function are aresult of its possessing a unit?

One characteristic of the unit within thecontext of measure functions is more scien-tific than mathematical: the arbitrary na-ture of selection of the unit. The selectionof kilometers to measure the distance be-tween Chicago and Chattanooga is a matterof judgment. Selection of a unit doespossess an affective component: an auto-mobile engine with a 2000-cubic-centimeterdisplacement may be more appealing thanan engine with a 2-liter displacement. Judg-ment of convenience is required and thisjudgment depends on many diverse factors,but how does a learner acquire this judg-ment? Arbitrariness in itself is a conceptrequiring some maturity and sophistication.It may well he that the cumbersome calcu-lations within the English system havehelped learners realize the arbitrary char-acter of units. With the convenience ofchanging units within the metric system,will children lose sonic perception of thearbitrary nature of unit selection?

A basic property of most measure sys-tems discussed heretofore has been theadditivity property. For a measure func-tion or mapping from a domain of elementst/ to a range space, we can write:

nr(t1, LJ d,) nr(d,) In(d,)

where

d, (-1 =

The essential intuitive concept that childrenmust acquire before dealing with the I111111-her concept is the conservation of thewhole, ci,u d,, on subdivision into parts. A

METRICATION, MEASURE, AND MATHEMATICS 131

child must accept, for example, that a linesegment will cover as much of another lineas the segment broken into two or morepieces. The typical child of kindergartenage does not readily conserve in this sense,whatever the measure function. The bestoperational strategy for helping the childappears to be simply to give him consider-able experience with conservation situa-tions. Some teachers attempt to protect theyoung child from conservation situations,arguing that he is not ready, since hecannot function correctly, This is an errorin judgment for two reasons. First, it is

futile; the child encounters many conserva-tion situations that are outside of the

teacher's control. Second, experience withinthe conservation, or, in measure-functionterms, additivity, context is necessary forthe eventual acquisition of the concept.Piagetian psychology argues for the neces-sity of experience; readiness as a limitationis more of a condition determining whenthe child should he held responsible forproducing and using the concept.

The child needs to manipulate situationsrequiring conservation and have the oppor-tunity to test his perceptions. The childprobably does not have adequate control ofconservation until his thinking matures tothe point that he can operate in contextsrequiring complementation or, in the rangespace of the measure function, subtraction.That is, in handling area on a geoboard,if the child is to find the area of the shadedtriangle in figure 19, then he looks at the

Fig. 19

square as a universe and realizes that thetriangle is the complement of the union ofthe three triangles labelled 1, 11, and III.Here the computational advantage in the

range space is, of course, clear, but thematurity of handling conservation in this"subtractive- or complementation form inthe domain space is a clear indication ofthe capability of the child to use conserva-tion in his reasoning within measure sys-tems.

Additivity and congruence, the two con-servation subconcepts for a measure func-tion, are factors in almost any learningconcerning measure. The unit concept de-pends directly on the child possessing a feelfor congruence. To measure the width of adesk by determining the number of paperclips that can be laid end to end across thedesk requires that the child have congru-ence under control. Moreover, additivity'srelation to conservation on subdivision intoparts is the fundamental psychological tiebetween the manipulative, physical worldand the mathematical model world of com-putation. As such, these two primitive sub-concepts for measure functions are psy-chologically among the more importantelements of the child's intuitive base formeasurement.

Children apparently have more difficultyin gaining control of the ideas of compari-son than in using units in measurement.Developmental psychologists have found,for example, considerable difference in theages at which a child can function in par-ticular measure systems; the child may beable to compare lengths up to two yearsbefore he can accurately compare mass.The conceptual area of comparison appearsto be complicated by the child's need forwords to express the comparison relations.Relational words and phrases such as "big-ger than," "less," "more than," and "notas large as" are more difficult to learn thanthose that name ob44.-.-as. Many develop-mental. psychologists have observed thatthe language factor is confusing and inter-feres with their attempt to design experi-ments and tests to ascertain precisely howchildren acquire control of comparison.

Some distinct problems with which learn-ers must cope on the path to a matureunderstanding of comparison have bum

132 A NI ET R IC HANDBOOK FOR TEACHERS

found. One striking difficulty is the com-parison of objects that are separated; thechild goes through a phase in which t,voobjects in close proximity can be comparedcorrectly, but the same two objects, whenseparated by distance or a visual blrrier,present the child with an inordinate amountof difficulty. fins is analogous psychologi-cally to the mathematical situation of mov-ing from measuring length on one line tomeasuring length on different lines; thechild must apparently create a mechanismfor comparing each of the objects to a thirdstandard. "Ile accurate comparison in bothsituations ( with two objects together andseparated) proceeds without any assign-ment of number or employment of a mea-sure function. The child progresses througha stage of rudimentary covering similar toa primitive use of unit in which one objectis compared directly to another. If morethan one dimension is involvedfor ex-ample, width and length in comparing thesize of rectangular reuionsthe child con-sistently fixates on a single dimension be-fore maturing to comparison on a moreappropriate basis. To date, no instructionalstrategy other than repeated experiencesfollowed by routine evaluation of the ac-curacy of the comparison appears to cor-rect this difficulty.

Another distinct difficulty of the child isbuilding inferential capability into his sys-tem for comparison. Acquisition of thetransitive property for measure systemsconies later than the pairwise comparisonbut oilers the child a portion of the reason-ing base that he needs. Seriation or thecapability for ordering a finite set of ob-jects by size further extends the capa-bility of the child in dealing with com-parison.

Learning situations for comparison seemto automatically include consideration ofthe subconcepts related to conservation.Asking a child to compare two sheets ofpaper, /I and 13. by placing one on theother, as in figure 20, and at the same timeto ignore the uncovered portion of 13 is

difficult. Indeed, the child's attention should

be focused on the uncovered portion. Forpairwise comparison, experience with addi-tivity's base of conservation is inescapablygiven, and congruence inextricably appearsas part of the experience of the child inusing a third unit for comparison of twoobjects.

Fig. 20

The Piagetian model of how childrendevelop geometrical concepts identifies theimportant intuitive components of the phys-ical, manipulative base for the measurefunctions. Although our discussion has beengeneral in the sense that it has cut acrossdifferent ratio-scale measure functions, eachof the primitive subconcepts has played arole in the examples given. The Piagetianmodel identifies psychological analoguesfor each of the subconcepts.

The emphasis on Piagetian psychologyis not all that helpful to the teacher inmany respects. It does not, for instance,specify a best sequence for children's cur-ricular encounters with ideas. Indeed, it isdifficult to specify a sequence, since theconcepts are so intertwined. It does, how-ever, strongly affirm that readiness is afactor in teaching measure. First, measurefor any characteristic function is a system,a structure of subconcepts. Consequently,it should be noted that system ideas arccharacteristic of mature learners in Piaget'smodel. If you accept his model for con-ceptual development, then you should ex-pect control of a particular measure func-tion to come at the upper elementary orjunior high school ages.

Second, the particular primitive sub-concepts will come earlier, before the childhas a grasp of the total system of the func-tion. This means that experiences with eachsubconcept must he a part of children's

METRICATION, MEASURE, AND MATHEMATICS

geometrical experience early in their math-ematics instruction. The Piagetian model ofdevelopment entails strong implications forinstruction in terms of readiness. The im-plication is that children mast be providedopportunities for manipulative experiencewith each primitive subconcept within avariety of measure functions. Conservationor additivity should be examined for length,volume, area, time, and mass. Rate andforce, as well as length, area, and volume,should he used as contexts for exploratoryexperiences in comparison. The idea of aunit should he exploited as a source of ac-tivities for children in a variety of measure-ment contexts; otherwise the child willnever be ready for the totality of themeasure-function system of concepts.Rather than readiness providing an argu-ment against curricular experience in thecase of measure, it provides a strong brieffor extensive experience with a wide varietyof activities.

Third, readiness considerations indicatethat computational facility for the measurefunctions is at a late stage in the learner'sevolution of measure ideas. Computation isa system or structure idea. The child musthave some understanding of operations andrelations within both the domain space andthe range space and, most importantly, ofthe functional connection between the two.

The child's coping with measure con-cepts in school is not limited to activitiesthat are designed to teach measure. Activ-ity-oriented and manipulative-based mate-rials designed to teach computational con-cepts and skills probably constitute themajority of the child's experiences withmeasure concepts in most classrooms. De-veloping addition and subtraction facts onthe number line (see fig. 21) is an effec-tive means of building conservation and

--y 3

I I

0 51< 5

Fig. 21

133

the additivity properties needed for facilityin measure. To use the rectangular arrayto develop the multiplication fact of 2 x3 == 6, as shown in figure 22, requires

3

Fig. 22

the child to count square and provide:.some feeling for units. To ask a child touse a number line to find how many 3s arein 29 informally deelops an awareness ofthe Archimedian property. (See lig. 23.)

4-111 1 11;911 F'1-1-1 T10 3 6 9 12 15 16 21 24 27 30 33 33

Fig. 23

The teacher should recognize that pro-viding children with these measure experi-ences is sound pedagogy for building con-cepts of measure. Albeit informal, in thatteaching measure concepts is not the ex-plicit primary objective of such lessons,these secondary objectives provide an im-portant component of instructional plan-ning of which the teacher should be awareas the primary goal of computational in-struction is addressed.

There are, of course, two ever-presentdangers of which the teacher should beaware. First, the teacher must not relyexclusively on physical models or objectsfor computational objectives if those mod-els rely on measure concepts to makethem work. Multiple embodiments of theconcepts give children different bases foracquiring the concepts depending on theiraptitudes, interests, and abilities. A varietyof types of models, some of which aredeveloped on a nonmeasure base, areneeded by those children who do not yethave full control of measure ideas. Dis-crete counting activities for establishingaddition facts supplant the use of the num-ber line for the child who does not yet

134 A METRIC HANDBOOK FOR TEACHERS

possess a full concept of length (see fig.24), but the teacher should use the num-ber-line activity as well as the discretemodel, since it provides ar. important as-similatory: base for the conservation con-cept. As such, the teacher is providing awod intuitive foundation for both additionand measurement.

C.7c-)C79c7

4 3

7

Fig. 24

Second, if the teacher is not careful inthe selection of models for computation,then the teacher risks establishing eitherproactive or retroactive interference withlearning; that is, learning can interfere withlearning yet to come or can cause for-getting of something learned previously.The take-away model for substractionmight he used to establish the fact 128 = 4 by exhibiting a ruler and breakingoff four units to throw away, as in figure25. For the child who as yet does not

0E1 1 1 ill 1.[LFig. 25

12

possess the conservation concept, an inter-ference with his acquiring it may be estab-lished. Or consider the case of the studentwho possesses a fairly firm grasp of theprimitive suhconcept of units; her teacherapproaches multiplication of fractionsthusly: 3/4 x 2/5 means you find 2/5 of

a unit and then 3/4 of what is left, A draw-ing like that in figure 26 is exhibited onthe chalkboard. It is modified as indicated,with an accompanying dialogue indicatingthat at A the square is a unit, a whole outof which we want 2/5. When we getthe 2/5 at stage C, we know what to dosimply look at the rectangle as a wholeunit (like we did at A) and take 3/4 of it.Many teachers then return to the originalunit and use one of the small rectangles ofstage E as a unit to cover the originalsquare A. Eventually, this yields the de-sired 6/20. Clearly, the "of" approach ormodel of multiplication of fractions leavessomething to be desired. For the studentwho does not have the unit concept undercontrol, this is particularly true. In alllikelihood, a proactive interference is estab-lished between learning about units of mea-sure in the context of fractions and theinstruction in which measure is the primaryobjective. Parenthetically, it might be notedthat the "of" approach to multiplication offractions has been the favored approachin elementary school texts.

Texts contain many sections devotingexclusive attention to objectives concern-ing measure, but the incidental learningconcerning measure, which accompaniesinstruction directed to objectives, shouldnot he ignored by teachers and curriculumdevelopers. Potential for this incidentallearning depends on the selection of themodels or physical situations that providethe embodiment for the number concepts.The extent to which such incidental learn-ing can be and should be relied on as asignificant component of a child's experi-ence with measure is an open question.Clearly, such activities cannot be reliedon exclusively as the means to teach

B C

Fig. 26

D E

-4.0

METRICATION, MEASURE, AND MATHEMATICS 135

measure; specific activities and lessons onmeasure are necessary. However, researchappears to indicate unequivocably thatmultiple embodiments of numerical con-cepts are preferred, since not all childrenhave sufficient grasp of measure concepts.This is to say, any instructional programfor computation that makes exclusive useof the number line should be viewed withextreme suspicion.

This section has been concerned withsome of the psychological problems ofteaching and learning about measure con-cepts. Primarily, it has addressed the taskof building intuitions for measure in theclean, pure world of the mathematicalmodel. The problems of learners with themeasuring skills and approximation judg-ments have not been considered.

Metrication and the teachingof measure

What are the implications of convertingto the metric system for the teaching ofmeasure concepts? Will acquiring theprocesses and skills of measurement beaffected by the metrication of schoolmathematics and science materials? Thesequestions are uppermost in the minds ofmany teachers of junior high and ele-mentary school children today.

The questions are significant. Clearly,text publishers have begun the shift to ametric base for instruction in school scienceand mathematics. Materials based on themetric system are becoming readily avail-able to the teacher, so, for the most part,material availability is not at issue. (It isperhaps an issue in some locations, de-pending on the availability of funds forinstructional materials.)

The preceding sections have focused oninstruction in measure rather than mea-surement. The use of the word measurewas reserved for the mathematical-modelcomponents of instruction. The wordmeasurement was restricted to processes,skills, and ideas in the real world of prac-tical and scientific use. The emphasis hasbeen on the mathematical structures within

the mathematical models; the functionsthat characterize the measures for themodels provide the skeleton of instruc-tional objectives and establish the transferpotential for measure sysiems. The dis-cussions have been independent of par-ticular measurement systems; for instance,discussion of the area function does notdepend on the measurement system beingEnglish or metric, but it would he quitelusty and misleading to state that metrica-tion has little or no implication for theteaching of measure concepts or that con-version will have small impact on instruc-tion for the mathematical-model ideas.Some representative implications are dis-cussed below.

I. Children need to develop metric in-tuitions. Children learn measure conceptsin the environment of the real world.Language, tools of measure, gross esti-mates of magnitude, and the like are the"stuff" of instruction. The child needs per-ceptual bases for learning, because mean-ingful experiences stem partly from famili-arity with the perceptual environment.If the child has no feeling for whethertwenty -five degrees Celsius is cold or hot,the child does not have available one (ofmany) stimuli when he learns about tem-perature scales. The larger the number offamiliar stimuli, the more likely the childwill enjoy meaningful learning. The teacherwho wants to capitalize on the motiva-tional potential of children's enchantmentwith the very large and the very smallcannot presume children's familiarity withmetric measurements if the children havehad little experience in the use of themetric system. In short, the particularmeasurement system used in instruction,along with its vocabulary and tools, pro-vide the environmental components theteacher uses. Temporarily, at least whilethe country is in the process of conversion,metric usage will not he a component ofmany children's away-from-school experi-ences. Consequently, the teacher must sup-plement and extend children's experiences

I 36 A METRIC' HANDBOOK FOR TEACHERS

with the metric world, In terms of assuringsuccess in teaching measure concepts,providing extensive opportunities for build-ing metric intuitions is one of the mostsignificant strategies a teacher can follow.

2. Children need to acquire a usableunderstanding and computatioludwith deinud fractions, The payoff ofmetrication for most individuals will hethe ease of conversion from one measure-ment scale to another, but understandingof the conversion process is possible onlywith understanding of decimal fractions. Itis tempting to advocate earlier encounterswith decimals for children; certainly theacquisition of understanding of decimalsat an earlier age would facilitate instruc-tion, but this requires that researchers andcurriculum developers in mathematics ex-plore questions of the scope and sequenceof children's experiences with decimals.The instructional sequences most com-monly used develop the mathematics ofdecimal fractions from children's under-standings of common fractions. Explora-tory evaluations of curricula that movedirectly from whole-number concepts orinteger concepts to decimal fractions with-out the intervening stages of common-fraction instruction need to he conducted.In short, understanding of decimals hasassumed greater importance as a goal inschool mathematics.

3. hit' metrication of American .vchoo[vand society does not deCretISO 1110 im-portance nJ Children's Or 1 1 conceptWO shills for nondecimal fractions. Chil-dren need to acquire a basic understandingof rational numhers in all their forms inorder to cope with important mathematicalideas. As with the decimal-fraction cur-riculum of the previous implication, thereare significant questions concerning thesequencing of children's encounters withrational numbers that need exploration. Itmay well he that instructional emphasison common fractions should be delayeduntil after decimal-fraction understandingsare acquired.

4. Aletrication should allow teachersthe freedom to include nu re and betterineasurement activities in instructionalplans, Children have needed extensive ex-perience in conversion within the Englishsystem because of the complexities of com-putation with common fractions. Withoutthe distraction of this computational level.teachers should he able to direct instruc-tion to the powerful concepts of measuremore frequently. Although most author-ities agreed that careful attention to thefundamentals of measure paid off in the in-creased achievement of students, theyfound- -and teachers agreedthat specificinstruction on conversion was m_LLssarywhen teaching the English system. Con-version within the metric systk.sin will nothe a distraction and instead will reinforcebasic concepts of numeration. Conversionbetween the English and metric systemswill rarely he necessary except for a fewtechnicians and should not he emphasizedin the school curriculum. It should henoted that the foundational decisions ofconvenience of the scale or unit needspecial attention. The fundamental ideasof the ratio-scale transformations of dila-tion and translation, along with the con-comitant outcome of ratio invariance, havenever received the attention they deservedat the secondary school level. Removal ofcomputational distractions should provideteachers with the opportunity to reordertheir priorities in planning, instruction.

5. Chi/droi need to apply meavure con-((TIN a 11e1 mectsurement processes to a

wiely of problems involving the metricyStem. Convincing children of the efficacyand efficiency of the metric system is per-haps hest accomplished through the useof the metric system in "real" situations.Rather than a frontal. tiresome assault onthe components of metric measurement, abalanced program of direct instruction andfrequent application of the ideas and skillsin realistic problem situations is the beststrategy. The use factor has been foundhighly motivating by many teachers. Use

mETRIcATIoN, MEASURE, AND MATHEMATICS

provides the argument for the importanceof the metric system. Max Bell's Mathe-matical Uses and Model.s. in Our ErerydayWorld ( 1 972 ) provides numerous anddiverse examples of different types ofmeasure problems that teachers and stu-dents can find in observing the worldaround them. Repeated attention to thecategories of problems represented in theBell book strengthen problem-solvingskills, build familiarity with metric mea-surement, and provide for a strong generaleducation component of it child's experi-ence with measure.

In conclusion, we reaffirm the themestated in the beginning: teachers shouldnot make a fuss about metrication. Metri-cation is reasonable and rational., teachersshould not lead children to anticipate non-existent difficulties. Teachers and cur-riculum developers can take advantage ofthe move to metrication to reconsiderand redirect the nature of instructionin measure concepts and measurementprocesses. This redirection should encom-pass building it tuitions for the functionaland homomorpkic character of measure.

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