CREATIVITY AND MATHEMATICS: USING LEARNING JOURNALS Alf Coles ad Gemma Bafield report o the power of jourals to support learig The I i this article is Alf s voice, later i the writig whe we cosider some pupil s work we have iterspersed a commetary i italics, idicatig Gemma s voice. This commetary liks what we preset about the work of three pupils to the wider class. I 2010, I was privileged to be ivited to work i Gemma s classroom i Colere Church of Eglad primary School, where the headteacher wated to develop creative approaches to the teachig of mathematics. We were successful i applyig to the Natioal Cetre for Excellece i Teachig Mathematics to set up a project as a Mathematics Kowledge Network. The project was the start of what is still a o-goig collaboratio betwee the charity xx=creativity (xx) ad the Uiversity of Bristol, where I work. I this article we report o oe aspect of the project - the use we made of learig jourals for pupils. xx place artists i schools, i collaboratio with a cultural cetre, to set up projects with studets ad staff, for more iformatio see www.xx=creativity.org.uk for more details. The ispiratio for xx comes from the work of the Reggio Emilia early-years settigs i Italy (Rialdi, 2006). xx has adopted the followig priciples from the work of these pre-schools: childre are see as iate ad creative kowledge builders, explorers ad co-costructors of their learig educators ad artists are eablers ad compaios i the childre s learig withi a culture of listeig learig is focused o the process of the childre s exploratios, ot the ed product documetig childre s learig joureys is our method for evaluatig ad reflectig upo the childre s thikig ad learig developig a creative learig commuity of teachers, artists, co-workers i cultural cetres, parets ad childre ivolvig family ad commuity i life-log learig (xx, 2011) Learig here is liked to exploratio, process, jourey, listeig ad commuity. My ow backgroud, as a teacher ad researcher, has its roots i the work of Gattego who, i a echo of the first priciple, writes of the powers of childre (1971, p.16). Gattego based his pedagogy o exploitig these powers, oe of these powers is stressig ad igorig, which he sees as the essetial aspect of the process of abstractio. I other words, all of us egage i abstract thought wheever we cosider two examples ad ask, what is the same ad what is differet?, a phrase used by Laurida Brow. I coversatio with the director of xx, pey Hay, we coceived of my work with the primary school i terms of a xx project, that is, I would be a artist/mathematicia. I the first istace, the school wated to set up a project with their two Year 3/4 classes, oe of which was Gemma s. The headteacher wated to develop pupils creativity with a particular focus o the capacity for logical ad systematic thikig. pey ad I met with staff to talk through some of the processes ivolved i a xx project. I the met with Gemma ad the other Year 3/4 teacher to discuss a rage of possible startig poits which led to us decidig o a problem from the publicatio A Addedum to Cockroft (Brow & Waddigham, 1982) ivolvig placig books o a shelf. This task starts off with the problem: If I have 3 idetical blue books ad 2 idetical red books, how may differet ways ca I arrage them o my shelf? We agreed I would start both classes off with this task, supported by the class teachers ad a teachig assistat. My backgroud is i teachig secodary mathematics. I my ow classroom I developed ways of workig with 11 year olds, ad older, such that studets became familiar with the process of askig questios, makig cojectures, testig out their ow ad each others ideas ad tryig to prove statemets. At the start of this project I had little idea how relevat these ways of workig would be with much youger childre. Agai, i discussio with Gemma ad the other Year 3/4 teacher, we agreed that I would offer pupils a purpose for the project of them becomig a mathematicia. I the cotext of this work beig liked to xx, we wated to develop pupil creativity withi mathematics. We took creativity to mea: 6 Mathematics Teachig 228 Joural of the Associatio of Teachers of Mathematics
pupils askig their ow questios ad beig supported to work o them; pupils spottig ad describig patters; pupils makig predictios ad reasoig; pupils choosig their ow methods of represetatio. I eded up takig te oe-hour sessios with the two classes, visitig oce a week ad talkig with Gemma after each class to debrief ad pla the followig week. The arragig books task lasted for four sessios. We the worked o a secod task for four sessios. This secod task was pick s Theorem http://rich.maths.org/1867, ad ivolves pupils explorig the relatioship betwee three variables: the area of a shape draw o a square dot grid; the umber of dots o the edge of the shape; ad the umber of dots iside the shape. I the fial two sessios we spet the time with pupils lookig back over the work we had doe together ad it is these sessio where we used learig jourals. What happeed! As we bega work o this project we had o expectatios of how log it would last, for how log I would cotiue to lead lessos, or as I commeted above, how the pupils would respod. We were struck, from the first sessio, how aturally ad ethusiastically pupils seemed to take to the idea of becomig mathematicias ad tryig to say the word cojecture! I what follows, we report o the way we ra the last two sessios, lookig back over the previous eight hours work. part of the work of xx is to put o professioal developmet courses for all educators ivolved i their projects. Alog with the other Year 3/4 teacher, I atteded a sessio i which we were offered a iput about the power of learig jourals as a mechaism for gettig pupils thikig about their ow learig. We were show some beautiful examples of detailed collages doe i jourals, documetig the process of learig. Ispired by this iput, I discussed, with Gemma ad the headteacher, the idea of usig jourals i mathematics ad we all agreed we should try this out. I preparatio for offerig the jourals to pupils, I bega to keep my ow joural about my work with the Year 3/4 classes. A joural starts off as a blak sketch-book. I cut out ad pasted i pages from my plaig books ad added aotatios or highlights. I also prited off copies of email discussio betwee all the educators ivolved i the project ad agai aotated. The process of collectig ad collatig this iformatio ievitably sparked off ew thoughts that I also wrote i the joural. I the first sessio that we gave jourals out to pupils I showed them this example of what I had doe. The ivitatio to the classes was to tell the story of their ow work ad learig o the two tasks we had doe together. All the work pupils had doe up to this poit had bee o paper. We made available scissors ad glue. We suggested they cut out sectios they wated but emphasized that we wated them to thik about how to be orgaized i doig this, ad also that the poit was partly to help them make progress o ay questios they had left uaswered from the tasks. We had plaed oe sessio o the jourals, but the pupils seemed so egaged by the activity of creatig collages of their work that we cotiued ito a secod sessio, which eded up beig the last oe with these classes. We offer below three examples of what pupils did i their jourals, ad discuss evidece that, as a way of reflectig o their work, usig the jourals provoked ew learig. Evidece of learig through a the use of a joural The followig images are of differet Year 3 pupils work, from their learig jourals. The prior attaimet of these pupils relative to the rest of their class, ad atioal expectatios, was broadly high i Case 1, middle i Case 2, ad low i Case 3. These three examples also display the rage of differet ways i which the jourals fuctioed to support pupils ow thikig ad learig. Case 1 The pupil writig at the top of page 8, is: 4 squer shapes ad squer shapes, I iterpret this as meaig: 4 square shape ad square shape which was the laguage used i the lessos o the pick s Theorem project. E refers to the umber of dots o the edge of a shape ad I the umber iside. May pupils oticed that they seemed to get the same sets of values for E ad I for shapes of the same area. The pupil, cut out ad arraged her shapes i a ew way compared to what she had doe before, that is, the shapes were draw more radomly o her paper as part of the pick s Theorem project hece, i part, perhaps why each shape was cut out separately. I other words, i her origial work she had ot bee orgaized as she is here ad had looked at shapes with differet areas rather tha cosiderig separately oes with area 4 ad area. She also draws i the learig joural, for the first time i her work o this project, tables of results. It is possible to see that the tables of results are May 2012 www.atm.org.uk 7
Image 1: The Learig Joural of a Year 3 Studet - pick s Theorem project draw directly i the joural ad ot cut out from paper. Havig begu a table of results, it ca also be observed that she exteds the table beyod the results that she obtaied, i order to make a predictio. For example, o the left had page, with 4 square shapes, all her examples have either 0, 1 or 2 dots iside; but she has exteded her table to make predictios for E whe I = 3, 4 ad. The latter two beig impossible shapes i fact, sice E eeds to be at least 3 to make a shape. There is evidece o this page, therefore, that returig to the pick s Theorem project with the learig joural has supported this pupil i demostratig mathematical thikig that is sigificatly higher level tha she demostrated i her origial work, ow makig a table of results ad a predictio that she had ot doe before. I fid it itriguig to muse o whether this is simply a fuctio of havig more time o the project. My huch is that the joural was sigificat i allowig the pupil to approach her ow work aew i some way, ot costraied by how she had tackled it up to that poit. She would have see examples of other pupils makig tables of results, ad we discussed some as a whole group; the joural seemed to allow her the opportuity to icorporate this thikig ito what she had doe. It is oly my ow cojecture, but I woder if, without physically re-orgaisig the shapes ito oes with the same area i the way she did, she would have distilled out such a table of results from her work. A opportuity to revisit learig allowed the childre to reorgaise their thikig based o what they had foud out. Some foud it challegig to start a problem i a systematic way. They chose a radom startig poit. So returig to a problem ad havig time to look through their workigs was valuable. These childre ofte chose to completely reorgaise their work. I may cases this led to the childre spottig patters ad solutios for the first time. Case 2 The ext example from a learig joural is also take from the pick s Theorem task. The shapes this pupil arraged are a mix of area 3 ad 4. The table of results is for area 4. I iterpret the writig at the top of page 9 to be: Shape project ad at the bottom to be: The patters ad assume this is a ufiished setece, I emphasized watig pupils to write dow ay patters they oticed. For this pupil, although the table of results is draw o the learig joural as with Case 1, he had show evidece i his previous work o the project of havig completed a similar table, but for square shapes. It is perhaps just possible to see i the table that there has bee rubbig out. Durig the lessos o the jourals, the pupil approached 8 Mathematics Teachig 228 Joural of the Associatio of Teachers of Mathematics
may times, but i the words of the old spiritual, obody else ca cross it for us, we must cross it by ourselves. Beig shoved or dragged across does o good. (Holt, 1990, p.160) I case 1, the learig joural seemed to support a extesio of mathematical thikig, i this istace the joural seems to fuctio more to revisit ad reiforce a previous awareess a re-crossig of a lie perhaps. Revisitig work usig the jourals led to a deepeed uderstadig of the learig that had take place. Some childre eeded to revisit the same learig a umber of times before they fully uderstood what they had foud out. Case 3 The third ad fial example of the use of a learig joural is from the iitial red ad blue book, or combiatios task. I offered the stadard otatio c m 2 to stad for the umber of Image 2: The Learig Joural of a Year 3 pupil - pick s Theorem project me askig me to check this table. I remembered this pupil had completed a table successfully for his area shapes, so I was expectig this table to correspod to the shapes o the page above. Iitially I could see o coectio. The first lie, which you may just be able to make out, had E = 10 ad I = 10. I asked the pupil where he had got his umbers from for the table ad poited out his shape where E = 10 ad I = 0. He retured to his seat ad cotiued workig, rubbed out some figures ad re-etered values i his table. He came back to me, but still his umbers did ot correspod to the shapes o his page. It was o the fourth occasio of this pupil comig to check his work with me, ad my reportig that it did ot seem to fit his shapes, that he wrote the values above, that do ow all match a shape he had draw. If I had bee asked to assess this pupils mathematical thikig at the ed of the pick s Theorem project, I would have bee coviced that he uderstood how to create a table of his results. Yet here was evidece that this pupil eeded to re-eter the problem of doig this. I am remided of a quotatio from Joh Holt that I have lived with as a teacher for may years: All of us must cross the lie betwee igorace ad isight may times before we truly uderstad. Not oly must we cross the lie ways of arragig books i total, 2 of which are red. After workig as a class o this startig challege, pupils the chose their ow problems to explore - iitially stickig to books i total; for example, fidig the umber of combiatios for c m i.e., books, 1 red ad so, 4 blue, 1 or the umber of combiatios for c m 3 i.e., books, 3 red ad so, 2 blue. I the image below, the pupil text at the top I read as: I kow that c m has oly 10 aswers. 2 The text from the bottom is less clear. My trascriptio of what he wrote is: T s cojectur c m= 1 I ew c m would equl 1 becuse ther Is o other coler. I iterpret this as: T s cojecture c m = 1 I kew c m would equal 1 because there is o other colour. The pupil who created this page is uderachievig i mathematics ad literacy agaist age expectatios. There is evidece here of him makig a cojecture, usig algebra ad offerig reasoig to justify his ideas. I his work i the first lessos o this project he had recogized that if there were books ad all of them were red, i.e., c m, the there will be oly oe possible way of arragig them. I this project, I was aware of watig to offer algebra as a mechaism for expressig ay geeralities pupils oticed. May 2012 www.atm.org.uk 9
Image 3: The Learig Joural of a Year 3 pupil - Combiatios project I the lesso whe T made his observatio about books ad reds, I asked the class to predict the umber of combiatios with 6 books, all of them red, 7 books all red, 10 books, etc, ad offered the otatio as a way of capturig the awareess that there will be oe combiatio o matter how may books, if they are all red. What I fid strikig i the image above is the way pupil T shows evidece of begiig to cosider issues of proof ad justificatio. At the top of the page he writes: I kow that c m has oly 10 aswers - ad I iterpret the arrow as idicatig that he sees as a justificatio of this statemet, the fact that he has writte the 10 solutios out. At the bottom of the page, the pupil cuts out his cojecture from work he had doe o paper i.e., that c m = 1 but the as at the top, adds a justificatio. He adds, writte directly o the joural page, that he kows his cojecture because there is o other colour. As with Case 1, I see evidece here of the use of the learig joural supportig the pupil i demostratig sigificatly higher levels of mathematical thikig tha he displayed i his origial work o the task. Agai, as with Case 1, I caot help myself cojecturig that loger time o the project would ot ecessarily have provoked such a shift i his recorded thikig. I imagie further time would have see him cotiue to explore differet combiatios ad patters. By revisitig his ow cojecture i the learig joural, ad with the offer to all pupils to try ad write out ad explai their thikig, this pupil is able to capture somethig of his ow reasoig that he had ot bee able to record at the time. I fid it itriguig that the pupil i Case 3 was 2 the lowest attaiig of the three ad yet the oe who arguably had bee the most mathematical. I am left to woder at the extet to which his difficulties i mathematics lie i problems with literacy, ad whether these difficulties ca act to mask the quality of thikig he is clearly able to demostrate. Discussio Whe the staff ivolved i the project ad I came to reflect, ourselves, o the te sessios ad their impact, oe of the thigs we were most struck by was the power of these jourals i supportig pupils to revisit ad take o their work i a maer that seemed to provoke ew learig ad awareesses. Gemma ad the other Year 3/4 teacher cotiued the use of the jourals with their classes for the remaider of the year. I have bee at pais ot to use the word reflectio up to this poit i the article i relatio to what the pupils were doig. It is perhaps ow relatively commo place for pupils to be ivited to traffic-light their ow cofidece levels, or fill out self-assessmet sheets about their feeligs o what they ca do/uderstad. I have persoally always foud it hard to egage pupils i as meaigful a way as I would like with such processes. I cotrast, the use pupils made of the learig jourals was alive with iterest, egagemet ad, as I hope to have show above, learig. It is ot a easy task to try ad distill out what was sigificat i the use we made of the jourals i terms of provokig the positive respose from pupils. No doubt, if pupils had ot bee egaged with the tasks i the first place, there would have bee little iterest to documet them, ad I am aware we have bee relatively silet o the role of the teacher i sustaiig such iitial egagemet but that is a whole differet story. There did seem to be somethig appealig to pupils about beig give a attractive blak sketch-book ad I tried to make my modelig of the use of the joural as aesthetic as I could. There was space for creativity i terms of how pupils chose to set out their jourals ad may of them seemed to take great care i doig this. The activity of choosig what of their past work to iclude i documetig their learig forced pupils to cosider what was the same, or differet, across what they had doe, i.e., the activity provoked abstract thikig - i Gattego s sese - ad it seems to me possible that it provoked abstractios about their work that pupils may ot have cosidered if simply give more time o the tasks i the first place. 10 Mathematics Teachig 228 Joural of the Associatio of Teachers of Mathematics
The jourals provided a chace for the childre to choose how to preset their learig. This meat it was meaigful to them. The jourals gave the childre complete owership of their learig. The fact that there was o wrog way of usig the jourals ad presetig their learig meat that the childre, particularly the lower attaiig, felt quite liberated. Some childre were t happy with their iitial workigs. They were disappoited with the way they had laid out their work ad were uable to see ay liks or patters. The use of the joural provided a ew ethusiasm. Beig give a secod chace to revisit a problem allowed these childre to sort out their thikig. There seems to me to be tetative evidece therefore, from this project, that it ca be powerful to ivite a activity of lookig back o past work after a short time gap. This may allow a re-etry ito the problem uecumbered by the approaches take at first ad may support either a re-workig of a past awareess, or a extesio of past thikig. Aother factor, perhaps, is the use of a differet medium for this reflective work, compared to the oe of the origial productio, i.e., where a distictio ca be easily made betwee what was doe before ad what was added. Startig a problem for some childre is a little dautig. So kowig that there are opportuities to retur to previous work gives them some security. They do t eed to get the aswer right first time! The childre leart the importace of lookig back at what they had foud ad usig what they had foud to ask ew questios. This prompted ew learig ad a deepeed uderstadig. The work i this school acted as a pilot for a wider project, curretly i three schools durig 2011-12 ad from 2012-13 i five schools, that is beig fuded geerously by the Raye Foudatio. I hope to report agai from further outcomes of this work. Alf Coles works at the Uiversity of Bristol, ad Gemma Bafield teaches at Colere Church of Eglad Primary School Refereces xx (2011) philosphy ad aims. http://www.xxcreativity.org.uk/?id=136 accessed 12th October 2011. Brow L., ad Waddigham J. (1982) A Addedum to Cockroft. Avo, RLDU Gattego C. (1971) What we owe childre. Lodo, Routledge Holt J. (1990) How childre fail. Lodo, pegui Rialdi C. (2006) I dialogue with Reggio Emilia: Listeig, researchig ad learig. Oxford, Routledge Refereces for: Equatios with differet techology: differet tools, differet views - page 14. Refereces Hoyles, C. & Noss, R. (2003). What ca digital techologies take from ad brig to research i mathematics educatio? I: Bishop, A.J., Clemets, M.A., Keitel, C., Kilpatrick, J. & Leug, F.K.S. (Eds.), Secod Iteratioal Hadbook of Mathematics Educatio (pp. 323-349). Dordrecht, Netherlads: Kluwer Academic publishers. Kiera, C. & Drijvers, p. (2006). The co-emergece of machie techiques, paper-ad-pecil techiques, ad theoretical reflectio: A study of CAS use i secodary school algebra. Iteratioal Joural of Computers for Mathematical Learig, 11(2), 20-263. Trouche, L. & Drijvers, p. (2010). Hadheld techology: Flashback ito the future. ZDM, The Iteratioal Joural o Mathematics Educatio, 42(7), 667-681. Vlassis, J. (2002): The balace model: hidrace or support for the Solvig of liear equatios with oe ukow. Educatioal Studies i Mathematics, 49, 341 39 Yeo Shu Mei, Thog Chee Hig, & Kho Tek Hog (2008). Algebra discs: digital maipulatives for learig algebra. Paper preseted i TSG22 of the ICME-11 coferece, Moterrey, Mexico, 2008. Retrieved o August 26th, 2011, from http://tsg.icme11.org/tsg/show/23#ierdocumets. May 2012 www.atm.org.uk 11