Sergei Abramovich a & Ilya Levin b a Department of Mathematics Education, The University of. Aviv, Tel Aviv, Israel Published online: 09 Jul 2006.

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1 This articl was downloadd by: [Tl viv Univrsity] On: January, t: : Publishr: Taylor & Francis Informa Ltd Rgistrd in England and Wals Rgistrd Numbr: Rgistrd offic: Mortimr Hous, - Mortimr Strt, London WT JH, UK Intrnational Journal of Mathmatical Education in Scinc and Tchnology Publication dtails, including instructions for authors and subscription information: Spradshts in taching and larning topics in calculus Srgi bramovich a & Ilya Lvin b a Dpartmnt of Mathmatics Education, Th Univrsity of Gorgia, drhold Hall, thns, G, US b School of Education, Tl viv Univrsity, Ramat viv, Tl viv, Isral Publishd onlin: Jul. To cit this articl: Srgi bramovich & Ilya Lvin () Spradshts in taching and larning topics in calculus, Intrnational Journal of Mathmatical Education in Scinc and Tchnology, :, -, DOI:./ To link to this articl: PLESE SCROLL DOWN FOR RTICLE Taylor & Francis maks vry ffort to nsur th accuracy of all th information (th Contnt ) containd in th publications on our platform. Howvr, Taylor & Francis, our agnts, and our licnsors mak no rprsntations or warrantis whatsovr as to th accuracy, compltnss, or suitability for any purpos of th Contnt. ny opinions and viws xprssd in this publication ar th opinions and viws of th authors, and ar not th viws of or ndorsd by Taylor & Francis. Th accuracy of th Contnt should not b rlid upon and should b indpndntly vrifid with primary sourcs of information. Taylor and Francis shall not b liabl for any losss, actions, claims, procdings, dmands, costs, xpnss, damags, and othr liabilitis whatsovr or howsovr causd arising dirctly or indirctly in connction with, in rlation to or arising out of th us of th Contnt. This articl may b usd for rsarch, taching, and privat study purposs. ny substantial or systmatic rproduction, rdistribution, rslling, loan, sublicnsing, systmatic supply, or distribution in any form to anyon is xprssly

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3 INT. J. MTH. EDUC. SCI. TECHNOL.,, VOL., NO., - Spradshts in taching and larning topics in calculus by SERGEI RMOVICH Dpartmnt of Mathmatics Education, Th Univrsity of Gorgia, drhold Hall, thns, G, US Downloadd by [Tl viv Univrsity] at : January and ILY LEVIN School of Education, Tl viv Univrsity, Ramat viv, Tl viv, Isral (Rcivd Octobr ) Th us of spradshts in introducing studnts to th concpt of a limit of a squnc is dmonstratd and th possibl computr-basd scnario as th nhancmnt of th taching/larning procss of calculus is xmplifid. It is shown how th spradsht's oprational capability assists visualizing th olzano-cauchy principl of convrgnc and lads vntually to th possibility of mploying computr tchnology in dciding th convrgnc of positiv sris.. Introduction It is hard to ovrrat th impact of lctronic tchnologis on mathmatics ducation in gnral [,] and of th us of th spradsht program to assist in th study of various topics at diffrnt mathmatical lvls in particular [-]. Th aim of this articl is to illustrat th spradsht's ffctivnss as a pdagogical tool in calculus lssons whn taching th concpt of a limit of a squnc and addrssing som rlatd issus. Th concpt of a limit of a squnc is of grat importanc and prmats th whol cours of calculus. Taching xprinc and rsarch [ ] show, howvr, that usually this concpt is not clar nough to most studnts. Unfamiliar in comparison with pr-calculus notions, and thus abstrus whn first ncountrd, th xact dfinition of a limit of a squnc involvs th charactrs and N accompanid by xistntial and univrsal quantifirs. This, actually, signifis a mov to advancd mathmatical thinking []. Th fact that in avrag conditions larnrs hav such difficulty in comprhnding this dfinition is causd by th vry form of th dfinition which uss symbols of prdicat calculus, with hardly any visual imagry. Th main concrn of th work of th authors in this dirction, originally prsntd as a confrnc papr [], is th issu of modlling, visualization and xploration. s is known, a spradsht maks it possibl to modl squncs in th form of a column (row) of numbrs by rplicating a formula which dtrmins a squnc. This provids an opportunity, givn a crtain squnc, to offr studnts a visualization of th bhaviour of a finit but sufficintly larg numbr of its mmbrs in a numrical form that is simpl to undrstand. Th spradsht is a powrful tool, with its rmarkabl capacity of rcurrnt counting and opration, and givs th opportunity to challng and stimulat studnts this, togthr with th rational -X/ $ Taylor & Francis Ltd.

4 S. bramovich and I. Lvin Downloadd by [Tl viv Univrsity] at : January for taching itration squncs xpoundd by Wigand [] dtrmind th form of a squnc rprsntation in favour of th itration formula in this articl. W assum that studnts hav basic skill in oprating a spradsht and dfining functions in clls []. Thus w can assum studnts to b abl to implmnt th following computr-basd scnario during a lsson, and undr tachr guidanc: () Dvlopmnt of th spradsht tmplat () Obsrving th bhaviour of th squnc on th spradsht tmplat through a tachr-studnt dialogu that will vntually lad to th xact dfinition of a limit of a squnc () Vrification of th dfinition of a limit of a squnc followd by th formulation of its ngation for procding to discuss th abov scnario, on rmark advocating th spradsht as a pdagogical tool is ndd. Th point may b raisd that a cannd program mad up by th instructor bfor th lsson could hav mor advantags sinc it dos not rquir studnts to b skilld in oprating th spradsht. ut this is not so. In th spirit of Lawlr [] th spradsht is an opn intractiv larning nvironmnt whr studnts can xrcis thir own crativity. Morovr spradsht-orintd taching, using this tchnological tool as an xplorr [], boosts studnts' constructiv thinking [] by giving thm opportunity to ngag in xploratory mathmatics. In contrast, th us of a cannd program actually is an xampl of passiv rndring of mathmatics knowldg.. Rprsnting th squnc in th spradsht Th rmarkabl fatur of th spradsht softwar is that it can rplicat formula ntrd into clls. So an itration squnc dfind by th itration formula can b rprsntd in th spradsht as follows: () Th givn first trm of a squnc is ntrd in an arbitrarily chosn cll of a crtain column of th spradsht. () Following th givn itration formula, th scond trm of a squnc is ntrd in th nxt cll of this column blow th chosn cll. () Th spradsht options Copy and Past rplicat th itration formula down this column. ftr ths stps hav bn takn th column is immdiatly filld up with th trms of a squnc.. Dfinition of a limit of a squnc Considr th squnc X n+ =^ + () Not that formula () dos not xprss th squnc in a closd form as a function whos indpndnt variabl varis ovr th st of all positiv intgrs. Formula () only prmits on to comput any mmbr of th squnc in trms of a prvious on. Hnc a tachr should spcify th valu of th first trm (initial valu). Th vry choic of an initial valu is arbitrary in gnral. So lt X = - ()

5 Spradshts in taching!larning calculus Downloadd by [Tl viv Univrsity] at : January. n : Xn C Xn E-.E-.E-.E-.E-.E-.E-.E-.E-.E-.E-.E-.E- Figur. t th vry bginning a spradsht with only two columns, say and, is ndd (s Figur l(tf)). In column currnt numbrs of th trms of th squnc ar dfind. Thn, th valus of ths trms will b locatd in column. To do this on must dfin th squnc X n in column. Th first trm () is dfind in cll. Following itration formula () th spradsht function =()/ + is dfind in cll and computs th valu of X. This function is copid down into cll. s a rsult th computd valus of th squnc X n immdiatly occur on th scrn. Th rsults of modlling indicat that th third trm alrady has as its intgral part. Furthrmor, whil th intgral part dos not chang, th figurs in th fraction part flash across th lins of th tmplat and gradually vanish. Th following imaginary dialogu btwn a tachr and a studnt is an xampl of a dsirabl taching/larning procss of th concpt of a limit of a squnc by using th possibility of modlling, visualization and xploration within a spradsht tmplat. Tachr: What is th bhaviour of squnc X n whil n incrass? Studnt: Squnc X n approachs numbr whil n incrass. Tachr: What dos it man that squnc X n approachs? Studnt: X n gts closr and closr to. Tachr: What dscribs th dgr of closnss of two numbrs a and b, or in othr words th distanc btwn thm? Studnt: Th closnss of two numbrs a and b, or th distanc btwn thm, is ithr a b or b a, whichvr is not ngativ, i.. th absolut valu \a-b\ rprsnts th distanc btwn a and b.

6 S. bramovich and I. Lvin Downloadd by [Tl viv Univrsity] at : January Hr a tachr must brifly intrrupt th dialogu in ordr to crat a third column C in th spradsht (s Figur (b)) for computing th absolut valu \X n \ as th dgr of closnss btwn X n and in accordanc with th studnt's dfinition of closnss. To this nd th spradsht function = S(-) ntrs into cll C] and is copid down into cll C. ftr th computations in th column C hav bn carrid out, th dialogu continus again. Tachr: What dos it man that squnc X n mor and mor approachs th numbr? Studnt (having xamind column C): Th valu that xprsss th dgr of thir closnss, i.. th absolut valu \X n gts smallr and smallr. Tachr: What do you man by 'smallr and smallr'? Studnt: Howvr small this absolut valu may sm to us, at th following stp it bcoms vn smallr. Tachr: What do you man by 'howvr small it sms to us'? Is it tru that vryon has his own concpt of smallnss? Studnt: Tachr: Of cours, vryon has his own concpt of smallnss. Wll, in accordanc with our sns of smallnss lt us point out thr valus of a crtain numbr, say s, = -, = - and = - and chck whthr th condition \X n < holds tru or not. t this point, th tachr introducs th complmntary columns (s Figur l(c)) for calculating valus (TRUE or FLSE) of th prdicat \X -\<. (Th symbols fc and C indicat TRUE and FLSE rspctivly.) To this nd, th spradsht functions = IF($C--<, "", ""), = IF($C-- <, "", ""), = IF($C-<, "tf ", "C"), ar dfind in clls D, E, F, rspctivly, and ar copid down as shown in Figur l(c). With rcours to th spradsht on may not that whn = - th valus of this prdicat bcom TRUE bginning from n = ; whn = -, bginning from «= ; and whn = -, bginning from n =. It turns out that for any givn positiv numbr on can always com across such a numbr N (which dpnd on such that whn is smallr thn N is largr) that for any n>n th inquality \X n < will b tru (though w ar unabl to try all s, working with th spradsht prmits us to dscrib our imprssions). In such a cas it is usually said that numbr is th limit of th squnc X n. ftr this dtaild prliminary discussion of th rsults of modlling, studnts ar abl to giv th in fact alrady statd xact dfinition of a limit of a squnc. Dfinition. L is calld th limit of th squnc X n if for any givn positiv numbr thr xists a positiv intgr numbr N such that \X L\< for all n gratr than N.. Vrification of th dfinition of a limit Th nxt qustion that th tachr could propos to studnts is as follows: may any othr numbr, say, b th limit of squnc (), ()? To answr this qustion studnts ar askd to chck th abov dfinition for numbr. This rsults in choosing an arbitrary positiv numbr (th words 'for any' allow for this possibility), say = -, and thn trying to sk for this a numbr N such that th absolut valu \X n L\ by «= N has bcom and for all n gratr than N rmains lss than.

7 Spradshts in taching I larning calculus Downloadd by [Tl viv Univrsity] at : January n Xn c Xn Figur. Th spradsht tsting th inquality D Xn- -.< is shown in Figur and indicats that inquality () holds tru only with n = and n =, but is impossibl to rtain () whn n incrass. Howvr, though such an N for = - is not to b found, it might b possibl to do this for othr valus of. Fortunatly, thr is no furthr nd to sk bcaus it is sufficint to spcify only on for which th dfinition of a limit of a squnc dos not hold tru as this dfinition rquirs implmnting inquality () which, though bginning from a crtain numbr n = N, must tak any givn positiv. In doing so th tachr lads studnts to th conclusion that numbr is not th limit of squnc (), (). In ordr to formulat what it mans that numbr L is not th limit of th squnc X n it would b hlpful to mphasiz that having arrangd within th spradsht th vrification of inquality () for squnc (), (), studnts larnt that if in som row of th tmplat th symbol! has appard (inquality () holds tru) thn ncssarily thr xists a row blow containing th symbol C! (i.. inquality () is not valid). In othr words w wr abl to spcify a numbr = - such that for any N (th numbr of th row) thr xists n>n (th row with a largr numbr) for which inquality () is not valid, i.. X n ^- with this n. nd this is lik saying that numbr is not th limit of squnc (), (). t this point it is asy to formulat Dfinition. L is not th limit of th squnc X n if thr xists an > such that for vry N> thr is a numbr n>n such that th inquality \X n L\^ holds tru. y comparing dfinition and dfinition on can obsrv that substituting in dfinition th words 'for any...' with 'thr xists an ', th words 'thr xists... N' with 'for vry JV', th words 'for all n' with 'thr is...«' and th inquality \X n L\<E with \X n L\^, rsults in dfinition. ()

8 S. bramovich and I. Lvin This complts th scnario of th discussion of th dfinition of a limit of a squnc. Howvr, having compltd our initial objctiv, th larning procss can b xtndd with th hlp of th acquird skills. Downloadd by [Tl viv Univrsity] at : January. Variation of th first trm and cofficints of a linar itration squnc On of th most important practical applications of th thory of squncs is in th discussion of th itration procsss usd in computing. Gnrally, th analysis of itration procsss is concrnd with dciding whthr a crtain squnc convrgs and if so, to what limit it tnds. Morovr, it is of importanc hr to know how fast th convrgnc is and whthr som itration procdurs (at last from a givn class) will provid answrs mor quickly than othrs. It should b rcalld that w chos th first trm () arbitrarily. So, it is natural to start with th variation of X. Studnts dtct from th outst that such a chang has no influnc upon th limit. s studnts usually lik to 'play' with larg numbrs, th tachr can xploit this by asking th following qustion: Lt us multiply th starting point () by th factor of. How many stps will th computr rquir to 'achiv' th squnc () to th limit? Most likly, studnts' gusss will b wrong as long as thy hav not mad prvious calculations. Howvr, quick calculations on th computr will show that th first trm's valu xtnsion by six ordrs lads to a diffrnc of thirtn stps. Othrwis, th valu of a first trm dos not affct th spd of convrgnc of squnc () to th limit. Nxt, studnts may turn to th family of linar itration squncs X n + =ax n + b () which dtrmins th itration procss dpnding on ral paramtrs a and b. On possibl goal of invstigating this on th basis of th spradsht could b to stablish which of ths paramtrs is rsponsibl for which fatur of th procss. It is asy to compos xprimnts which rsult in th influnc of only th multiplicativ paramtr a, both on th xistnc of th limit and on th spd of convrgnc whil th additiv paramtr b contributs only to th valu of th limit to which squnc () tnds. Ths xplorations, whn prformd undr tachr guidanc, will lad studnts to th conclusion that, givn a ral numbr b and th starting point X t, thr xists a limit of squnc () with a[<l, and th smallr \a\, th fastr th convrgnc. Th tachr can dmonstrat this to studnts by using th graphical analysis of th itration procss [] as wll as by transforming X n into a progrssion []. On mor visual approach in dciding th convrgnc of squnc () which dos not involv th concpt of a limit will b shown blow in sction. To xprimnt with paramtrs th spradsht is programmd as follows (s Figur ). In row diffrnt valus of th paramtr a ar dfind by tabulating a from - (cll l) with stp (cll ) up to - (cll El), or in othr words th spradsht function = + $$ is dfind in cll Cl and is copid right into cll El. In cll th valu of paramtr b is dfind. In column bginning from cll currnt numbrs of th trms of th squnc ar dfind whos valus ar dfind in column by ntring th first trm into cll, spradsht function =$ + $$ into cll and rplicating this function into cll E.

9 Spradshts in taching!larning calculus Downloadd by [Tl viv Univrsity] at : January a b XI STEP c Figur. D E Modlling squncs with diffrnt bhaviour to prvnt misconcptions It has bn notd by many authors [] that taching th concpt of a limit of a squnc through spcific xampls such as monotonic squncs could caus studnts' misconcption which, in turn, may lad to rronous oprations with limits. For instanc, whn askd to find th limit of a squnc studnts somtims act on a mistakn assumption that such a limit xist. Procding from th slant that on pictur is worth a thousand words a tachr could introduc to studnts squncs with diffrnt and quit unxpctd bhaviours using a spradsht as a tool of visualization. Thus, modlling th quadratic squnc x - d x ~ () for diffrnt valus of th ral paramtr d by tabulating d with appropriat stp, provids th visualization of diffrnt typs (divrgnt, convrgnt, oscillating, priodic, chaotic) of squnc () bhaviours as shown in Figur. Particularly, column D indicats that it is irrlvant for th xistnc of a limit whthr th valus of X n ar locatd on on sid of th limit sinc, as statd in th dfinition, it is only ssntial that th variabl should finally diffr from its limit by an arbitrarily small amount. Finally, this xampl calls studnts' attntion to th importanc of th thorm of th xistnc of limit for a boundd monotonic squncs. Th spradsht in Figur is programmd similarly to th spradsht in Figur. Valus of th paramtr d ntr into row, valus of th squnc () ar providd by th spradsht function =( )/ + ($l)/ dfind in th cll and rplicatd into cll G.

10 . bramovich and I. Lvin Downloadd by [Tl viv Univrsity] at : January n\d C D Figur. E F G E+.E+.E+.E+ E+ E+ #NUM! #NUM! #NUM! #NUM!. Visualization of th olzano-cauchy principl of convrgnc Th clbratd thorm stats: In ordr that th variabl X n has a finit limit it is ncssary and sufficint that for any numbr > thr xists a numbr N such that th inquality \X H -X m \< is valid, providd n>n and m>n.. s can b sn, givn th variabl, inquality () srvs as th xampl of an inquality dpnding on two positiv intgral variabls. Similarly to th ability to numrically modl quations of partial diffrncs [], a spradsht can modl inquality () by calculating its valus (TRUE or FLSE) for any positiv numbr. This has th advantag of nabling studnts to discovr whthr th squnc convrgs or not without involving th concpt of th limit, whos xistnc thy want to prov. Modlling inqualitis of typ () can b ralizd on th spradsht quit asily. y way of an xampl a tachr could propos that studnts dvlop th spradsht implmnting inquality () for squnc () with th starting point (). This spradsht (s Figur(a) whr a = -, =, = -) is programmd as follows. In row bginning from cll and in column bginning from cll, positiv intgral valus of n and m ar dfind rspctivly. In cll th first trm () is dfind. Th spradsht function = + $ is dfind in cll (C) and copid down (writ) into cll (P). Th spradsht function = IF(S(C$ -$)<EPS, "gfc", "") is dfind in cll C and calculat th valu of th prdicat \x x \<- This function is rplicatd into cll P. Th spradsht shown in Figur (b) corrsponds to th following tripl: a = V, b =, =. ()

11 Spradshts in taching!larning calculus With rcours to Figur (a) and () on may not that whn = - a =, b = th valus of inquality () bcom TRUE bginning from «= and m = (in th olzano-cauchy thorm N=), whn a=l-, b = irrspctiv of th valus of inquality () appar not to form th rctangular pattrn filld with symbols ft (such an N dos not xist). Th studnts should b ncouragd to xprimnt with various valus for a and and b challngd to xplor th dpndnc among a, and N whn th visualization of th principl of convrgnc has bn originatd from modlling on th tmplat. Ths xprimnts, actually, can giv studnts an apprciation of th rcalculation procss by providing a visual imag of statmnts lik 'th smallr, th gratr N\ 'th gratr, 'th smallr N', th smallr a in absolut valu, th fastr th convrgnc', and so forth. Downloadd by [Tl viv Univrsity] at : January. pplying th visualization of th olzano-cauchy thorm in dciding th convrgnc of positiv sris Finally, w would lik to dscrib a possibl computr-basd scnario in dciding th convrgnc (or divrgnc) of a positiv sris, which a tachr could arrang during a lsson. This layout includs th following points: () Rprsntation of th squnc of partial sums of th givn sris in th form of an itration squnc. () Modlling th olzano-cauchy principl of convrgnc with rspct to this squnc. EPS a b n\m IS Of C. D. «fc E. r - F. «t G. it «fc H. i. Figur (a). j. «fc K. «fc L. ft l M. N. Mm\i. P ftim

12 S. bramovich and I. Lvin Downloadd by [Tl viv Univrsity] at : January EPS a b n\m EPS n\m «. C. c c D. D. E. E. F. «F J G c: c: C c: c; H ^ ^ i. Figur (b). G. H.. i Figur (a) j it c: j. K ct < K. it L L. k M M. ct ik ik N N p <i... c; k «k ik p ik

13 Spradshts in taching/larning calculus Downloadd by [Tl viv Univrsity] at : January k a k C bk Figur (). D a k >bk It It ft ft ft < () nalysis of th spradsht tmplat and making conjcturs concrning th problm of whthr givn sris will convrg or fail to convrg. () Inspction of th appropriat sris by comparing trms within a spradsht for applying a comparison tst. () Th argumnt of convrgnc (divrgnc) by a comparison tst. Th abov scnario for th sris «which turns out to b th 'major sris' for th harmonic sris is implmntd on th spradsht tmplats shown in Figurs (a) and (b).. Conclusions Th rapid dvlopmnt of tchnological tools in mathmatics ducation affcts both taching and larning procsss. On th on hand, as computr softwar bcoms mor sophisticatd and provs to b wll-adaptd to particular ducational

14 S. bramovich and I. Lvin Downloadd by [Tl viv Univrsity] at : January purposs, it capturs mor and mor parts of th curriculum. On th othr hand studnts of all ags and abilitis ar intrigud by computing dvics irrspctiv of thir particular assignmnt, if only thy provid an xciting pursuit. Th ducational task, thn, should b only to stimulat studnts' radinss to larn mathmatical idas through appropriat computr tchnology. In this articl th authors try to xplicat th ffctivnss of a spradsht softwar for both taching and larning th concpt of a limit of a squnc. Thr may b an objction that th usfulnss of th computr as a tchnological tool for calculus lssons is not compltly apparnt. Of cours, a spradsht simply calculats th valus of X n and thn shows what numbr L thy sm to approach sinc a computr is capabl of xamining only a finit collction of mmbrs of a squnc irrspctiv of th tim of computing. Howvr, and w should lik to strss this, a spradsht allows studnts to construct tmplats of numbrs dsignd to illustrat how it sms that as n grows larg, X n gts clos to L. Morovr, watching this apparnt approach and rsponding simultanously to th prcis and dirctd qustions of a tachr, studnts ar abl not only to arriv at comprhnsion of a fairly abstract concpt but, bttr still, to formulat th xact dfinition of this concpt almost indpndntly. Finally, on mor pculiarity of th spradsht that contributs to its pdagogical advantag is its analytical invstigation and visualization capability. Easily constructd as an xplorr, th spradsht, in combination with tachr guidanc, givs studnts th opportunity to apprhnd th importanc of th thory of squncs in applications, by conjcturing and thn tsting th hypothss concrning th convrgnc of positiv sris. This howvr xcds th fram of th prsnt articl, and th authors hop to dvot a sparat papr to this mattr. Rfrncs [] FEY, J. T.,, Educ. Studis Math.,,. [] KPUT, J. J.,, Tchnology and mathmatics ducation. In Handbook of Rsarch on Mathmatics Taching and Larning (a projct of th N.C.T.M.), ditd by D.. Grouws (Nw York: Macmillan), pp. -. [] RGNRIGHT, D. E.,, Mathmatical pplications of Elctronic Spradshts (Nw York: McGraw-Hill). [] HELY, L., and SUTHERLND, R.,, Exploring Mathmatics with Spradshts (Oxford: lackwll). [] CPPONI,., and LCHEFF, N.,, Educ. Studis Math.,,. [] CLEMENTS, R. R.,, Int. J. Math. Educ. Sci. Tchnol.,,. [] MXIM,. R., and VERHEY, R. F.,, Using spradshts to introduc rcursion and diffrnc quations in high school mathmatics. In Discrt Mathmatics cross Th Curriculum, K- Yarbook, ditd by M. J. Knny and C. R. Hirsch (Rston, V: N.C.T.M.), pp. -. [] SCHWRZENERGER, R. L. E., and TLL, D. O.,, Math. Taching,,. [] ROERT,.,, Rsrchs n Didactiqu ds Mathématiqus,,. [] DVIS, R.., and VINER, S.,, J. Math. haviour,,. [] TLL, D. O.,, Th transition to advancd mathmatical thinking: functions, limits, infinity and proof. In Handbook of Rsarch on Mathmatics Taching and Larning (a projct of th N.C.T.M.), ditd by D.. Grouws (Nw York: Macmillan), pp. -. [] RMOVICH, S., and LEVIN, I.,, Us of spradshts in taching thory of limits. In ook of bstracts of Short Prsntations (ICME-), coordination: M. Millur (Québc: Univrsité Laval), p.. [] WEIGND, H.-G.,, Educ. Studis Math.,,.

15 Downloadd by [Tl viv Univrsity] at : January Spradshts in taching I larning calculus [] CO, D., MCGUFFEY,., and DODGE, M.,, Microsoft Excl Companion (Rdmont, W: Microsoft Prss). [] LWLER, R. W.,, Larning nvironmnts: now, thn, and somday. In rtificial Intllignc and Education, vol., ditd by R. W. Lawlr (Norwood, NJ: lx Publishing), pp. -. [] LEVIN, I., and RMOVICH, S.,, J. Comput. Math. Sci Taching,,. [] PPERT, S.,, Situating constructionism. In Constructionism, ditd by I. Harl and S. Paprt (Norwood, NJ: blx Publishing), pp. -. [] NNRD, D. N.,, Making connctions through itration. In Discrt Mathmatics cross Th Curriculum, K- Yarbook, ditd by M. J. Knny and C. R. Hirsch (Rston, V: N.C.T.M.), pp. -. [] SPENCE, L. E.,, Math. Tachr,,. [] RMOVICH, S., and LEVIN, I.,, J. Comput. Math. Sci. Taching,,.