check our website

1. Article Review II
Review and answer all article review questions.
Do they sound familiar?¥Is the diameter the short one or the long one?¥I forget. I think
obtuse is the wide kind of angle.¥Eight is a multiple of 24, I think, . . . or is it a factor?At the
middle school level, students encounter a lotof new mathematics vocabulary, some of it
confus-ing. Sometimes I wonder whether we overwhelmstudents with terminology! Yet we
want them tospeak the language of mathematics, using standardterms that others recognize
and understand. More-over, we want the vocabulary to make sense to them.One strategy
that supports studentsÕ languagelearning is examining word origins. Once we asteachers
are familiar with the roots of mathemati-cal terms, we can share the etymologies as we in-
troduce new vocabulary and help students use theroots as bridges to meanings. One bonus
is that themathematical terms are often linked with commonEnglish words, including words
in other disci-plines. The links promote studentsÕ general vocab-ulary development, as well
as mathematical flu-ency. For example, in studying geometry, studentsoften confuse
diameterwith radius. Consider theroots: radiuscomes from the same root as ray. Justas a ray
of sunshine starts at the sun and goes outfrom that point, so too does the radius of a
circlestart at the center and radiatefrom there. Evenradiois related because broadcast
waves radiatefrom a tower to our receivers. Diameter,in con-trast, comes from roots
meaning measureandthrough. The diameterof a circle is a measurethrough its center.
Another term, diagonal,sharesthe same prefix for through,as well as gon,mean-ing
angle.These roots remind students that a diag-onal is a segment that goes through an angle
orcorner of a polygon. Note that polygonitself has thesame root; it means, literally, many
angles.Table 1(pp. 494Ð97) gives more examples for some com-mon terms in the middle
school curriculum. (Text continues on page 497)Listen to some remarks that I have heard in
mathematics classrooms. RHETA RUBENSTEIN,rhetar@aol.com, teaches at
SchoolcraftCollege, Livonia, MI 48152. She enjoys using word origins,mental mathematics,
and the theory of multiple intelligencesto make mathematics accessible to her students.R H
E T A N. R U B E N S T E I NWord Origins:Building Communication ConnectionsBuilding
Communication ConnectionsVOL. 5, NO. 8 .APRIL 2000493 This material may not be
copied or distributed electronically or in any other format without written permission from
NCTM. Copyright © 2000 The National Council of Teachers of Mathematics, Inc.
www.nctm.org. All rights reserved.
494MATHEMATICS TEACHING IN THE MIDDLE SCHOOLTABLE 1Roots, Meanings, Related
Word, and Notes for Some Middle School Mathematics VocabularyTERM, ROOT,

2. LINKTOMEANINGRELATEDWORDSTEACHINGNOTESNUMBERdenominatornominate: to
name someone Teach denominator before nom:namefor an office numerator. First, we must
The denominator of adenomination: the name of know a fractionÕs name, thenfraction tells
the name ora religious sect, a coin or bill, we learn how many of the partskind of fraction;
e.g., a playing card, and so onof the fraction are of interest.fifthsfor five equal parts.Students
who know Frenchcan recognize nomforname.numeratornumeral:a symbol for aThe
numerator is thenum:numbernumberÒnumbererÓ of a fraction,The numerator of
aenumerate:to count, as iswhereas the denominatorfraction tells the numberdone in a
national censusis the Ònamer.Ó In of parts of interest; e.g.,numerous: many; in
largemeasurement, the samein 3/5 we are interestednumberpattern applies: We namein
three of those partsthe number before thenamed fifths.name, or denomination,ofthe unit, as
in five miles,ten square inches, or eight cubic feet.reciprocalreciprocal trade:twoWhen you
multiply twore:back, as in backwardcountries buying andreciprocals, you get
thepro:forwardselling from one another;number 1. You have In writing the reciprocal
products moving back Ògone back and forthÓ andof a fraction, the numerator and
forthreturned to the identity and denominator Ògo backreciprocity:agreements to honor for
multiplication.and forth.Óone anotherÕs credentialspercentcentury: 100 yearsPer,or for
each,is a helpful per: for eachcent:100th of a dollartranslation in other
contextscent:100centennial:100th anniversarythat signal division, suchPercent means for
eachas miles per gallon, cents 100or divided by 100orper gram, or feet per second.simply
hundredths.factorfactory:a place whereStudents often confuse factors factor:maker,
doer,products are madeand multiples because they performerfact:something that hasare
both parts of multiplication. Two factors when multiplied been doneIt helps to point out
that justmake a product.manufacture:to make byas factories produce products, sohand
(French main) too do factors produce products.from raw
materialsmultiplemulticultural:representing Again, to clarify the confusionmulti:manymany
culturesbetween factors and multiples, pli:foldpliŽ:in ballet, a bendinghave students
remember,Fold a piece of paper in half, or folding of the kneesÒEvery number has ain half
again, and in half again.pliable:flexible or bendablemultitude of multiples.ÓThe resulting
number of piecesis eight times, or eightfold, thenumber of original pieces.
VOL. 5, NO. 8 .APRIL 2000495GEOMETRYperimeterperiphery: the edge orThe Greek word
forper:aroundboundary of somethingperimeterbegins with themeter:measurediameter:
measure of aletter π,which for anyThe perimeter of acircle through the centercircle
represents the figure is the measurequotient of its perimeter around it.divided by its
diameter.circumferencecircumnavigate:to sailA circle drawn around acircum:around or
bendaround the globepolygon and through itsaroundcircumspect:to lookvertices is called a
ferre:to bring, to carryaround; to be suspiciouscircumscribed circle.The circumference of
acircumlocution:to talkScribemeans to write orcircle is the measure youaround a subject,
not draw,so circumscribedneed to get around the directly about it.means drawn
around.entire circle.polygonpolynomial:an algebraicIn the Old English
frompoly:manyexpression with manywhich gonis derived, the gon:angle or kneenames or
termskin kneewas pronouncedA polygon is a closed knee: a joint that bends, so it sounded
more like gfigure with many angles.or makes an anglein gon.polyhedrasanhedrin:a

3. historicalThe root for hedra is poly: manyjudicial body in whichsedra,from which
wehedra:bases or seatsmany representativesget sit, seat,and Polyhedra are three-are
ÒseatedÓresidence.We call thedimensional shapes thatresidence: a place
wherepolyhedronÕs seats itshave many bases or seats,one stays or is seatedfaces,but seator
baseisalso called faces.more literal.acuteacute vision:sharp visionWe can demonstrate
acute:sharp or pointedacupuncture:procedureacute pain by bending An acute angle
measuresusing sharp needlessharply to make an acutebetween 0 and 90 acrobat:one who
performs angle with our bodies.degrees, which is fairlynear the sharp or
pointedÒsharp.Ópart of a tentobtuseobscure:literally, toAnother word of interest,ob:against
or towardconceal against;but possibly one you tuse:beatsomething that is lackingwould not
mention in When you beat a knifelight; hidden; not easy class, is obscene,whichagainst a
surface, itto understandmeans off stage. In Greekbecomes dull or blunt.theater, actions that
wereAn obtuse angle obscene were not seen!measures between 90and 180 degrees, which
is fairly Òdull.ÓALGEBRAvariablevariable weather: conditionsMany educators suggestwer:a
raised spot on thethat are changinggiving students earlyskin; a change fromvariation(in
statistics):experiences with normalhow values are dispersed,variables that vary (Pegg andA
variable in algebra isor changing, from oneRedden [1990]; Demana and a symbol that can
take onextreme to the otherLeitzel [1988]. Avoid limitingvarious, or changing,early
experiences to Òfindingquantities.an unknown.Ó➤
496MATHEMATICS TEACHING IN THE MIDDLE SCHOOLalgorithmalgebra:One of al-Arabic
is also the sourceBen Musa al-KhowarizmiKhowarizmiÕs booksof our word zero. It is was a
ninth-centurybegan with the Arabicfrom the word sifr, formathematician who wroteword
al-jabr,meaningempty. The Arabs gotbooks about what we refer the reunion of broken the
word from the to today as arithmeticand parts. In solvingHindus, who first algebra. The end
of his name, equations, we are conceived of a place-valueal-Khowarizmi, meaning from
working from parts ornumeration system with the town of Khowarizm,is the pieces of
information toa symbol to stand in asource of the word algorithmwholes, or finding
theplace with no value.for a step-by-step procedure, missing pieces.which he often showed
in his works.coordinatecooperate:to operate orLongitude and latitude co:together
withwork togetherfrom another system ofordinate:a straight rowcoordinates:clothes
thatcoordinates that tell aWhen you put things in order, go togetherlocation. In other
words,they are in a straight row.where in the world are we?Coordinates are an ordered set
of numbers used to locate a point.axisaxle:a pin or rod on Just as a car wheelÕsaxis:axle or
pivotwhich a wheel revolvesturning requires an axis, so An axis is a line around which too
does locating points in a a coordinate system Òpivots.Ócoordinate first require
axes.originOrient:where the sunIn China, the name for Japan, origin:the beginning; therises
or originates; Yat Bun,literally means day sourcerefers to the Far Eastroot, or the origin of
the day. Notice how the Chinese look east to the sunrise in Japan, just as Europeans look
east to the sunrise in China. Similarly, Europecomes from a Semitic word meaning (land of
the) setting sun.DATAstatisticsstatus: condition or standing When you are done
withstatus:a condition orof a person or thingstatistics, you know
howstandingstature:natural heightthings Òstand.ÓStatisticsmeans the collection, when a
person is standingorganization, analysis, andinterpretation ofquantitative

4. data.graphtelegraph:writing from aGraphand its cousin, gram,graphein:to write, long
distance; literally,are pervasive in mathematicsto scratchfar writingand science. ChallengeA
graph is a diagram orautograph:writing oneÕsstudents to find other examples,written
picture ofown name; literally, self-writingsuch as diagram, histogram,quantitative
information.graphology:study of handwritingscattergram, pictograph, crab:an animal that
scratches bar graph, electrocardiogram, the ground as it walksencephalogram, spirograph,
computergraphics, and so on.
VOL. 5, NO. 8 .APRIL 2000497averagedamage:loss resultingStudents can be helped
toawar:fault or blemishfrom injury, as well asvisualize averageusingFrom the Arabs, who
were the legal compensationmaterials. For example, extensive traders, averagefor such a
lossshow students stacks oforiginally meant goods (Damage was originallyseven blocks,
two blocks,damaged in shipping. Later,a synonym of average.)and three blocks. Ask the
meaning became thethem how to make financial loss due to damaged equally tall stacks.
Helpgoods and, eventually, thethem see that the average the portion of the loss borne by is
the height needed foreach investor. Now averageeach stack to have anmeans an equal
portion.equal number of blocks.Although word origins can often be found ingood
dictionaries, one especially helpful resource isSteven SchwartzmanÕs book The Words of
Mathe-matics (1994),which I highly recommend. In it,Schwartzman provides etymologies of
hundreds ofmathematical terms, along with examples of relatedroots in common language.
Much of table 1isbased on SchwartzmanÕs information. I suggestkeeping a copy of The
Words of Mathematics, aswell as your favorite dictionary, handy in class.Students need
some help getting started with ety-mologies. Usually, middle schoolers are just begin-ning
to realize that language has a history and thatknowing word roots can help them build
vocabu-lary. You may want to work with an English teacherto help the students build a
foundation for languagestudies. One way to begin is simply to share an ety-mology when a
new concept is first introduced. It isusually easiest to tell students the meanings of theroots
and help them see those roots in other wordsbut not to dwell on the foreign-language roots
them-selves. Unfortunately, many roots are Indo-Euro-pean, which is no longer a living
language. Invite students to brainstorm to identify cog-nates, or words that appear to be
related, thencheck dictionaries to verify their suggestions. Stu-dents for whom English is a
second language mayshare related words in other languages, too. Oncestudents get the idea,
they can be invited to digdeeper using dictionaries and the Internet. Ety-mologies can be
part of study-skills activities(UCSMP 1995), can serve as sources for studentjournal entries
(Thiry 1990), can be used as com-ponents of student projects, or can be included instudent-
developed bulletin boards or cartoons.Thompson and Rubenstein (forthcoming)
providemore specific ideas for using etymologies.Incorporating word origins into the
curriculumproduces multiple benefits. The main advantage isthat mathematical terms often
mean precisely whatthey say. When we recognize the Latin, Greek, Indo-European, Arabic,
or other lan-guage roots from which wordsstem, their meanings are clearer.Another benefit
is that studentswho are verbal learners, as in themultiple intelligences of Gardner(1983),
often find verbal connectionsparticularly helpful in gaining ac-cess to the mathematics.
Etymolog-ical links help English-as-a-second-language students learn commonand technical
vocabulary in co-herent ways. Parents, lookingahead to college-entrance examinations, are

5. usuallypleased to know that language development is partof the curriculum. Finally, the
study of word originsis a wonderful opportunity for teachers to collabo-rate across
disciplines. Along with reading and writ-ing across the curriculum, why not integrate lan-
guage across the curriculum? Word origins appearin all disciplines. Teachers who are aware
of termsthat students are learning in other classes can useetymologies to support student
learning in otherareas.➤
Incorporatingword originsinto the curriculum produces multiple
benefits
498MATHEMATICS TEACHING IN THE MIDDLE SCHOOLDo you have the Òwrite stuffÓto
become an MTMSauthor?If you are engaged daily withall the challenges and rewards
ofteaching middle school mathematics,then you do! Find out more by re-questing a writing-
tips packet fromus. Telephone Kathleen Lay at (703)620-9840, ext. 2193; or send e-mail
toklay@nctm.org. Ask for the ÒMTMSWriterÕs Packet.Ó You can also re-quest it by mail
atÒMTMSWriterÕsPacket,Ó NCTM,1906 AssociationDrive, Reston,VA 20191-9988.The
Write Stuff!kMost of all, word origins are fun! Learning how seemingly unrelated wordsin
our lives are connected is often surprising and exciting. For example, didyou ever realize
that angleand anklesound alike? This similarity is not surpris-ing when one discovers that
they both stem from a root meaning a small bend;in other words, where the foot meets the
leg, an ankle makes an angle! To findmore interesting and enlightening connections, begin
exploring word origins,sharing them with students, and enlisting students as fellow
explorers. ReferencesDemana, Frank, and Joan Leitzel. ÒEstablishing Fundamental Concepts
through Nu-merical Problem Solving.Ó In The Ideas of Algebra, KÐ12,1988 Yearbook of
theNational Council of Teachers of Mathematics (NCTM), edited by Arthur F. Cox-ford,
61Ð68. Reston, Va.: NCTM, 1988.Gardner, Howard. Frames of Mind: The Theory of Multiple
Intelligences.New York:Basic Books, 1983.Pegg, John, and Edward Redden. ÓProcedures for,
and Experiences in, IntroducingAlgebra in New South Wales.Ó Mathematics Teacher83
(May 1990): 386Ð91.Schwartzman, Steven. The Words of Mathematics: An Etymological
Dictionary ofMathematical Terms Used in English. Washington, D.C.: Mathematical Associa-
tion of America, 1994.Thiry, Jean M. ÒMathematics Pictionary: The Sounds of Silence.Ó
MathematicsTeacher 83 (March 1990): 200Ð201.Thompson, Denisse, and Rheta
Rubenstein. Learning Mathematics Vocabulary: Po-tential Pitfalls and Instructional
Strategies. Forthcoming.University of Chicago School Mathematics Project (UCSMP). Study
Skills Hand-book. Glenview, Ill.: Scott-Foresman & Co., 1995. C
Article Reflection
Read the article then respond to the following prompts:
Discuss the main points (summary) of the article in about 100 words. How do they align
with constructivist principles?
Discuss three mathematics words and their etymologies that you find very relatable to you
as a teacher and explain how you will integrate them in your teaching.
Describe your "Aha" moments when reading the article.
Describe at least two important benefits of incorporating word origins in the curriculum
based on the article.
Describe the instructional shifts a teacher must implement to effectively apply the big ideas
in the article.