Sustainability Development in Mathematics Education—A Case Study of What Kind of Meanings Do Prospective Class Teachers Find for the Mathematical Symbol “2/3 ”?

sus ainabili y A icle Sus ainabili y De elopmen in Ma hema ics Educa ion—A Case S udy o Wha Kind o Meanings Do P ospec i e Class Teache s Find o he Ma hema ical Symbol “2 3”? Jo ma Jou senlah i 1,* and Päi i Pe kkilä 2 1Facul y o Educa ion and Cul u e, Tampe e Uni e si y, 33014 Tampe e, Finland 2Facul y o Educa ion, Uni e si y o Jy äskylä, 40014 Jy äskylä, Finland; [email p o ec ed] *Co espondence: [email p o ec ed] Recei ed: 26 No embe 2018; Accep ed: 14 Janua y 2019; Published: 16 Janua y 2019   Abs ac : In his a icle, ou ocus is on sus ainable de elopmen in ma hema ics educa ion om he poin o iew o eache aining. The aim was o de elop p ospec i e eache s’ con en knowledge and pedagogical con en knowledge o school ma hema ics. As a case s udy, we chose he ma hema ical symbol “a/b”, and examined how p ospec i e class eache s in Finland connec i o he concep s o ac ion, a io, di ision, a ional numbe o p obabili y. Ma hema ics ex books o en ha e a cen al ole in lessons, and hey a ec s ongly how pupils unde s and concep s and he ela ionships be ween hem. We chose languaging as a mul i-semio ic app oach o in e p e ing wha kind o meanings he p ospec i e class eache s ga e he ma hema ical symbol “a/b”. The esul s show ha some o hese concep s a e di icul o see a he same ime om he gi en ma hema ical symbol. The concep o a io is pa icula ly di icul o p ospec i e class eache s o in e p e . Pic o ial p esen a ion suppo ed he in e p e a ions. Ma hema ics lea ning ma e ials and eache educa ion should de elop in acco dance wi h he esul s o he s udy. Keywo ds: ac ions; mul i-semio ic app oach; languaging; p ospec i e class eache ; con en knowledge; pedagogical con en knowledge 1. In oduc ion 1.1. Sus ainable De elopmen o Educa ion The concep o sus ainabili y is o en associa ed wi h ecological and economic issues, bu i also applies in he educa ional ealm. Fo example, e en wi hou an explici awa eness o he concep , al eady in 1657 Comenius dedica ed a chap e o his book “Didac ica Magna” o he “ ounda ions o las ing eaching and lea ning” [ 1 ]. The Uni ed Na ions (UN) has de eloped an agenda o 2030 Agenda o Sus ainable De elopmen , whe e educa ion is seen as one o he main co ne s ones o sus ainable de elopmen [ 2 ]. The main pu pose in he ou h goal o his agenda is o ensu e inclusi e and quali y educa ion o all and p omo e li elong lea ning. I is commonly known ha ma hema ics ac s as a ga ekeepe in educa ion: hose s uden s who ha e a good knowledge o ma hema ics a e mo e likely o ha e be e sco es and end wi h be e educa ional ajec o ies han hose who unde pe o m in his subjec [3]. Ou ocus is on sus ainable de elopmen in p ospec i e class eache s’ ma hema ics educa ion. Following Shulman [ 4 ], we dis inguish h ee ca ego ies o eache ’s con en knowledge: subjec ma e con en knowledge, pedagogical con en knowledge and cu icula knowledge. These ca ego ies a e Sus ainabili y 2019,11, 457; doi:10.3390/su11020457 www.mdpi.com/jou nal/sus ainabili y Sus ainabili y 2019,11, 457 2 o 15 applied o he eaching o p ima y school ma hema ics. We need o ake accoun o all h ee ca ego ies in de eloping ma hema ics eaching owa d g ea e sus ainabili y in achie ing be e lea ning esul s. In he Finnish Co e Cu iculum 2014 (FCC) [ 5 ], he wo d sus ainable is men ioned 176 imes. In he gene al pa o FCC, sus ainable de elopmen is seen as a “school’s ole in building a sus ainable u u e ha can be s eng hened in he o ganiza ion o educa ion”. The basic idea o sus ainable de elopmen in he FCC can be seen in he ligh o he UN’s agenda o 2030 Agenda o Sus ainable De elopmen , in which one o he main pu poses is o ensu e inclusi e and quali y educa ion o all and p omo e li elong lea ning. The FCC has a majo ole in de e mining he con en s o ma hema ics ex books, which in Finland a e he mos impo an ac o in luencing he s uc u e o ma hema ics lessons a p ima y school [ 6 ]. Thus, he ex books a e an impo an ool o ensu e sus ainabili y de elopmen in ma hema ics educa ion. 1.2. The Bases o Sus ainable De elopmen in Ma hema ics Educa ion Ma hema ical con en knowledge equi es unde s anding o he ac s o concep s o a domain. This means ha eache s mus unde s and he s uc u e o ma hema ics. Teache s mus no only be capable o de ining ma hema ical concep s, bu hey also mus be able o explain why a ma hema ical concep is wo h knowing and unde s anding, and how i ela es o o he concep s in bo h heo y and p ac ice. The second ca ego y o con en knowledge is pedagogical con en knowledge, which includes he eaching dimension o ma hema ical con en knowledge. In o he wo ds, pedagogical con en knowledge equi es he eache s o ha e he skill o eaching ma hema ics ha make i unde s andable o o he s. Because he e is no single mos e ec i e way o eaching, eache s mus ha e a hand al e na i e app oaches, some o which migh be based on he esul s o cu en esea ch, whe eas o he s a e bo n o expe ience. Pedagogical con en knowledge also in ol es an unde s anding o how pupils lea n ma hema ics, and wha makes he lea ning o a speci ic opic easy o di icul . The hi d ca ego y o con en knowledge is cu icula knowledge, which includes eache s’ skills and abili ies o ela e he ma hema ical con en o a gi en lesson o opics o issues being discussed in o he classes. A he same ime, eache s need o ha e a g asp o he ma hema ical opics and issues augh in he ma hema ical subjec a ea du ing he p eceding and la e yea s in school, and he eaching ma e ials ha embody hem. In building a sus ainable ma hema ics eache educa ion, and class eache educa ion, we unde s and ha bo h con en and p ocess a e impo an in eaching p o essionals. Ma hema ical con en knowledge includes knowledge o ma hema ical s uc u es, pedagogical knowledge o he gene al and speci ic opics o ma hema ics and specialized cu icula knowledge [4]. I a eache has oo na ow a pedagogical con en knowledge and cu icula knowledge, hen he eache migh see ma hema ics only as “ ules” and “p ocedu es”. In ha case, hey migh concen a e on ma hema ics eaching only in e ms o calcula ing, p ac icing and pe ec ing such ules and p ocedu es, wi hou ying o unde s and he unde lying concep s and s uc u es [ 7 ]. In his way, he eache does no build a solid pic u e o ma hema ical con en knowledge o hei s uden s. 1.3. A Case: F ac ions and Meanings o Symbol “a/b” We conside he concep o ac ion and he symbol “a/b” as a case om school ma hema ics. The esea ch (e.g., [ 8 – 11 ]) shows ha ac ions a e one o he mos challenging a eas in school ma hema ics. Pupils, p ospec i e class and subjec eache s, and some imes e en expe ienced eache s, ha e ouble wi h ac ions, especially unde s anding ac ions as numbe s ha ex end he whole numbe sys em o a ional numbe s (e.g., [ 12 , 13 ]). To unde s and his ex ension, one mus ha e a clea pic u e o he di e en meanings o ac ions. In Finnish ma hema ics ex books, hese di e en meanings o ac ions a e in oduced sepa a ely wi hou clea con ex , and a e o en augh emphasizing a p ocedu al pe spec i e. F ac ions ha e in e es ing ea u es compa ed o o he numbe s: wo ac ions could ha e di e en nume a o and denomina o , bu hey a e equal (equi alen ac ions). Sus ainabili y 2019,11, 457 3 o 15 The concep ual pic u e o he ac ion and i s di e en meanings ecei e less a en ion han hey should [14]. When mo ing om in ege s o a ional numbe s, a concep ual change needs o ake place in he pupils’ hinking. A concep ual change means a undamen al change in a pupil’s p e ious hinking [ 15 ]. Acco ding o Vosniadou [ 16 – 18 ], pupils end o o m an “inconsis en syn hesis” o ea lie knowledge and new lesson ma e ial, as he ea lie da a s uc u e is o en unconscious as well. In ma hema ics, such a phenomenon has been obse ed in he case o a numbe concep , whe e he na u e o he na u al numbe and he ules o ope a ions ha e been ound o ans e w ongly, o example, o he hinking and unde s anding o a ional numbe s [15,19]. Nowadays, ma hema ics ex books play a powe ul ole in he eaching and lea ning o ma hema ics. I is commonly known ha eache s us ex books, and base a majo pa o hei eaching on hem [ 20 , 21 ]. This gi es suppo o Pe kkilä’s [ 22 ] indings, who showed ha , in Finnish p ima y-le el ma hema ics, ex books and eache guides play a c ucial ole in eache s’ wo k. Teache guides usually include pedagogical ips and ins uc ions, ma hema ical con en knowledge, academic yea plans o ma hema ics, exac lesson plans, e c. Teache s e y o en ollow he sugges ions o eache guides on wha o each and how o each, and usually he eaching ma e ials also p esc ibe he o de in which he con en should be augh . Consequen ly, e en i ex books a e w i en o he pupils, hey can also be media o s be ween he in ended cu iculum and he implemen ed cu iculum [ 19 , 20 ]. In p inciple, p ospec i e class eache s ha e enough con en knowledge and cu icula knowledge o each o he concep s o ac ion, a io and di ision, bu he e could be p oblems in he unde s anding o how he concep s a e ela ed o each o he , and how we should ake accoun o i in ma hema ics eaching. This is an impo an pa o pedagogical con en knowledge, so i is in e es ing o gain a pic u e o how p ospec i e class eache s unde s and he di e en meanings o ac ions. Du ing he i s six g ades, i is class eache s who a e esponsible o he ma hema ics eaching. They o en eel hemsel es unce ain in ma hema ics [ 22 ], which is why hey end o ollow he con en s o ex books and eache guides s ic ly, which hen in luences he pupils’ iew o ma hema ical concep and knowledge. Due o his, pupils’ unde s anding o he di e en meanings o ma hema ical concep s and meanings—in his case, he di e en meanings o ac ions—may be based on he con en o ex books. In he cou se o he a icle, we e e o p ospec i e class eache s as he s uden s, and we use “2/3” as s anding o he symbol “2 3”. 2. Theo e ical Backg ound 2.1. Mul iple Meanings o Symbol “a/b” in School Ma hema ics Because o he abo e, i is in e es ing o examine he concep ual in e p e a ions o symbol “a/b”, o which he ma hema ics ex books gi e meanings depending on he con en a ea ma hema ics. I is no ou in en ion o discuss in his a icle he heo e ical ela ionships be ween he concep s o which symbol “a/b” is connec ed. Resea che s ha e ound di e en subcons uc s which a e connec ed o he symbol “a/b”. Each subcons uc c ea es a con ex o he ac ion ha gi es i a con ex ual meaning. These subcons uc s also e e o he ex ension o whole numbe s o a ional numbe s. Exis ing s udies (e.g., [ 23 – 25 ]) ha e conside ed he meanings o he symbol “a/b” om he poin o iew o he ac ion concep . Pan zia a and Philippou [ 23 ] highligh ed he ac ha pupils’ di icul ies in ac ion lea ning is linked o i s mul i ace ed cons uc . F om his poin o iew, hey desc ibed he ollowing i e subcons uc s: pa -whole, a io, quo ien , measu e, and ope a o . S ewa [ 24 ] also e e ed o he mul i aced cons uc o ac ions in in oducing he ollowing subcons uc s in he hesis: pa -whole ela ionship: he a io o he pa o he whole; measu emen : he posi ion o a numbe on he numbe line; ope a o : a ans o ma ion; quo ien : di ision; and a io: ela ionship be ween wo quan i ies. Pa k, Güle , and McG o y [ 12 ] app oached ac ions in hei esea ch amewo k om he poin o ou subcons uc s o he de elopmen o ac ions ound in his o y: pa -whole: concep ualizing a pa Sus ainabili y 2019,11, 457 4 o 15 o a whole as a new uni ; measu emen : inding ac ions om whole numbe s h ough measu emen and p opo ions, add essing he need o a common uni o measu emen o wo quan i ies; di ision: inding he algeb aic solu ion o an equa ion A X = B whe e A and B a e whole numbe s and A is nonze o; and se - heo e ical: de ining a ional numbe s as a se o o de ed pai s consis ing o whole numbe s. The di e en meanings o he ac ion concep in oduced abo e gi e qui e a good pic u e o how in ica e his concep is [ 12 , 23 , 24 ]; i can be assigned a ious meanings acco ding o he con ex in which he concep is used. In Finnish ma hema ics ex books [ 14 ], he symbol “2/3” has i e main in e p e a ions depending on con ex : (A) ac ion (A1 “ wo hi ds o a gi en whole” o A2 “ wo o h ee pa s”); (B) a ional numbe (“ wo hi ds”); (C) a io (“a a io o wo o h ee”); (D) di ision (“ wo di ided by h ee”); and (E) p obabili y (“The e is a wo- hi ds chance o i happening”). The i s one ( ac ion) is di ided in o wo subca ego ies acco ding o he e bal exp essions, o example “ wo hi ds o a gi en whole” o “ wo o h ee pa s”. We use he six subcons uc s as he heo e ical amewo k o he meanings o symbol “2/3”. A. The symbol “a/b” as a ac ion and as a pa ou o a sum o pa s. The pa -whole ela ionship is conside ed as he a io o he pa o he whole [ 23 , 24 ]. Beckmann [ 26 ] de ined he symbol “a/b” as a ac ion as ollows: I a and b a e whole numbe s and b is no ze o, and i an objec , collec ion, o quan i y can be di ided in o b equal pa s, hen he ac ion “a/b” o an objec , collec ion, o quan i y is he amoun o med by pa s (o copies o pa s). The pa -whole ela ionship can be in e p e ed in wo ways depending on he con ex (c . he abo e de ini ion): Fi s ly (A1), i we ake wo hi ds (“ wo hi ds o a gi en whole”) o a pizza, we ake wo equal slices ou o a pizza cu in o h ee equal pieces. Secondly (A2), i we ake wo pa s ou o h ee (“ wo o h ee”), we ein o ce he idea o di iding quan i ies in o h ee equal pa s and choosing wo o hem. Typically, ex books desc ibe he a io o a pa o he whole as an a ea model (e.g., as a pie model), and a pie model o “2/3” is ead o example as wo hi ds o he pizza. In addi ion, he pa -whole app oach may apply when eache s use a ea models di ided in o equal pa s, o men ion ha a pa o a collec ion o objec can be exp essed as a ac ion wi h equal pa i ioning and coun ing [12]. B. The symbol “a/b” as a a ional numbe . Acco ding o Pa k, Güle and McG o y [ 12 ], he oo s o he de ini ion o a ional numbe s a e based on a se - heo e ical app oach. Ra ional numbe s Q a e de ined as ollows: nm n  m,n∈Z,n6=0o(1) Fo example, in ege s a e a subse o Q, and on he o he hand Q is a subse o eal numbe s. F ac ions a e hose a ional numbe s which a e w i en in he o m m/n, in which m and n a e in ege s (n 6= 0). This kind o de ini ion is seen in many ex books, ollowed by explana ions o wha he symbols mean and how o manipula e wi h hem [ 12 ]. The a gumen is ue also in Finnish ma hema ics ex books, especially in g ades 7–9. In he ex books o g ades 3–6, a ional numbe s occu as ac ions and decimal numbe s, and in g ades 7–9 hey occu in he calcula ions o ac ions and decimal numbe s. In addi ion, in he ex books, a ional numbe s can be ep esen ed on a numbe line [ 27 ]. To p esen “2/3” on he numbe line, we di ide he segmen om 0 o 1 in o 3 segmen s o equal leng h. Then, s a ing om 0, we coun 2 o hese segmen s and s op he ma k co esponding o he igh endpoin second segmen o ob ain he poin assigned o he a ional numbe “2/3”. C. The symbol “a/b” as a a io. In he ex books, pupils encoun e he concep o a io connec ed o he ma hema ical symbol “2/3”, o example as ollows: a) “ he boa d o a ce ain associa ion should ha e a a io o 2 o 3 o women o men” (meaning wo women membe s o e e y h ee men); (b) “eggs cos wo eu os a dozen Sus ainabili y 2019,11, 457 5 o 15 ( his means ha e e y dozen eggs cos s wo eu os); and (c) “ wo pa s o luid A and h ee pa s o luid B make a o al mix u e o i e pa s”. Each o hese examples illus a es a a io [ 24 ]. The a ios a e w i en in Finnish ex books a bo a:b. D. The symbol “a/b” as a di ision The symbol “a/b” can also e e o di ision whe e i ocuses on he ope a ion [ 24 ]. In Finnish ex books, he di ision ope a ion is ep esen ed ei he by a hyphen “-“o by a di ision sign “:”. The di ision ope a ion can be ei he di ision by pa s o di ision by con en s depending on he con ex [ 28 ]. In p ac ice, he di ision app oach could be pe o ming di ision wi h ba e numbe s o a pa i i e di ision. The di ision “a/b”, in which a uni o a con inuous quan i y is sha ed equally by b ecipien s, men ioning ha he esul s o di ision a e no always whole numbe s. As a esul o a di ision, a ecu ing decimal ep esen a ion can o en be ounded up, which is no longe he exac alue ha he ac ion numbe ep esen s. Due o his, one needs ac ions o exp ess he esul s [12]. E. The symbol “a/b” as a p obabili y The symbol “a/b” can also mean p obabili y: Fo an expe imen wi h sample space S wi h equally likely ou comes, he p obabili y o e en A is gi en by P(A)=n(A):n(S)(2) whe e n(A) is numbe o elemen s o Aand n(S) is numbe o elemen s o S. The consequence o he de ini ion is ha 0≤P(A)≤1 (3) Fo example, when you oss a coin, he p obabili y o ge ing heads is 1 2. 2.2. The Role o Ma hema ics Tex books and Teache s Ma hema ics ex books a e a powe ul esou ce in ma hema ics classes (e.g., [ 11 , 29 ]) because hey in e p e he ma hema ics cu iculum o bo h pupils and eache s. Finnish class eache s ely on he use o ma hema ics ex books in eaching because hey belie e ha doing so ensu es ha hey will ollow he na ional cu iculum [ 22 ]. A simila phenomenon can be seen in he eaching o Finnish ma hema ics eache s, as hey o en see he ex book as a conc e e cu iculum ha de ines he con en s o he eaching, he goals and he didac ical solu ions [ 25 ]. In many cases, ma hema ics lessons a e planned and implemen ed h ough he ma hema ical asks and ac i i ies ound in he ex books. Fo his eason, ma hema ics ex books may be di ec ly connec ed o he s uc u e o eaching p ac ice and can hus ac as an implemen able cu iculum [30–32]. In Finland, eache s a e esponsible o he choice o ex books. The ex book has a s ong in luence on eache ’s didac ical choices, bu he eache s ill has he op ion o no using i in he lessons. Fo example, eache s can build up he whole lea ning p ocess based on pupils’ indi idual wo k wi h he ma hema ics ex books, o hey may use he ex books simply as a sou ce o exe cises and homewo k asks. Fo ma hema ics lea ning, ma e ials included in he ex books comp ise ac i i ies such as eading explana o y ex s and acqui ing new con en , looking h ough wo ked examples, games, sol ing asks, e c. Pupils do no ac ually ge he oppo uni y o ully exploi he ma hema ics ex book as a mul i ace ed lea ning esou ce, because eache s mos ly use he book as a c ucial esou ce o exe cises. Ma hema ics ex books a e o en laid ou in such a way ha each page-sp ead is designed o be deal wi h du ing one lesson. In addi ion, he eady-made es s a e included as ma e ials o he eache s. Due o his, he eache is an implemen e o a lea ning p ocess ha was designed and egula ed by ex book au ho s [25,30,33]. Sus ainabili y 2019,11, 457 6 o 15 Typically, in he Finnish ex books e e y opic s a s wi h he de ini ions o a new concep o con en ; o en he e a e wo ked examples, ollowed by a se o asks which could include ma e ial o se e al pagesp eads ( ha means sequen ial lessons). Fo example, ex books usually desc ibe he a io o he pa o he whole as an a ea model (e.g., as a pie model), and a pie model o “2/3” would be ead o example as wo hi ds o a pizza. Especially in he hi d g ade, when pupils ge o know he symbol “a/b”, in ex books, his symbol is in e p e ed in he con ex o a ac ion. As p e iously men ioned in he In oduc ion, ex books o en emphasize a p ocedu al pe spec i e conce ning he eaching o ac ions, in which case “ ex book d i en” augh pupils will no ge a clea pic u e o he ac ion concep . As men ioned abo e, Finnish ma hema ics eaching is “ ex book d i en”. This o en means ha he na u e o ma hema ics and he con en s o school ma hema ics appea o pupils only h ough he medium o ex books. I bo h he ex books and he lessons emphasize calcula ions, he e is no oom le o he pupils o exp ess hei own hinking, o example by means o languaging. The ex books emphasize he pa -whole app oach [14], bu he o he meanings o he ac ion concep do no come h ough clea ly. This is why i is in e es ing o disco e wha kind o meanings p ospec i e class eache s gi e o he symbol “a/b”. 2.3. Meaning Making o he Concep The main pu pose o using na u al language (mos o en he s uden ’s mo he ongue), and isual ep esen a ions in lea ning ac i i ies (e.g., in s udying new ma hema ical concep s and doing exe cises) is o de elop he s uden ’s own meaning-making p ocesses. Fo example, Mo gan [ 34 ] showed ha w i ing and he use o na u al language in he solu ions o ma hema ical p oblems may in ac boos lea ning in ma hema ics, de elop ma hema ical unde s anding, change he pupil’s a i ude owa ds ma hema ics o he be e , and help he eache ’s e alua ion. I we b oaden he adi ional meaning o “language”, we can unde s and he di e en ep esen a ions o he concep as exp essions by e sa ile languages: ma hema ical symbolic language (MSL), na u al language (NL) and pic o ial language (PL) [ 28 , 35 – 37 ]. We call he p ocess in which o example a pupil exp esses hei hinking as “languaging” [ 38 ]. We desc ibe languaging in ma hema ics as e e ing o exp essing one’s ma hema ical hinking by di e en modes, ei he o ally (by na u al language), o in w i ing (by na u al language, ma hema ical symbolic language, o pic o ial language) [ 28 ]. We can he e o e ecognize h ee languages which ma hema ics ex books use as meaning-making ools o ma hema ical concep s and p ocedu es. Howe e , we see ha he h ee languages should also be used by he pupils; i has been shown ha languaging o ma hema ical hinking helps pupils o s uc u e hei hinking and in ha way o unde s and ma hema ical concep s and p ocedu es [28,34,37,39]. One aspec o unde s anding a concep is an awa eness o how i is ela ed o o he concep s, and an abili y o use i meaning ully in new con ex s. In school ma hema ics, we can ecognize many ep esen a ions o concep s: ma hema ical symbolism, exp essions by na u al language, isual display and manipula ions by conc e e ma e ials (e.g., [ 10 , 37 ]). F om he poin o iew o a concep , we can speak o ep esen a ions o he concep [10,40]. When a pupil exp esses hei ma hema ical hinking, he o she can use di e en mul imodal app oaches (e.g., MSL, NL and/o PL). Theo e ically, he mul imodal languaging model is ela ed o mul ili e acy [ 41 ]. When a eade makes meanings o a ma hema ical ex he languages can be seen as a mul i-semio ic app oach, whe e he di e en languages make i possible o cons uc many kinds o meanings o concep s in e sa ile con ex s [ 28 , 36 , 37 ]. In his a icle, we concen a e on connec ing ma hema ical symbolic language and pic o ial language by in e p e ing s uden s’ exp essions in na u al language. Sus ainabili y 2019,11, 457 7 o 15 3. Resea ch Ques ions and Da a Collec ion 3.1. Resea ch Ques ions Du ing Finnish elemen a y school, ou pupils go h ough all he meanings o he symbol “a/b”: a ac ion, a a ional numbe , a a io, a di ision o a p obabili y. Howe e , in school ma hema ics, he e a e a ely si ua ions whe e hese meanings a e conside ed simul aneously. Ou pu pose is o s udy wha meanings Finnish s uden s ind ou o ha lis . Especially i we conside p ospec i e class eache s, we can unde s and he di icul ies in he concep ual lea ning o ac ions, and ind new de elopmen a ge s in eache educa ion. The e o e, i would be in e es ing o examine he meanings which p ospec i e class eache s (ou s uden s in his case) ha e gi en o he ma hema ical symbol “2/3”. In his a icle, we concen a e on he ollowing esea ch ques ions: 1) Wha meanings do s uden s gi e spon aneously o he symbol “2/3”? 2) Wha ela ionships do s uden s ind o he gi en pic u es and he symbol “2/3”? 3) Wha kind o in luences has he mul i-semio ic app oach o s uden s’ in e p e a ions? 3.2. Da a Collec ion Ou da a in ol ed 102 i s -yea s uden s o wo uni e si ies in Finland, 71 om he Uni e si y o Tampe e and 31 om he Uni e si y o Jy äskylä. The e we e 85 emale and 17 male s uden s. The da a we e collec ed du ing he ma hema ics didac ics cou se by ques ionnai es in he sp ing o 2017. The ques ionnai e had h ee pages: 1) On he i s page, he s uden s ga e hei opinions in he open ques ion abou wha di e en meanings (e.g., ac ion, di ision, a io, e c.) he ma hema ical symbol “2/3” can ha e. Se e al lines we e p o ided o he answe s. 2) On he second page, he s uden s we e asked o desc ibe in na u al language (Finnish) how he pic u es A–D (Figu e 1) a e connec ed o he ma hema ical symbol “2/3”. The s uden s could gi e one o mo e desc ip ions o he connec ions. 3) On he hi d page, he e was a ma hema ical p oblem, including he pic u e and he ma hema ical solu ion (Figu e 2). The s uden ’s ask was o explain he solu ion p ocesses by na u al language, and o disco e whe he he e we e any mis akes in he p esen a ion. Sus ainabili y 2019, 11, x FOR PEER REVIEW 7 o 14 1) Wha meanings do s uden s gi e spon aneously o he symbol “2/3”? 2) Wha ela ionships do s uden s ind o he gi en pic u es and he symbol “2/3”? 3) Wha kind o in luences has he mul i-semio ic app oach o s uden s’ in e p e a ions? 3.2. Da a Collec ion Ou da a in ol ed 102 i s -yea s uden s o wo uni e si ies in Finland, 71 om he Uni e si y o Tampe e and 31 om he Uni e si y o Jy äskylä. The e we e 85 emale and 17 male s uden s. The da a we e collec ed du ing he ma hema ics didac ics cou se by ques ionnai es in he sp ing o 2017. The ques ionnai e had h ee pages: 1) On he i s page, he s uden s ga e hei opinions in he open ques ion abou wha di e en meanings (e.g., ac ion, di ision, a io, e c.) he ma hema ical symbol “2/3” can ha e. Se e al lines we e p o ided o he answe s. 2) On he second page, he s uden s we e asked o desc ibe in na u al language (Finnish) how he pic u es A–D (Figu e 1) a e connec ed o he ma hema ical symbol “2/3”. The s uden s could gi e one o mo e desc ip ions o he connec ions. 3) On he hi d page, he e was a ma hema ical p oblem, including he pic u e and he ma hema ical solu ion (Figu e 2). The s uden ’s ask was o explain he solu ion p ocesses by na u al language, and o disco e whe he he e we e any mis akes in he p esen a ion. Figu e 1. The second ques ion on he second page o he esea ch ques ionnai e: How he pic u es A– D desc ibe ma hema ical symbol “2/3”. The i s and he second page o he ques ionnai e we e made in he cou se unde eache s’ con ol. The solu ions o he hi d page we e made as homewo k. Figu e 2. The hi d ques ion on he hi d page o he esea ch ques ionnai e: Explain wha has been done in he p esen a ion and conside he co ec ness o he solu ion p ocess. Figu e 1. The second ques ion on he second page o he esea ch ques ionnai e: How he pic u es A–D desc ibe ma hema ical symbol “2/3”. Sus ainabili y 2019,11, 457 8 o 15 Sus ainabili y 2019, 11, x FOR PEER REVIEW 7 o 14 1) Wha meanings do s uden s gi e spon aneously o he symbol “2/3”? 2) Wha ela ionships do s uden s ind o he gi en pic u es and he symbol “2/3”? 3) Wha kind o in luences has he mul i-semio ic app oach o s uden s’ in e p e a ions? 3.2. Da a Collec ion Ou da a in ol ed 102 i s -yea s uden s o wo uni e si ies in Finland, 71 om he Uni e si y o Tampe e and 31 om he Uni e si y o Jy äskylä. The e we e 85 emale and 17 male s uden s. The da a we e collec ed du ing he ma hema ics didac ics cou se by ques ionnai es in he sp ing o 2017. The ques ionnai e had h ee pages: 1) On he i s page, he s uden s ga e hei opinions in he open ques ion abou wha di e en meanings (e.g., ac ion, di ision, a io, e c.) he ma hema ical symbol “2/3” can ha e. Se e al lines we e p o ided o he answe s. 2) On he second page, he s uden s we e asked o desc ibe in na u al language (Finnish) how he pic u es A–D (Figu e 1) a e connec ed o he ma hema ical symbol “2/3”. The s uden s could gi e one o mo e desc ip ions o he connec ions. 3) On he hi d page, he e was a ma hema ical p oblem, including he pic u e and he ma hema ical solu ion (Figu e 2). The s uden ’s ask was o explain he solu ion p ocesses by na u al language, and o disco e whe he he e we e any mis akes in he p esen a ion. Figu e 1. The second ques ion on he second page o he esea ch ques ionnai e: How he pic u es A– D desc ibe ma hema ical symbol “2/3”. The i s and he second page o he ques ionnai e we e made in he cou se unde eache s’ con ol. The solu ions o he hi d page we e made as homewo k. Figu e 2. The hi d ques ion on he hi d page o he esea ch ques ionnai e: Explain wha has been done in he p esen a ion and conside he co ec ness o he solu ion p ocess. Figu e 2. The hi d ques ion on he hi d page o he esea ch ques ionnai e: Explain wha has been done in he p esen a ion and conside he co ec ness o he solu ion p ocess. The i s and he second page o he ques ionnai e we e made in he cou se unde eache s’ con ol. The solu ions o he hi d page we e made as homewo k. The idea o he ques ionnai e was based on ou heo e ical backg ound; om he answe s, we wan ed o ge an unde s anding o he s uden s’ pe cep ions o he meanings o he ma hema ical symbol “2/3”, and on he o he hand how a mul i-semio ic app oach suppo s he in e p e a ions. 4. Resul s The da a we e analyzed by mixed me hods. We used he IBM SPSS s a is ic 24 p og am o ypical s a is ical analysis (e.g., in compa ing dis ibu ions). The quali a i e analysis was made by heo y guided con en analysis (e.g., ca ego iza ions). We used he classi ica ion in o he six ca ego ies p esen ed in he heo e ical pa o he a icle, and on he o he hand he ca ego ies gene a ed by he answe s. In he i s ques ion o he ques ionnai e, s uden s’ (N = 102) spon aneous unde s anding o he ma hema ical symbol “2/3” was in mos cases as a ac ion (77 s uden s ga e “ wo hi ds o a gi en whole” and 43 s uden s “ wo o h ee pa s”), which a e ypical also in he Finnish ma hema ics ex books (Figu e 3). Ra io, a ional numbe o p obabili y we e men ioned he leas . We ha e chosen ypical e bal exp essions as examples om e e y ca ego y (Figu e 3): 1) Di ision: “Two di ided by h ee: wo pizzas o be di ided in o h ee people.” (S uden 89), 2) F ac ion 1: “In he es I go wo hi ds o answe s co ec ”. (S uden 55), 3) F ac ion 2: “Two ou o h ee. Pekka has ea en wo pizza pieces om h ee.” (S uden 87), 4) P obabili y: “Can be used o desc ibe p obabili y.” (S uden 76), 5) Ra ional numbe : “0.66 . . . ” (S uden 92), 6) Ra io: “You’ll ge good juice i you pu 2 dl concen a e and 3 dl wa e ” (S uden 57). When we coun ed how many di e en meanings each p ospec i e class eache (N = 102) ound in Ques ion 1 on he i s page (spon aneously answe ed) and in Ques ion 2 on he second page (pic o ial guided, see Figu e 1) o he ques ionnai e, we go he dis ibu ions o he o als in Figu e 4. The co ela ion (Pea son = 0.245) be ween he dis ibu ions in Figu e 4is signi ican (p< 0.05). The second dis ibu ion (pic o ial) is signi ican ly (p= 0.000) be e han he i s one (Wilcoxon signed ank es : Z=−8.168). Sus ainabili y 2019,11, 457 9 o 15 Sus ainabili y 2019, 11, x FOR PEER REVIEW 8 o 14 The idea o he ques ionnai e was based on ou heo e ical backg ound; om he answe s, we wan ed o ge an unde s anding o he s uden s’ pe cep ions o he meanings o he ma hema ical symbol “2/3”, and on he o he hand how a mul i-semio ic app oach suppo s he in e p e a ions. 4. Resul s The da a we e analyzed by mixed me hods. We used he IBM SPSS s a is ic 24 p og am o ypical s a is ical analysis (e.g., in compa ing dis ibu ions). The quali a i e analysis was made by heo y guided con en analysis (e.g., ca ego iza ions). We used he classi ica ion in o he six ca ego ies p esen ed in he heo e ical pa o he a icle, and on he o he hand he ca ego ies gene a ed by he answe s. In he i s ques ion o he ques ionnai e, s uden s’ (N = 102) spon aneous unde s anding o he ma hema ical symbol “2/3” was in mos cases as a ac ion (77 s uden s ga e “ wo hi ds o a gi en whole” and 43 s uden s “ wo o h ee pa s”), which a e ypical also in he Finnish ma hema ics ex books (Figu e 3). Ra io, a ional numbe o p obabili y we e men ioned he leas . We ha e chosen ypical e bal exp essions as examples om e e y ca ego y (Figu e 3): 1) Di ision: “Two di ided by h ee: wo pizzas o be di ided in o h ee people.” (S uden 89), 2) F ac ion 1: “In he es I go wo hi ds o answe s co ec ”. (S uden 55), 3) F ac ion 2: "Two ou o h ee. Pekka has ea en wo pizza pieces om h ee ". (S uden 87), 4) P obabili y: “Can be used o desc ibe p obabili y.” (S uden 76), 5) Ra ional numbe : “0.66…” (S uden 92), 6) Ra io: “You’ll ge good juice i you pu 2 dl concen a e and 3 dl wa e ” (S uden 57). Figu e 3. The meanings he s uden s (N = 102) ga e o he ma hema ical symbol (Ques ion 1 on he i s page o he ques ionnai e). When we coun ed how many di e en meanings each p ospec i e class eache (N = 102) ound in Ques ion 1 on he i s page (spon aneously answe ed) and in Ques ion 2 on he second page (pic o ial guided, see Figu e 1) o he ques ionnai e, we go he dis ibu ions o he o als in Figu e 4. The co ela ion (Pea son = 0.245) be ween he dis ibu ions in Figu e 4 is signi ican (p < 0.05). The second dis ibu ion (pic o ial) is signi ican ly (p = 0.000) be e han he i s one (Wilcoxon signed ank es : Z =− 8.168). 63 77 43 1 30 38 0 10 20 30 40 50 60 70 80 90 Di ision F ac1 F ac2 P ob Ra Num Ra io F equency Meanings F ac1: “ wo hi d o a gi en whole” F ac 2: “ wo o h ee” P ob: p obabili y Ra Num: a ional numbe Figu e 3. The meanings he s uden s (N = 102) ga e o he ma hema ical symbol (Ques ion 1 on he i s page o he ques ionnai e). Sus ainabili y 2019, 11, x FOR PEER REVIEW 9 o 14 Figu e 4. Dis ibu ions o o als o di e en meanings he s uden s (N = 102) gi en spon aneously (Ques ion 1 on he i s page o he ques ionnai e), and o he gi en pic u es A–D (Ques ion 2 on he second page o he ques ionnai e), o he ma hema ical symbol “2/3”. Fo example quan i y “2” means, ha 44 s uden s ound exac ly wo di e en meanings o he symbol “2/3” in Ques ion 1 and 17 s uden s in Ques ion 2. The pic u es (see Figu e 1) associa ed wi h he ma hema ical symbol “2/3” ga e mo e e ec i e meanings o he s uden s han a pu ely ma hema ical symbol. Pic u e A (Figu e 1) had he mos , and pic u e B he ewes in e p e a ions o he six ca ego ies (Table 1). The concep o a io seems o be he mos di icul o connec wi h he ma hema ical symbol “2/3”. An in e es ing ea u e in he in e p e a ions o pic u e B was ha 13 s uden s supposed ha pic u e B does no connec in any way o he gi en ma hema ical symbol, ha ins ead o “2/3” he e should be “2/5”, and ha he in e p e a ion o he connec ion is a ac ion illus a ion. Table 1. F equencies o he s uden s’ (N = 102) in e p e a ions how ma hema ical symbol “2/3” is connec ed o pic u es A–D (Figu e 1, Ques ion 2 on he second page o he ques ionnai e). Pic u e A B C D F ac1 59 0 18 45 F ac2 38 0 0 0 Ra io 2 43 0 0 Di ision 1 1 56 1 Ra ional 3 0 1 41 O he 16 49 15 8 In he answe s o he hi d ques ion on he hi d page o he ques ionnai e, s uden s exp essed ma hema ical meanings on he basis o a pic u e (Figu e 2) in which di e en languages (MSL, NL, and PL) gi e possibili ies o mul imodal in e p e a ions. In he da a analysis o he hi d ques ion, we used a classi ica ion in o ca ego ies gene a ed by he answe s. The answe s o he hi d ques ion (Figu e 2) we e di ided in o ou hemes (Table 2): (1) meanings o he symbols ” ” and “ ” (52 e e ences in he answe s); (2) concep ual app oach o he exp essions ( e e ences in he answe s); (3) p ocedu al app oach o he exp essions (81 e e ences in he answe s); and (4) do no ind any p oblems in he p esen a ion (35 e e ences in he answe s). Be o e making he hemes, we di ided he answe s in o ca ego ies, and, inally, pu hese ca ego ies in o he app op ia e hema ic a eas as ollows: he i s heme (1a, 1b, and 1c; see Table 2), he second heme (2a and 2b; see Table 2), he hi d heme (3a, 3b, 3c, and 3d; see Table 2), and he ou h heme (4; see Table 2). Many s uden s 0 10 44 36 12 2 5 17 40 38 0 5 10 15 20 25 30 35 40 45 50 "0" "1" "2" "3" "4" F equency Quan i y Spon aneous Pic o ial Figu e 4. Dis ibu ions o o als o di e en meanings he s uden s (N = 102) gi en spon aneously (Ques ion 1 on he i s page o he ques ionnai e), and o he gi en pic u es A–D (Ques ion 2 on he second page o he ques ionnai e), o he ma hema ical symbol “2/3”. Fo example quan i y “2” means, ha 44 s uden s ound exac ly wo di e en meanings o he symbol “2/3” in Ques ion 1 and 17 s uden s in Ques ion 2.