TEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Buma Abramovitz 1, Miryam Berezina 1, Abraham Berman 2

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1 TEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Bum Armoviz 1, Mirym Berezin 1, Arhm Bermn 1 Deprmen of Mhemics, ORT Brude College, Krmiel, Isrel Deprmen of Mhemics, Technion IIT, Hf, Isrel Asrc We give wo emples of using homework online ssignmens o direc sudens o prove, on heir own, heorems in clculus One emple is Green's heorem in recngle The second is he sic heorem of clculus 1 Inroducion In his pper we descrie n pproch of eching sudens o prove independenly some mhemicl semens I coninues our ques for wys of improving sudens undersnding of he heoreicl pr of he sujec This pper is sed on [1] I conins wo emples descriing our mehod One emple (he sic heorem of clculus) is new The oher (Green's heorem) is n dopion of n emple from [1] o online homework This pper nd [1] re pr of series of works on educing undergrdue engineers o pprecie nd undersnd he heoreicl pr of he mhemicl courses h hey ke The previous resuls were pulished in ([ 5], [7]) I is well known h heorems nd heir proofs re n imporn pr of mhemics educion The prolem of proving heorem nd is impornce were disscused in [6, 8, 9, 11, 1, 17, 19] For more deiled ckground see [1] This work is influenced y he conceps of "rnspren proofs", (see [15, 16]) nd "generic proofs" (see [14]) In his pper we descrie how we ech our sudens o prove wo cenrl heorems of clculus: he fundmenl heorem of clculus nd Green's heorem in recngle We choose suile generic emple for which he proof is essenilly he sme s in he generl cse Anoher conriuion of our pproch is direcing he sudens hrough he differen seps of he proof y using cive homework We use he echnology of we ssignmens since checking rdiionl homework is prolemic in lrge clsses A homework on Green's heorem The im of his homework is o help sudens o prove Green's Theorem: Theorem If L L= y,, y, oriened in he is he oundry of he recngle ( ) counerclockwise nd PQ, C 1 ( L) hen { } P (, y) d + Q(, y) dy = ddy The firs wo ssignmens re prolems h sudens re used o solve Assignmen 1 is he firs sep in he process of sudens lerning o prove Sudens hve o wrie down he correc nswer, wihou ny eplnion If his nswer is wrong, hen hey re re-direced o solve his ssignmen once more Assignmen 1 Find he inegrl yd + y dy where L is he oundry of he squre {(, ), 1} L= y y, oriened in he counerclockwise

2 1 The correc nswer is I = If sudens ge differen nswer hey go ck o ssignmen 1 Assignmen Find he inegrl J = ddy Py, = yqy,, = y nd ( ) ( ) where L is he squre from ssignmen 1 1 The correc nswer is J = Assignmen Wh is he connecion eween I nd J? Mrk he correc nswer in he lis elow: I = J I = J c I = J d I = J The poin of his ssignmen is h he sudens noe h he correc nswer is d The ne ssignmens re preprrion for proving Green's heorem, since he direc compuion of he inegrls is no possile Assignmen 4 + y y Find he inegrl e d + e dy where L is he oundry of he recngle {(, ), } L= y y, oriened in he counerclockwise Epress he nswer in erms of definie inegrls of eponenil funcions of one vrile Mrk he correc nswer y y ( ) ( ) + + e e d e e dy + y y + e e d e e dy ( ) + ( ) y y e e d e e dy c ( + ) + ( ) + y y + d ( ) + ( ) e e d e e dy y y The corec nswer is ( ) ( ) + + e e d e e dy This ssignmen cn e divided ino four uiliry ones, y clculing inegrls on he sides of he recngle This ssignmen is noher sep in he preprion process I is no possile o clcule he line inegrl, u i is possile o presen i s sum of wo definie inegrls, s i is done in he proof of Green's heorem Assignmen 5 Find he inegrl J = ddy where L is he recngle from ssignmen 4 nd ( ) + y,, (, ) y Py = e Qy = e Epress he nswer in erms of definie inegrls of eponenil funcions of one vrile Mrk he correc nswer

3 y y ( ) ( ) + + e e d e e dy y y ( ) ( ) + + e e d e e dy y y c ( + ) ( ) + + e e d e e dy + y y + d ( ) + ( ) e e d e e dy y y The correc nswer is ( ) ( ) + + e e d e e dy This ssignmen cn lso e divided ino wo uiliry one: o compue L ddy nd o compue P ddy Sudens were requesed o find he connecion eween I nd J The correc nswer is I = J The ls ssignmens 6 nd 7 re he proof of he heorem Assignmen 6 P, y d + Q, y dy is he oundry of he recngle where L Find he inegrl ( ) ( ) {(, ), } =, oriened in he counerclockwise nd PQ, C 1 ( L) L y y Epress he nswer in erms of definie inegrls of funcions of one vrile Mrk he correc nswer P P d Q y Q y dy (,) + (, ) + (, ) + (, ) (,) (, ) + (, ) + (, ) P P d Q y Q y dy P P d Q y Q y dy c (,) (, ) + (, ) (, ) P P d Q y Q y dy d (,) (, ) + (, ) (, ) This ssignmen cn e divided in four uiliry ones, y king inegrls on he sides of he recngle P P d Q y Q y dy The correc nswer is (,) (, ) + (, ) (, ) Assignmen 7 Find he inegrl J = ddy where L is he recngle from ssignmen 1 nd PQ, C 1 ( L) Epress he nswer in erms of definie inegrls of funcions of one vrile Mrk he correc nswer

4 P P d Q y Q y dy (,) + (, ) + (, ) + (, ) (,) (, ) + (, ) + (, ) P P d Q y Q y dy P P d Q y Q y dy c (,) (, ) + (, ) (, ) P P d Q y Q y dy d (,) (, ) + (, ) (, ) P P d Q y Q y dy The correc nswer is (,) (, ) + (, ) (, ) A homework on he sic heorem of clculus The im of his homework is o prepre sudens o prove he sic heorem of he Clculus To help he sudens o prove he heorem we offered sudens he following ssignmens Assignmen 1 If ( ) g = d Find g( ) Find g ( ) 4 g = g = 4 Assignmen 1 is simple prolem, h sudens know o solve Assignmen The correc nswer is ( ), ( ) Define g ( ) = e d Find g ( ) Hin: Use he definiion of he derivive Here sudens re unle o sr y clculing g( ) They hve o use he definiion of he derivive, s in he proof of he heorem To compue he derivive we need he difference g( ) g( ), so we sr wih he ssignmen In ssignmens 6 we denoe ( ) g = e d Assignmen If hen g( ) g( ) is equl o e d e d The correc nswer is () Assignmen 4 e d e d c e d d e d e d

5 If, y using he properies of he definie inegrl (he funcion for every rel ) g( ) g( ) cn e wrien s: e d e d The correc nswer is () Assignmen 5 If e d c e d d e d e is coniuous, y using he men heorem of he definie inegrl here eiss poin c( ) such h c( ) nd g( ) g( ) is equl o: c ( ) c( ) e ( ) e c c ( e c( ) d e ) ( ) The correc nswer is () Assignmen 6 Under he ssumpions nd wih he noions of ssignmen 5, i follows h lim c = nd ( ) ( ) g( ) g g ( ) = lim Is equl o e e c e d e The correc nswer is (c) Thus, sudens hve proved h g ( ) = e The ne ssignmen is he proof of he heorem Assignmen 7 Le ( ) f e funcion coninuous on he inervl ( αβ, ), ( αβ, ) define g ( ) = f ( ) d For ( αβ) find ( ), g fied poin, Hin: Use he definiion of he derivive Hopefully, his poin sudens re redy o prove he heorem If he echer feels h he sudens need more help hn ddiionl ssignmens cn e offered 4 Discussions The ide of his pper is h he sudens cn e ugh o prove, on he heir own, heorems in clculus y using series of ssignmens The limiion of our pproch is: i is suile o proofs h re sed on clculion We give wo emples where he ssignmens re given s online homework References 1 B Armoviz, M Berezin, A Bermn (14), Lerning o prove: from emples o generl semens, Inernionl Journl of Mhemicl Educion in Science nd Technology, in press B Armoviz, M Berezin, A Bermn, L Shvrsmn (9), How o undersnd heorem? Inernionl Journl of Mhemicl Educion in Science nd Technology, 4(5), pp B Armoviz, M Berezin, A Bermn, L Shvrsmn (9), Proofs nd puzzles, in A Rogerson (Ed) Proceedings of he 1h inernionl conference Models in developping mhemics educion, Dresden, Germny, pp 5-9

6 4 B Armoviz, M Berezin, A Bermn, L Shvrsmn (7), Lgrnge s heorem: wh does he heorem men?, in: D Pi-Pnzi,, G Philippou, (eds), Proceedings of CERME 5, Lrnc, Cyprus, pp B Armoviz, M Berezin, A Bermn, L Shvrsmn (1), A lended lerning pproch in mhemics, in: A A Jun, M A Huers, S Trenholm, nd C Seegmn (eds), Teching mhemics online: emergen echnologies nd mehodologies, IGI Glol, pp -4 6 N Blcheff (), The resercher episemology: dedlock from educionl reserch on proof, in: Fou Li Lin (ed), Proceedings of inernionl conference on mhemics Undersnding proving nd proving o undersnd, Tipei, NSC nd NTNU, pp M Berezin, A Bermn (), Proof reding nd muliply choice ess, Inernionl Journl of Mhemicl Educion in Science nd Technology, 1 (5), pp M D De Villiers (6), Rehinking proof wih geomeer skechpd, Key Curriculum Press 9 G Hnn (), Proof, eplnion nd eplorion: n overview, Educionl Sudies in Mhemics, 44(1), pp5-1 G Hnn, M De Villiers, F Arzrello, T Dreifus, V Durnd-Guerrier, H N Jhnke, F L Lin, A Selden, D Tll, O Yevdokimov (9), Discussion Documen, In: F L Lin, F J Hsieh, G Hnn, M De Villiers (eds), Proceedings of he ICMI Sudy 19 conference: proof nd proving in mhemic educion, vol, Tipei, Tiwn 11 L Hely, C Hoyles (1998), Jusifying nd proving in school mhemics, London: Universiy of London, Insiue of Educion: Technicl Repor 1 C Hoyles (1997), The curriculr shping of sudens pproches of proofs, For he lerning of Mhemics 17(1), pp C Hoyles, C, D Kuchemnn (), Sudens' undersnding of logicl implicions, Educionl sudies in Mhemics, 519(), pp U Leron, O Zslvsky, O (9), Generic proving: reflecion on scope nd mehod, In: F L Lin, F J Hsieh, G Hnn, M De Villiers (eds), Proceedings of he ICMI Sudy 19 conference: proof nd proving in mhemic educion, vol, Tipei, Tiwn 15 N Movshoviz Hdr (1988), Simuling presenions of heorems followed y responsive proofs, For he Lerning of Mhemics, 8(), pp1-16 A Mlek, N Movshoviz Hdr (9), The r of consrucing P- rnspren proof In: F L Lin, F J Hsieh, G Hnn, M De Villiers (eds), Proceedings of he ICMI Sudy 19 conference: proof nd proving in mhemic educion, vol, Tipei, Tiwn 17 M A Mrioi, (6), Proof nd proving in mhemics educion In: A Guierrez, P Boero, (eds), Hndook of reserch on he psychology of mhemics educion: ps, presen nd fuure, Roerdm/Tipei, Sense Pulishers, pp D Tll (1989), The nure of mhemicl proof, Mhemics Teching, 17, pp 8-19 D Tll (9), Cogniive nd socil developmen of proof hrough emodimen, symolism nd formlism In: F L Lin, F J Hsieh, G Hnn, M De Villiers (eds), Proceedings of he ICMI Sudy 19 conference: proof nd proving in mhemic educion, vol, Tipei, Tiwn