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1 Deakin Research Online This is he published version: Widjaja, Wany 2010, Modelling he cooling of coffee : insighs from a preliminary sudy in Indonesia, in MERGA 2010 : Shaping he fuure of mahemaics educaion : Proceedings of he 33rd annual conference of he Mahemaics Educaion Research Group of Ausralasia, MERGA, Fremanle W. A., pp Available from Deakin Research Online: hp://hdl.handle.ne/10536/dro/du: Reproduced wih he kind permissions of he copyrigh owner. Copyrigh : 2010, MERGA

2 Modelling he Cooling of Coffee: Insighs From a Preliminary Sudy in Indonesia Wany Widjaja Sanaa Dharma Universiy <wany_widjaja@yahoo.com > This paper discusses an aemp o examine pre-service eachers mahemaical modelling skills. A modelling projec invesigaing relaionships beween emperaure and ime in he process of cooling of coffee was chosen. The analysis was based on group wrien repors of he cooling of coffee projec and observaion of classroom discussion. Findings showed ha pre-service eachers were able o model he process of cooling of coffee as a decreasing exponenial funcion. Difficulies wih inerpreaion of he consan rae of cooling and reinerpreaion of mahemaical model were idenified. Mahemaical modelling has gained an increased aenion in mahemaics educaion communiy. Almos wo decades ago, Blum (1993) repored ha modelling was no ye par of he core mahemaics eaching componen in mos counries. Since hen, he siuaion has been improved. Burkhard (2006) repored ha many counries including Germany, he Neherlands, he Unied Saes, and Ausralia, have aken in a significan porion of mahemaical modelling in heir mahemaics curriculum. Publicaions on mahemaical modelling in various journals (see e.g., Barbosa, 2006; Blum, 2002; Blum, Galbraih, Henn, & Niss, 2007; English & Waers, 2006) have documened he global rends of incorporaing more modelling in school mahemaics. The growing ineres owards modelling is srongly suppored by he Organisaion for Economic Cooperaion and Developmen s (OECD) Programme for Inernaional Suden Assessmen (PISA), a major inernaional assessmen sudy which promoes mahemaical and scienific lieracy. PISA (OECD, 2003) conends ha mahemaical lieracy is he overarching lieracy (de Lange, 2006, 15) ha comprises quaniaive lieracy, numeracy, and spaial lieracy. The OECD definiion of mahemaical lieracy depics a broader specrum of wha consiues mahemaics as i goes beyond school mahemaics curriculum: Mahemaical lieracy is an individual s capaciy o idenify and undersand he role ha mahemaics plays in he world, o make a well-founded judgmen, and o engage in mahemaics in ways ha mee he needs of ha individual s curren and fuure life as a consrucive, concerned and reflecive ciizen. (OECD, 2003, p.72) The close relaion beween mahemaical lieracy and mahemaical modelling is well documened. Recenly, Sacey (2009) noed ha mahemaical modelling plays he mos imporan par of mahemaical lieracy. This is consisen wih an earlier repor by Kaiser and Willander (2005) saing ha innovaive mahemaical modelling projecs wih realisic conexs resuled in a greaer comprehension of mahemaical conceps. Furhermore, hey proposed he inclusion of modelling and applicaions in learning mahemaics saring as early as primary school level. Despie he growing ineres in mahemaical modelling a school level, fewer sudies a eacher educaion level have been documened. Concerns abou his rend were voiced by Burkhard (2006) and Doerr (2007). Boh highlighed he imporance of preparing preservice eachers wih skills of doing mahemaics which requires broader eaching sraegies. A profound change of view abou wha consiue mahemaics, he role of L. Sparrow, B. Kissane, & C. Hurs (Eds.), Shaping he fuure of mahemaics educaion: Proceedings of he 33rd annual conference of he Mahemaics Educaion Research Group of Ausralasia. Fremanle: MERGA. 626

3 eachers and sudens is necessary o suppor modelling in pracice. Hence, eacher educaion plays a crucial role in raining fuure eachers o acquire modelling skills and is pedagogy. A reform movemen o improve mahemaics eaching in Indonesia, called PMRI (Ekholm, 2009; Sembiring, Hoogland, & Dolk, 2010), has been pu in pracice for a decade. Inspired by Freudenhal s noion of mahemaics as a human aciviy (Freudenhal, 1983, 1991), PMRI pus emphasis on mahemaics as a meaningful aciviy ha enables learners o grasp real-world phenomena. Consequenly, mahemaical modelling is an inegral elemen of learning mahemaics. Recen sudies in PMRI classrooms (Dolk, Widjaja, Zonneveld, & Fauzan, 2010; Widjaja, Fauzan, & Dolk, 2009) sugges he need o suppor eachers (boh pre- and in-service) wih skills of doing mahemaics and modelling mahemaics. This provides a srong impeus o inegrae mahemaical modelling as par of a course for mahemaics pre-service eachers in Indonesia. This paper will discuss preliminary findings from a modelling projec o invesigae relaionships beween emperaure and ime in he cooling of coffee experimen wih samples of Indonesian pre-service eachers. Seing and Paricipans Mehod This sudy was siuaed in a privae eacher raining insiue in Yogyakara, Indonesia. A modelling projec called The cooling of coffee (Keng-Cheng, 2009) was ranslaed ino Indonesian and assigned o 20 pre-service eachers (see Figure 1). This sudy aimed o provide pre-service eachers wih experience in mahemaical modelling. Pre-service eachers knowledge and difficulies displayed in compleing his projec will be examined. The design of projec also inended o engage pre-service eachers in a producive classroom discourse based on explanaions and jusificaions of heir mahemaical models. Having a differen pedagogical approach was expeced o enable pre-service in carrying ou modelling projec in he fuure professional life. Cooling of Coffee Projec How does a cup of ho coffee cool wih ime? Is i possible o model he cooling of coffee? 1. Lis facors or variables in he problem. 2. Collec daa o help consruc a model. Record your daa. 3. Wha is a suiable model? (Look for an exising model o develop one). 4. Wha is your mehod of soluion? 5. Carry ou your soluion mehod and inerpre he soluion. 6. Examine assumpions and sugges ways o refine he model Figure 1. The Cooling of Coffee Projec (Keng-Cheng, 2009, 36-37). The cooling of coffee projec was chosen because he saring poin of his problem was considered experienially real for pre-service eachers. The whole projec ook four weeks o complee. Pre-service eachers worked in small groups of four people o carry ou his projec in he firs wo weeks. The following wo weeks were devoed o poser presenaions and whole class discussion. Represenaives of each group explained heir sraegies and answered quesions from oher groups during he poser presenaion. 627

4 Framework for Analysis In analysing group wrien work and posers, a framework of didacical mahemaical modelling process by Kaiser and Blum (in Kaiser & Schwarz, 2006, p.197) will be employed. This paper will examine mahemaical knowledge ha pre-service eachers display and he difficulies hey face in compleing he projec. Pre-service eachers knowledge and difficulies will be discussed wih respec o he wo main didacical modelling processes in Figure 2, i.e., mahemaisaion and reinerpreaion of mahemaical soluions o he real-life siuaions. Figure 2. Didacical modelling process (in Kaiser & Schwarz, 2006, p. 197). Resuls and Discussion In his secion, daa from wrien projec repors of five groups in compleing he Cooling of Coffee projec will be analysed. The firs four seps of he projec (see Figure 1) i.e., idenificaion of facors, daa collecion, mahemaical model formulaion, and solving he mahemaical model will be considered as he mahemaisaion process. The process of inerpreing soluions, examining assumpions, and refining he model will be considered as he re-inerpreaion or validaion process. Mahemaisaion from he Cooling of Coffee Daa o he Exponenial Decay Funcion A lis of facors affecing he cooling process by various groups was idenified. The facors refleced he real life consideraions ha pre-service eachers brough prior o he experimen. These facors were hen esed during he experimen in cooling he coffee. All groups excep Group B idenified he room emperaure as one of he facors affecing he cooling process. Types of cups (a plasic cup, a ceramic cup, a glass cup, ec.), and size of cups (based on he diameer or heigh of he cups) were also idenified by all groups. Table 1 presens facors ha influence he cooling process as idenified by differen groups and he corresponding ways of collecing daa. A variaion in combining differen facors was observed, suggesing differen assumpions abou facors ha conrol he cooling process. For insance, Group C colleced daa on cooling of coffee using he same cups of differen sizes. This group also examined wheher ypes of cups (plasic, glass, and aluminium) have differen cooling raes. Group D and Group E wen furher o explore differen mixes of subsances hey pu in heir coffee. These groups colleced daa for coffee wihou sugar or wih sugar using differen ypes of cups (plasic, glass, ceramic). The facors idenified by differen groups were based on assumpions of daily life experiences. However, i was unclear how he combinaion of differen facors ranslaed o he mahemaical model. Figure 3 provides illusraions of some of he daa collecion process o consruc he mahemaical model. Table 1 Facors Affecing Cooling Processes and Daa Collecion Mehods Group Facors in he cooling of coffee Ways of collecing daa 628

5 A B C D E - room emperaure - ypes of cups (plasic or glass) - size of cups. - ypes of cups (plasic or ceramic) - size of cups - ypes of cups (plasic or glass) - size of cups - room emperaure - ypes of cups (plasic, aluminium, glass, ceramic) - size of cups - room emperaure - iniial emperaure - ypes of cups (plasic, ceramic, glass, ec.) - size of cups - ingrediens (coffee only; coffee and sugar; coffee, sugar and milk) - room emperaure Collec daa every minue unil he coffee reaches around 40 o Celsius. Collec daa every minue unil he emperaure of coffee reaches around o Celsius. The daa colleced wice o find he average of emperaure colleced of 2 experimens. Collec daa every 3 minues unil he coffee reaches around 30 o Celsius (room emperaure). Collec daa every 5 minues unil he emperaure of coffee reaches around o Celsius (room emperaure). Collec daa every 5 minues unil he emperaure reaches around 30 o Celsius (room emperaure). In general wo benchmarks were observed in ways of collecing daa, namely emperaure and ime for cooling a cup of boiling coffee. Four groups considered emperaure as a crierion for cooling of coffee. As seen in Table 1, hree groups (C, D, and E) used room emperaure (i.e., 30 o Celsius) as a benchmark emperaure for cooling off coffee whereas Group A seleced 40 o Celsius as he cool emperaure for coffee. I was unclear wheher 40 o Celsius was he acual room emperaure when Group A colleced heir daa. In conras, Group B used a lengh of ime (i.e., 60 minues) o derive a mahemaical model and o deermine he raes of cooling under differen circumsances. This choice led ulimaely o differen emperaures of coffee for differen ypes of cups. Ineresingly, in comparing differen raes of cooling for differen cups, Group B calculaed differences beween he iniial emperaure and he emperaure afer 60 minues of he cooling process began. 629

6 Figure 3.Daa collecion processes in Cooling of Coffee projec All five groups were able o relae he mahemaical model of cooling of a cup of ho coffee o an exponenial decay funcion. They derived heir mahemaical model firs by ploing heir daa ino a Microsof Excel spreadshee. By observing he plos of he daa, all groups noiced ha a decreasing exponenial funcion was he bes mahemaical model o fi his se of daa. Hence, all groups found a model of exponenial decay ha could explain he cooling of coffee process. Samples of group posers and he ineracion during poser presenaion are presened in Figure 4. Figure 4. Samples of group poser and ineracion in The Cooling of Coffee projec. Explanaions in deriving mahemaical models showed difficulies in undersanding and inerpreing he rae of cooling. In his case, he rae of cooling was represened by he value of k in general funcion of T() = A.e -k.. Of he five groups, only hree groups provided reasonable explanaions for deriving he mahemaical model and inerpreing he value of k. Two groups (Group D and Group E) uilized he rend line ool o approximae he exponenial funcions ha bes represened heir daa. However, in deriving he value of k, hese groups sared wih he general funcion y=a.e -k. and calculaed an average of iniial emperaure for differen ypes of cups and an average of emperaure a a paricular ime. This approach conradiced earlier effors o find differen mahemaical models in differen experimens. Calculaing averages for various cups under differen circumsances indicaed lack of meaningful inerpreaion of k. In conras, he oher hree groups worked ou he mahemaical model by relaing he emperaure changes over ime o he rae of change noion. Group A, and B expressed his as y y y y k. y. However 630

7 hey did no ake ino accoun he decreased emperaure in heir equaion, y dy dy dy expressing Lim k. y or k. y insead of k. y. Using inegraion, 0 d d d he funcion of y() = y 0.e k. was obained. The negaive value of k was deermined laer afer working wih he daa. Similarly group C also did no express decreased emperaure change over ime in he mahemaical model. This group was he only group ha showed no only knowledge abou he rae of change noion bu also knowledge of physics by referring o Newon s Law of Cooling. Using his law, he rae of change of he emperaure of an objec is direcly proporional o he difference beween he objec s emperaure and he room emperaure. Hence his group represened he change of emperaure over ime as dt/d = k(t 30) which led o he mahemaical funcion T () = (T 0 30)e k In his funcion, k represens he rae of cooling process, T 0 represens he emperaure of boiling coffee, represens he ime and T () represens he emperaure of coffee afer he cooling process going on for. Re-inerpreing he Exponenial Decay Funcion Back o he Daa In examining heir assumpions, all groups reviewed he role of facors, which were idenified a he beginning of he projec. By observing he graphs of experimens carried ou in 26 o Celsius and 29 o Celsius a he same ime, a conclusion abou he impac of room emperaure was drawn. I was noed ha when room emperaure is higher, he coffee akes a longer ime o cool. The influence of ypes of cups (plasic, glass, ceramic) in he cooling process of was also recorded. I was found ha a cup of coffee made of glass ook longer ime o cool of compared o he same cup made of plasic. Size of cups also affeced he rae of cooling. A cup wih larger base area ook shorer ime o cool as compared o a cup wih a smaller base area for he same heigh. Group D and Group E experimened wih differen mix of subsances. Group D found no difference in rends of cooling of coffee using four differen cups wih or wihou sugar. They concluded ha a cup of coffee made of aluminium cools he fases in boh condiions. However, his conclusion should be considered cauiously. I urned ou ha in carrying ou he experimen, conaining sugar was no he only differen facors in he experimen. The Group also used differen ypes of cups wih varying sizes so i was difficul o pinpoin which facors were he conrol variables. A reasonable inerpreaion for he value of k was ariculaed by Group B. This group observed ha he value of k was a consan. Noe ha Group B formulaed he mahemaical model as T () = (T 0 30)e k. + 30, hence hey found negaive value for k. Furhermore as k ges smaller, he coffee cools faser, indicaes a sensible inerpreaion of k as he consan rae of cooling. In addiion, Group B noiced ha he emperaure of coffee was close o he room emperaure a he end of he cooling process. The fac ha only one ou of five groups was able o ariculae a sensible inerpreaion of k suggesed ha re-inerpreaion of a mahemaical model is a challenging par of he modelling process. Similarly, refining he mahemaical model was found o be difficul for he sudens in his sudy. Ideas for refinemen were limied o general limiaions of he curren mahemaical model relaed o echnical aspecs of daa collecion. Lack of accuracy in reading he emperaure and calculaion error caused by rounding were offered as explanaions for discrepancies of emperaure based on mahemaical model and he real daa. All groups only proposed increasing he number of experimens o refine heir mahemaical model. 631

8 Concluding Remarks This sudy sough o enhance pre-service eachers modelling skills in a projec o invesigae he cooling of coffee. Analysis of wrien repors of he projec indicaed ha pre-service eachers in his sudy were able o obain a decay exponenial funcion as a model for he cooling of coffee. Relevan facors affecing he process of cooling including room emperaure, ypes and size of cups were idenified. However, pre-service eachers did no aend o he complexiies of combining differen facors (e.g., sizes of cups, ypes of cups) in mahemaical models. Evidence in his sudy showed ha re-inerpreing and linking variables in he mahemaical model back o he real-world daa was problemaic. Difficulies wih inerpreing he meaning of he consan rae of cooling (k) and incorporaing room emperaure ino he mahemaical equaion were observed. Sound mahemaical knowledge of calculus and funcions is essenial o suppor rich mahemaical discussion in modelling cooling of coffee. Findings from his preliminary sudy indicaed ha a modelling projec is a rich learning ool for pre-service eachers o do mahemaics and o see mahemaics in daily life. References Barbosa, J. C. (2006). Mahemaical modelling in classroom: A socio-criical and discursive perspecive Zenralbla für Didakik der Mahemaik, 38(3), Blum, W. (1993). Mahemaical modelling in mahemaics educaion and insrucion In T. Breieig, I. Hunley & G. Daiser-Messmer (Eds.), Teaching and Learning Mahemaics in Conex (pp. 3-14). Chicheser: Ellis Horwood Limied. Blum, W. (2002). Applicaions and modelling in mahemaics educaion: Discussion documen. Educaional Sudies in Mahemaics, 51(1/2), Blum, W., Galbraih, P. L., Henn, H.-W., & Niss, M. (2007). Modelling and applicaions in mahemaics educaion. New York Springer Burkhard, H. (2006). Modelling in mahemaics classrooms: Reflecions on pas developmen and he fuure. Zenralbla für Didakik der Mahemaik, 38(2), de Lange, J. (2006). Mahemaical lieracy for living from OECD-PISA perspecive Tsukuba Journal of Educaional Sudy in Mahemaics, 25(13-35). Dolk, M., Widjaja, W., Zonneveld, E., & Fauzan, A. (2010). Examining eacher's role in relaion o heir beliefs and expecaions abou sudens' hinking in design research In R. K. Sembiring, K. Hoogland & M. Dolk (Eds.), A decade of PMRI in Indonesia (pp ). Bandung, Urech: APS Inernaional. Ekholm, M., & van den Hoven, G. H.. (2009). PMRI - Majulah! Naarden: Ze&Prin. English, L., & Waers, J. (2006). Mahemaical modelling in he early school years Mahemaics Educaion Research Journal, 16(3), Freudenhal, H. (1983). Didacical phenomenology of mahemaical srucures. Dordrech: D. Reidel Publishing Company. Freudenhal, H. (1991). Revisiing mahemaics educaion, China lecures. Dordrech: Kluwer Academic Publishers. Kaiser, G., & Schwarz, B. (2006). Mahemaical modelling as a bridge beween school and universiy. Zenralbla für Didakik der Mahemaik, 38(2), Kaiser, G., & Willander, T. (2005). Developmen of mahemaical lieracy: Resuls of an empirical sudy. Teaching Mahemaics and Is Applicaions, 24(2-3), Keng-Cheng, A. (2009). Mahemaical modelling in he secondary and junior college classroom. Singapore: Prenice Hall. Organisaion for Economic Cooperaion and Developmen. (2003). The PISA 2003 assessmen framework mahemaics, reading, science and problem solving knowledge and skills. Paris: OECD. Sembiring, R. K., Hoogland, K., & Dolk, M. (2010). A decade of PMRI in Indonesia. Urech: Ten Brink. 632

9 Sacey, K. (2009). Mahemaical and scienific lieracy around he world. In U. H. Cheah, Wahyudi, R. B. Devadson, K. H. Ng, W. Preechaporn & J. C. Aligaen (Eds.), Proceedings of he 3rd Inernaional Conference on Science and Mahemaics. (pp. 1-7). Penang: SEAMEO RECSAM. Widjaja, W., Fauzan, A., & Dolk, M. (2009). The role of conexs and eachers' quesioning o enhance sudens' hinking. In U.H. Cheah, Wahyudi, R.B. Devadson, K.H. Ng, W. Preechaporn & J. C. Aligaen (Eds.), Proceedings of he 3rd Inernaional Conference on Science and Mahemaics. (pp ). Penang: SEAMEO RECSAM. 633