Understanding Goonrctry for rt Oltrrrrgitrg World
188
ScI|4, Michacl l)illt)vct ittg (;()tttcll )'
/ Inl dit'( Ilry\\xri,. lill)cry!illc, ( rrlil
From T.V. Craine & R.Rubinstein (eds.) (2009). Understanding
Geometry for a Changing World. Reston, VA: NCTM, pp. 189-203.
: K0y
Curriculurn Press, l9[39.
Stigler, James W., Patrick Coluales, Takako Kawanaka, Steffel1 Kloll, ancl Ana
Serano. The TIMSS l/ideotape Cl.tssrctom Stu,ty: Methods and Findings fi'on
an Exploratory Research Prcject on Eighth-Grude Mathematics Instruclion in
Gemany, Japan, ond the (Jnited State'. Washington, D.C.: National Center for
Educational Statistics, 1999.
Stigler, James W, and James Hiebert. The Teaching Gap; Best ldeas from the Wolkl's
Teachers for Improving Educatio i4 the Classroon. New York: The Free Press,
1999.
ADDITIONAL READING
Nationaf Center for Educational Statistics. Prrsuing Excellence: A Study of U.S. EighthGrade Mathematics 6nd Science Teaching, Learning, Cn"rictlum, and Achievement
in lnternational Context Washington, D.C.: U.S. Departrnent ofEducation, 1996.
Defining in Geometry
Michael de Villiers
Rajendran Govender
Nikita Patterson
U.S. Department of Education (USDE), Educational Publication Center. Be.fbre It s n)o
Late: A Report to the Nationfrom the Nalional Commision on Mathemdtics and
Science Tedching for the Tv)e ry-rtr Century. Washington, D.C.: USDE,2000
T\EFrNlrloNs are impodant in matlematics They are tools lor communrcaf,) rion, for reorganizing old knowledge, and lor building new knowledge
through proof. But definitions present many challenges to both leamers and
teachen. Mathematical definitions are very concise, contain technical terms,
and require an immediate synthesis into a sound concept image Too often,
however, definitions are presented before students' concepts have evolved naturally ffom existing knowledge. Consequently, students often resort to meaningless memorization. Over time many educators have thought about better ways
to teach defining.
Earty in the twentieth century the German mathematician Felix Klein
strongly opposed the practice of presenting mathematical topics as completed
axiomatic-deductive systems. He argued instead for the biogenetic principle in
teaching, which has also been advocated by Wittmann (1973)' P6lya (1981),
Freudenthal (1973), and many others. The biogenetic approach takes the stand
that the student should retrace, at least in part, the path followed by the origi[al
discoverers or inventors, or alternatively, retrace a path by which knowledge
could have been discovered or invented. Human (1978, p. 20) calls this the recons trLtctive
A blackline master ofthe Law of Sines worksheet that supports
this article is found on the CD-ROM disk accompanying this
qpproqch.
Wilh tlris te n lreconstructive] we want to indicate that contellt is not directly
ir1lro(lrcc(l lo pupils (as finislretl products of mathenatical activity), b l thal
Yearbook.
189
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tlulslirlc(l liotrr Aiitlirr:rrrsl
'l hc rccoltslrLlclivc appl.oaclt
nllows stLt(lcnts lo p rtit i/tt(,rrr./rr.t,/r. irr lltc
dcvcloprDct'lt of the conlcnt and thc relalcd nratlternatical proccsscs ol'dclining.
axiornatizit'tg, coniec{uriDg, and provillg. ConteDt is not prcsentcd as il Iillishc(1.
prcl:ablicaled product, but ralltcr the lcacher locuses on the proccsses by wltich
thc conteltl can be dcveloped or- recoltstructed. Note that !l l.econstructi\c
i.Ul
proach does not treccssarily imply learning by discovery. for it ntayjust involve a
reconstlL]ctive explanrtion by the teacher or thc textbook.
Mathcuraticians and nlathenlatics educators alikc have often criticized thc
direct teaching ofgeontetry tlefinitions wilh no emphasis on the ulldcrlying pro_
cess oll dcfining. Fot exalnple, as czrrly as 1908 Benchara Blandlbrcl (quoted in
Crilfiths arrd Howson fl97.1, pp.2l6 171)wrote,
fte it appearu a radicallv vicious mcthod, ceftainly iD geonctry. il.not in
other subjccls. to supply a child rvith reacly rnadc dcfinitions. to be subsequcnl_
To
ly nrcnrorizecl aficr-Lrcing more or-lcss car-efully explaincd. To do this is surcly
to thtow away delibclatcly oDc ol the most \,aluablc agents olintellcctual
discipline. The evolving of a rvorkablc deliDirion by thc chilcl,.s own acri\ it)
stinlulated by approp|iatc qrrestions, is both intercsting and highly
cducational-
Hans Freudenthal (1973, pp. 417 l8) strongly criticizecl the practice of dj_
rcctly providing geometric definitions on sirnilar gr.ounds. Ohtani (1996, p. g1)
rrgued that the pmctice of simply tclling dcfinitions to studcnts lunctions tojus_
tify the tcdcher's control ovcr stLldents, to attain a degree of unifbrmity, to avoicl
having to deal with students'ideas, and 1() circLlmvcnt ploblematic intetactioDs
with students. Vinner
including Batista (2009) hnve prescnted
arguurents and ernpirical data supporting the thesis that krrowing the dcfinition
of a concept does not guaral]tee understanding. For cxarlple, studcnts who have
been told, antl arc alrlc to recile, the slandard dcfinition of a parallelograrn nay
(
1991 ) and other.s
still not consiclcr reclrngles. squarcs, aucl rhornbi as parallelograms iftheir con_
ccpt imagc (i.c. thcir private mcntal picturc) of a parallelogmnr is onc in which
not all anglcs or siclcs arc allowed kt be equal.
Studeltts oflen meet n]athcntatics only tltrough the stlucture presented in
formal rnathematics textbooks. This str.Llcturcd approach can lcad to a common.
but lalse conception that only one correct dcfinition exists for.each cleflned ob_
ject in rrlathen'tatics. [n lact, several diflercnt. concct definitions nay exist for a
particular coltcept, so wc have a ccltaiu alrrount offi.eedont in oul choicc ofclcfi_
nitions. Thus dcfinitions can be considerccl ..arbitrar.y" (Linchevsky, Vinuer, and
Karscnty 1992, p. ,18). For exrmple, a rcctltnglc may be clefined as a parallelo_
grarn with a right anglc, a quadrilateral with tl.trce rigltt angles, or a quaclrilateral
l)(:llllllr{]
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trrllr lrrrLsol ,\rrrrrrr'tr\
tlr,)L11,ll r)l)l)r)srt. srrlr's.
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lr){)
oli(
lr\tlnx)i\s
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tltr'
itrtp|esstorr llllll r rr'( Irrrr1,I( (iIrl iln(l Illusl l)c (leline(l orrly oDe i!lry.
lrrrIllreIrrrrrrc. llrc lirrrrlrl ilpl)roitcll prollt()tcs lltc rrriscorrccplion tl)ltt DLttllcrnllics ul\.vrys sllrls wilh I dcfinilion. nd thill dcllnilions ol'Dlllthcnrrtical ol)iccts
alc rivcn a 7.,r'lr.,r'l. Sludents then do not letrlizc that dchnitions arc not discovcr
ics. but human "inventiolls." Studellts rarcly undcr-stand that a main plllpose ol'
clcfinitions is to prornole accLrrate mathelratical communication. (See also lllair
ancl Canacla [2009] regarding developing dcfinitions with students.)
The ideas shared in tllis articlc stcn fr-onr years of consideration of issucs
related to nralhematical definitions ancl to a variety of ellbrts to engagc sccondary school sttlclcnts and prospective teachers in meaningful work in this clornain
(de ViJliers 1986, 1994, l99ll^ 2003.2004). Ifteachels are 10 engagc studcnls irl
a recoustructivc approach in which they create aud critique lltcir o\\'r dcfinitions,
tlrcn tcachers themselves must nrst understand thc subtlc distiDctions anrong various types of definitions. Those disti|lctions arc thc major topic of this article.
We first consider the distinction bctwccn partitional ancl hielarchical nrethocls ol classilicalion. Next wc cxaminc thc charactetistics of a correct definitiol'l
based on necessary and sufficicnt conditions. Therr we look at economical definitions alld clcvclop clitelia fot good delinitions. Finally, we discuss how these
distinctions can hclp tcachers plan and in]plelner'rt slrategies 1(] develop their studcnts' lnatlrcnraticlll thinking.
Types mf ffi e{'in"llt[ms"es
Partitional and Hierarchical Definitions
A nrutual telatiolrship exisls betwecn classifying and defirrirrg. The classilication ol any set tll'conccpts irrplicitly ol explicitly irrvolves defining the concepts involvcd, whci cas defiuing concepts in a certain way automatically involves
their classilication. For cxarrplc, dcfining a lectangle as rr 4 uu(lriktteral with t\\)
ares qf s),t nct\) lllrough oliposile .\i(le.r'rvill then il.rply that a square is classificd
as a spccial rcctangle. However, if a rectangle is deflned as u qLuth iloterul with
1r,o axes 01 s.\nt nel q' lllroL!gh opposile,;itles, ltut no other line.t of .\.\rnnlclrl, thci
squarcs iuc clearly excluded ilon the set ofrectangles.
We describe a defirition such as the firsL as hierunltical (i.c., incJLtsivc) bocause it allorvs the inclusion of lnorc particular corlccpts as subscts of thc nrolc
general concept. The latler \,ve call a p.o'titiotlal (i.c.. cxclusivc) rlclinition bc
caLLSe the conccpts involvcd are considclccl disjoint f'r'our cach otllcr (i.c.. s(lLlillcs
ar€ not considered rcctanglcs).
lfstudcnls iuc givcn thc opportunity to crcatc clclinilions ol llrcir or,vn lbr
such gcolnclric conccpts as the qLtadrilltcrlls. tbc rcsLll( clll bc ir lii,cly ancl
lruitlul class cliscLrssion of why hierarchical tlcllnitions llc gcncrally prcfcllccl
t
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rrirrlr(,,rrrr(s. l)L Villiers (l()()1. l) lt) rrlt.rrtilies tlre lirll.rrirrg rrrr|rrrt.rrrl
il(l!irr)lil{cs ol lriclrrclricitl cllrssilic.rtr,,rr {,\cr I I I { I rlllsslltr.ttt(}n. A lllct.lu_
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chical rlclinitiorr
.
simplifies the deductive systern rntd the derivation olpropeflics ol.
lrote special concepts,
"
,
.
oftcn provides a
sorretilres
LrsefLrl
sLrggests
conceptuai schema dudng problen solring,
aiternative clellnitions and netv propositions, and
ptovides a useful global perspectivc.
Nevertheless, in sonte situations wc lecd partitiolal clefinitiolls to distin_
guish concepts cleally. For exanple, wc havc little choicc but to create partia
tional classification olconvex ancl corrcave quadrilaterals, becaLlse it is Dot mcaningful to view a concave quadrilateral as a special kincl ofconvcx quacllilateral,
ur \ icc versit tde \ illicrs lq94).
Furthermorc, lront a historical pcr.spective, hicr.archical cjcfinitions have not
always been fhvored. Folexarnple, in Book I of Elenents, Euclicl partitionccl
quadrilatcrals into fivc mutually exclusive calegorics:.rr7larc (both equilateml
ancJ righranglecl), o6lorg (r.ightangled but not equilateral), rhctnbus (eqtilateral
but llot right-angled ), rhonboid (opp<tsite sides and angles cqual to one another
but reithcr equilateral nor dght-angled ), ald, fi-apeziun (any othel quadrilateral).
Sirrilar'ly, Euclid did not consider an cquilateral triangle to be a special case ol
an isosccles trialgle.
Some of the challenges that deiinitions create for stlldents are illustmted
in the interviews and experienccs with chilclren in grades 9 to 12 over several
years reported in de Villiers (1994). Hefe is one example (1 :
irterviewer, .9 =
student):
1. lf wc clelinc a partllelogrant as any quad iateral
rvith opposite sides
parallel, carr wc then say that a rectalgle is a parallelogram?
,t.
Yes,
...
because a rcctanglc also has opposite sides parallel.... But I
don't like this cletinilion of pamllelogranrs. . . . I know we are taught this
definitiou at school and that squares and rectangles are par-allelograms
(pulls face). but I clon't like it....
How wotLld you definc parallelograrr.rs instead?
s.
)rrlrrrrrrl ttr ( icotrr,ltV
193
r
I rr,rrrl,l tlrL r,. lirte rlllll( r sllv rr l)lrlilllal()1'llllll rs lr tIIr;trItrIItIt I;rI
rr'rllr o||]osrtr'strles Irrtltllcl. l)tll tl(|l itll sitlcs ot ltrrglcs cqttltl
(
rlll(
('lcllly
lcads to nrore cconotrrical dcfinitir.rs ofconcepts and firrrnulalions ol.
tltcorenls.
.
1
I
As any quadrilateral with oppositc sidcs parallel, but not all angles
equal.
What about rhombi then? . .. Would you say a rhombus is a
parallelogram?
[{mm ... according to my clefinition, yes, ... but I don't like tirat
llris clcvcrrlh grildo studcllt (who lrapl'lcttctl to bc ll lop sttl(lcn( lll his
sclrtxrl) lrltl no plotrlcnt with clrarvit'tg correct conclusions 1'rrtlr givctl dclillitions
ancl rnaking hielarchical class inclusions but prcfelled not to do so. Moteovcr.
this sludcnl clearly exhibited the ability to lormulate a definition. Btrt the dellnitions hc pleferred were exclusive oncs. Battista and Clements ( 1 992, p. 63) havc
similally reportecl two cases oi studcnts who werc able to lollow the logic of a
hierarchical classjfication ofquadlilatemls ofsquarcs and rectangles but had difficulty aeceptir.rg it. Quitc often the origin of this problem can be traced back to
the elemcntaly school, and to dilect pl'ovision ol cxclusive definitiotts or static
exclusive images, which then beco1]le so fossilized in students'minds that by the
time they reacll the high school, the students are very resistant to change.
Another challcnge lor students a ses whcn tlley struggle 10 define a more
general figure fiom a n]ore specific one. For example, students may say, "A rhonlbus is a square with not all angles equal." On the one hantl, studellts ale trying 10
definc a rhombus hiclarchically as a special square. Yet on the othel ha[d, thcy
are endirg up with'A rhonbus is a quadrilatelal with equal sides, but with not
all angles cqual," which partitions rhombuses fionr squares. The authors have fre-
quently observed this type ofproblem and labelecl it as an "inverse hierarchical-
partitjon" definition because students are lfying to bc hiemrchical but instead
are partitioning. Textbook authors and tcachers often use this approach without
rcalizilg the conceptual diffictLlties it crcates. For example, consider the following inloduction to a rhombus: "The next shape we arc going to bc lookirlg at is
called a rioirrbas. We can think ofthis figure as a squarc that bas becn pulled out
of shape." This approach ercourages studcnts to view a rhombus incorrectly as a
special kind of square instead of viewing a square correctly as a special kind of
rhonrbus.
Correct Definitions
For our students to participate actively in thc construction of definitions,
tlley must know what qualifies as a correct definition. A definition that corltains
conditions (propelties) that are both necessary and stLfficient is said to be correc1. For a condition ir a given description to be necessary, it lrust apply to all
elenents ofthc set we want to deline. (The concept implies the property, so the
ploperty is lreoessary lor the concept.) However, for a condition to be sufficient,
it must elsurc that wl]enever it is met. we obtain all the elements of the set we
want to define. (The property inplies the conccpt, so the prope y is sufficient for
the concept.)
It is helpful to recall that logically in the biconditional p <=> 4, the condition p is vierved as necessary and sufficient for thc condition 4' meaning that one
l,rI(IIII:iIiIII(III|(J ( ,tr)nt(:tty l()l it (;lti l(,
t(J
W()tl(l
(irrl c()rr(lu(le llrlrl 11 lollorrs liorrr/r, llll(l \i(e \r.tsit. lltc tlelirrll!. rorrrlrliorrs lirr
l scl r)rLls1 bc bollt ttcccsslly arrtl strllicicrrt. lol crlrntplc. corrsitlcl llrc lollorvirrg
cantlitl|tc lilr a clclirrilioD: "A rcctanglc is ilny qua(lrilulct.ill lvillt oplxrsitc sitlcs
palilllel." Bclolv
are somc drarvings
ofa
quacJr.ilatcral cotnplying
wilh thc con(li
tion "oppositc sides parallel. "Ccftain ly thc sliltemcnt does apply 10 clcnrcnts ol'
the sel wc want to dcfine (see fig. 13.1(a).).Therclore wecan say ll\..tl ,ollo.titc
sides ptu'ullel" is a necessary condition for a rcctangle. However, Iooking at thc
three drawings. we notice the existence of elencnts (figs. l3.l(b) and 13.l(c))
that do not belong to the set we want to clefine. Thus,,opposite sidcs parallel,,is
[o1 suf'ficient to glrarantee that a pa].ticular quadr.ilateml is a rectaugle. Heltcc we
say that "oppositc sides palalleJ" is a necessary but ltot a sufl,icicnt condition tbr
n
EL\ D
l){rlr)ir{l lr ( ji(}ntllty
.1.
r\ l.rlr'r,rr
r
I
r r,
1$5
rt
lr r l;r I c r r r I
n rllr
|l
rIrt rrrIit rrIIrr
rlilrll'rlrls.
l l)c lirst \lirt( nll lll rs irllJ()rrccl bcclrrrsc il eorrllrirrs lrtr ur]r)cccssury l)r()l)cr'lv.
itt lhul isoseclcs llrrpczoitls rlo not in gcncrirl hlrvc pclpcndictrlll tliirgorirls. lhc
sccond sliltcnrcnt is llso an incollcct dclinition bccausc it does not coltltill sulIicicnt propcrlics to ciefinc a kite ancl is theretbre incomplele. fo[ cxaulplc. wc
citn qonstrlrct a cliagonal and another one perpendicular lo it, and thcll conncct
thc cndpoints, obtairring a quadrilateral as shown in figure 13.2, which is clcarly
not a kita. In general, to show that a definition is il'rcomplele, it sufficcs to grvc u
coLrnterexample that meets the putpolted definition but is not an cxarnplc ofthe
set offigures one wants to define.
rectangles.
(a)
(b)
Fig. 13.2. Nonexample of a kite
(c)
Fig. 13.1 . Example and nonexamples of a rectangle
Howevcr, having congruent diagonals that arc perpendicular bisectors
of
cach othcr is a sulficicnt condition for.a t.cctangle (in f'act, sufficienl fbr a squarc;.
But this colclition is llot necessary, becausc it does not apply to lnany rectangles,
including the one in figure 13.l(a).
Ncxt consider thc tbllowing: 'A rcctanglc is a quadrilateral with opposite
sides parallel and u,ith onc interior angle eqnal to 90 clegrees.,'This pr.operty
applics to every rectanglc (making it i.t neccssary condition), and any figure we
construct witlt it will be a rcctanglc (mal{ing it a suf'ficicnt concjitiou). Thcrctbre
this statenent provides a necessary ancl sufllcient condition for rectangles.
lncorrect Definitions
A defillition is incorrect if it coltains insrLfflcicnt or rulnecessary propertics.
for errrnpl,. con:idcr the loli,,wrtg:
l.
'An isosceles trapezoid
cliagonals."
is any quadrilatcr.ai with pcrpendicular
Notc that thc statemcnt "A kite is a quaclrilateral with perpendicular diagonals" is a correct statement about a plopcrty ofkitas, but it aontains too ]ittle inlormation to be uscd as a dcfinition. We therefore say that having "perpendicular
diagonaJs" is ir necessary but not a sut'fiaiert condition for kites.
The authors have observed that one common difficulty students have in producing correct counterexamples to inco|nplete definitions is that they oflen tl'y to
rcfute a clefirrition with a special case. For example, lbl the incouect definition 'A
rectangle is any quadrilateral with congruent diagonals " solne students will provide a square as a counterexample. But obviously a squarc is not a valid colLntcrexample. becalLse a square
lr a rectangle.
Therefore, studeuts shorLld already have developed a sound understlnding of
a hierarchical (inclusive) classification of quadriJatcrals /refire being errgagcd in
fonrally defining the quadrilaterals thcmsclves (Crainc and Rubenstein J993).
This developncnt carr bc fostered by using interactive geometry sottu'rre. ligures
creatcd with tlcxiblc wirc, or paper-strip models of quadrilaterals. lror exanrple,
if stuclcnts usc an intcractivi] geometry tool to collstrulct a quadrilateral with opposite sides pal.illlel, then they may notice that they can dmg it into the shape o1'a
rectangle, rhombus, or square as shown in figule I 3.3. Students cau lllcl1 bc askcd
to describe what lhis oulcorne tells tllcrn. Tcachcr-s should hclp students realize
(Jrruurs(urr(]tng (Jaomelry lor a unnnglng world
ll)ilL ll)is
(lc.rorsrr'ri'rr sl*lws
197
Doflnlng ln Goomolry
trrar rcct.r)glcs, rrr()rnbrscs. arcr sc;.a'cs lrr.c s1]cci,r
cnscs ol'a parallclogmm.
c
------>
c
I
D
----+
A
J
Fig. 13.4. Nonexample of a kite
condition logically implies that "one pair ol opposite angles are congruent" because triangles lD-B and ADC are clearly congruent (by side-side-side congruence), aud therefore m
lB:n
lC.
Fig. 13.3. Dynamic transformation of a parallelogram
Economical Definitions
A correct definition can be either economical or uneconomical. An economi_
cal definition has a minimal set of necessary and sufficient prope ies; that is,
it
has no superfluous information. Conversely, an uneconomical definition
contains
redundant properties.
For example, consider the lollowing candidates for defiritions
ofa kite:
1. 'A kite is a quadrilateral with two pairs ofcongruent adjacent
sides
and one pair ofopposite angles congruent.',
2. 'A kite is a quadrilateral with perpendicular
diagonals with one being
bisected by the other.',
The first one is correct, but uneconomical because it contains too much in_
formation. In other words, the conditions are necessary and sufficient, but
not all
ofthem are required. But which condition can be left out? When students evalu_
ate these conditions, they may realize that ifthey were to leave out,.two pairs
of
congruent adjacent sides," they would obtain an incorrect definition because
it is
possible to construct a quadrilateral with one pair ofcongruent,
opposite angles
that is not a kite (fig. 13.4).
However, we can construct a kite according to the condition ,.two pairs
of
congruent adjacent sides,' by placing a compass first atl and then at D
and draw_
ing circular arcs as shown in figure 13.5- In addition, we can easily show that
this
Fig. 13.5. Testing a definilion of a kite
The second candidate is a coffect, econon.rical definition because no superfluous information is supplied. In other words, we camot leave out any ofthe proper-
ties, and we can construct a kite from the given conditions. We can demonstrate
the latter, as shown in figure 13.6, by flIst constructing two perpendicular lines,
then placing th€ compass at O to cut off equal distances on the one line, and
picking two arbitrary points on the other to determine the vertices. ln addition,
students can show that the other properties of a kite can be logically deduced
lirrrrr llre girerr rorrrlilirrDs. lo|errrrrrIlc" irr liitrrrr' l\1)" trirnltl(s l()lt
clcillly c{)nglucnt (by si(lc-iu)glc-sidc eorrgIrrcl]ccl rrrrtl llrcrclirrc.l/l
(lrc srorc wNy thcy can shorv lh.tl DB - l)L'.
lurc
l l()('
Convenicrrt Dclittiliotts
.l(
ob,,,iorrslv rr 1'rxrrl rlclitrttiott. its uc ltltl'c scctt itr llte lirtcgoittg. ltr"rirls r''
(lull(liUlt irIIi'IIII Ii(rII: il n)us1 l)c ccotr{)!rricitL ll l lr goorl tlclinition illso Ilils olhcl
ch |uclclislics- A goocl. ot convcnicnt, tlcllnition is ollc thitt also allorvs tts to
dcducc thc olhcr propeltics ofthc concept cirsily;1hat is' it should bc tlcrlutlit'c-
. lrr
A valuable exelcise fbr sttrdents is to have then compare different clcfiniliorrs
accotcling to this cl'ilerion. For exanlple, thc definition ol'a rhol'nbus as a cluaclrilatcral with two axes of symn'letry through the opposite angles is lllore dedrrctivecconon.rical than the standard textbook delinitiol'l of it as a quadlilatelal wilh all
sicles congrucnt. For the former, llle othcr properlies (e g., perpendicular-, bisecting cliagorals, all sides congrucnt. and so on) ftrllow imnrediately from synnlctry,
wheleas with the latter, we havc to use congruelcy ald somewhat longel argu-
tnc ls rr, deducc
Ihe otllet Itloperlies.
A Teaching Sequence
children rt'tost easily lealn what a tablc or a chair is by sceing Inany differcnt examplcs ofthose colcepts, not by being provicled with fomlal definitions
ofa table and a chair. (See also Battista [2009] fol more aboul this phcnolnenon of
conccpt formation.) Similarly, without starting with a formal definition, children
can easily learn wllat a square, lectang]e, or kite is.
In an interactive georlletry ellvirolllllcnt, colcepts such as the special quadrilatel.als can be introducad in three stages. The first stage uses the softwarc to hclp
YoLrng
Fig. 13.6. Testing another definilion of a kile
This ninimality
principle that is, that definitions should be economical
is a crucial st[lctural element of nlathclnatics as a deduclive systen. As a matter
of fact. it shapes the way iD which matllcrnatics progrcsses when it is presented
deductivcly, fbr alter the clefinitiorr is prescnted, tllcolcrns tllzrt givc additional
information about the concept arc fbrmulatcd antl proved (Linchevsky, Vinner,
and Karsenty 1992, p. 54).
For cxarrple, ifa rhombus is definecl as a quadrilateral with foln congllrcnt
sidcs, thcn thc fact that thc diagonals are perpendicular bisectors ofeach other
can bc provcd as ir theorcrr. Conversely, ila rhombus is defined as a quadrilatelal
with diagonals thal arc perpendicrLlar bisectors of each other. then the fhct that
thc fbur sidcs arc corlgrucnt becomes a theorem. Both definitions are economical
insofar as thc dcfining contlitions contain no superfluous inlomration. This goes
back to the iclca tlrat clcfinitions arc arbitrary. Diflerenl sets o1'definitions produce
ditlerent thcorclns withill a system.
However, in a 1'cw instanccs in mathematics, dellnitions are nol minimal. A
familiar examplc is the way in which somc tcxtbooks define congruent tfiangles:
"Congruent triangles are triangles that have a1i pairs ofcorresponding sidcs congruent and all corresponding angles congruent." Wc know that it is sufficicnt to
require less than that lbl two triargles to be congrLLent, and thc fact that lcss is
required is cxpressed by each ofthe four postulates nonnally accepted in high
school textbooks for the congruelcy of triangles.
students learl what a spccil]c shapc is (a concept image). For examplc, a concept
such as an isoscelcs trapczoid can easily bc introduccd as slrown iu ligure 13 7 by
first having stlLdents construct a ljne AB, thcn colstruct a linc segment CD, and
I'
Fig. 13.7. lntroducing an isosceles lrapezoid by construction
l)(
sul)se(llrrlll) r(lle(l ( /) lll llK
elr
li|( l/)'to ('l)ti|ll ('/)' l]y ro
l|e(li|l1,.
\r.ttr.r\,
c\l)l(Jtc lltc pt.()l)crlics ()l rllit(lnlitlclll ( /)/)'( '.
whiclr thcy nrily bc told is callcd ln i.sostt,lt.s trqtt:oil (conrltirlc rlc Villicls
slu(lcn1s
tlrcrr bc lrslicti
1()
[2003])
By dragging thc figure and nlcasufing its attributes, studcnts cun tlcvclop
a
sound concept image ofan isosceles trapezoid as a cluac|-ilateral harrrrg nr.rny
properties, including conglucnt diagonals, two pair.s ofcongruent adjacent anglcs.
at least one pair of opposite sides congruent, at least one pair of opposite sidcs
parallel, and othcrs. Moreovcr, students can discover that they calt drag ttl isosccles tmpezoid into the shapc ofa rectanglc and square, but not a gcneral rltotnbus,
parallelogran, or kite, thus fomring thr: foutrdalion for a hierarchical vicw.
The second stage involves challenging students to wdte tllcir own correet,
economical delloitions for an isoscelcs trapezoici, and then to test tltose defini,
tioDS by lnealrs of constructiot'l and measuLement. Equivalcntly, studcnts oan be
challcnged to devise different ways ofconstrlrctitrg a dylamic isoscclcs trapezoid
that always renains an isoscelcs trapezoiil no matter how it is dragged. Such
constructions, as reported in Snith (1940), help develop an rurdcrstanding not
only of tlre difference between a premise anda tctrtclrislol but also ofthe carisal
relationship between thetrr, that the conditions lbrcc the resrLlt. For example, consider a circle with two parallel lines inteNecling it (see 1ig. 13.8). Then thc quadrilateral lonrred by the points wherc those two parailel lines intersect the circle
must have congruenl diagonals and congruent opposite sic.les and, hencc, is an
isosccles trapezoid. This result shows that the conditioll is sutficient to cnsure al.]
isosceles trapezoid, and that wc could define an isoscelcs trapezoid as any cyclic
quadrilateral with at lcast one paiI ofopposite sides parallel.
T
)l
lrrJ r t ( i(
r(
)l r !lty
201
llrc llrirrl .rlrrr'r'rtlrr'lrL:. llrr' li' ltl s]st!.nlit{izitliorr ol tlrt. l)tr)l)etlir.s i)l i
isoseclcs llrpezorrl. lrrr c,,:rrrrple. l)y sllulil{ liorl irrry eivcrr rlclirrilion. slrrrlcrrls
tlrcn llirvc l() (lc(lltcc tllc olltct ltr(,llcrtics logicllly lionr i( as tllcotcllts. A ltoltrrllr.
inilill choicc suggcstcd by studcnts. tcachcrs, and sourc lextboolts is thc firllow
ing: "An isoscclcs trapezoid is aDy quadrilateral in which at least one pair of
siclcs arc pamllcl and the olhcr pail of oppositc sides are congrucnt." However,
t
this delinition is incorre..t. For example, although the condilions can produce au
isosceles trapczoicl, the conditions can also produce a parallologran. Students
rrced to realize that a gencral pamllelogran cannot be consit{ered an isoscelcs
trapezoid, as parallelograrns do not necessarily have all pr.operties ofan isosccles
trapezoicl (e.9., congruent diagonals, two pairs o1'adjaccnt angles cougruent, cyclic, axis of symmetry, aDd so fbrth).
Although sttLdents usually attenpt to improve this clefirrition, after a rvhile
thcy find no satisfactory way of corrccting it. lf thcy forotulate it in such a way
as to exc|.rdc tlre parallelograms for examplc'i\n isosceles trapezoid is any
quadrilateral with one paiI of opposite sides parallcl, and anothe[ pair ofopposite
sides equal but lrot parallel" thcy exclude not only the (general) parallelograrr
but also rcctangles and squarcs.
After further discussion and clitical comparisot] ofvalious correct, econontical definitions for an isosceles trapezoid. students usually scttle with a convcnient
definition, such as "an isosceles tnpczoid is any quadrilatcral with an axis of
symmetry through a pair oloppositc sides." This definition is much easiel to rLse
to derive other proporties. Contrast it, tbr exanrple, with a definilion such as "al
isosceles trapezoid is any cyclic quadrilateral with at least one pair of opposite
sides parallel."
Gonclusion
Fig. 13.8. An isosceles trapezoid conslructed
from two parallel lines intersecting a circle
Students should be given the opportunity to engage in thc process of constructing defiuitions. Intet-active geontetry software is a tool that can ltelp prornote this goal (e.g., de Villjels [2003]). It is plausible to conjecture that, through
expcriences in which definitiolts are not supplied dircctly, students'undcrstancling ol geometric definitions, and ofthe concepts to which they relatc, will incrcase. Furthernlore, students are Iikcly to develop a better underslaltding olthe
nature 01'definitions as well as skili in clefining objects o[ their ow11. Irr particular.,
they n.uy come to realize that dcfinitions should be econoutical lnd that they
are a matter of choice. Rccent results lrom Covender and dc Villicls (20Ct2), dc
Villiers ( 1998, 2004), and Sicnz-Ludlow and Athanasopoulou (2007) do indicare
some improvement and positive gains in students'understrnlding ofthe [ature of
definitions, as well as in their ability to define geot'nctric ct)nccpts themselves.
202
[.,r
](l(irstiur(lin!J ( irx)n r(ilry lor
i
r(
II r: rI
I(|iI Iu
Woll(l
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the Tu,enty-se(\n.l Inletnaliono! Con/bt?nce for the Psycholegl, 6l Mulhe 14li.s
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Calil:
Key
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Technolog,35, t1o.5 \2004):703 24. Available flon rnysite.n]wcb.co.zalresidents/
profmd/vanhiele.pdi
Freudenthal- Haffi. Muthenatics as oi| E(luculion.t
l
(loonrrlr
203
y
:\
(,.1'llr(y llowsi)rr. i\lttllN ti a.
llcss. l ()7':l
\t[irtt,tnl ( nti,ttl,t
('urrrlrt rtige: ( rrrrrlrr rtlge I lrri!ursily
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National Council ol'Tcachers of Malhematics (NCTM). edited by Timolhy V
Craine, pp.9l 108. Reston, Va.: NCTM,2009.
dc
irr
(;rillillrs. ll. llrirrll,rrrr,l
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l)t:lirrirrll
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"
A Sketchpad 4 file for "Exploring the Hierarchical Classification of
Quadrilaterals" is available at:
http://dynamicmathematicslearning.com/quad-classify.gsp
An interactive "Hierarchical Classification of Quadrilaterals"
is available online at:
http://dynamicmathematicslearning.com/quad-tree-web.html
(icon'rctcrh SketchpiLd filcs that support this arliclc lrtc lirttntl ott
thc ( l)-lt()M tlisk accorrpanying 1his Ycalbook.