Actes Proceedings. Université de Montréal Juillet

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1 Actes Proceedigs 3. La créativité et les activités mathématiques Pour expliquer le succès au travail et das la vie courate das ce vigt-et-uième siècle, o évoque fréquemmet la créativité mathématique et l iovatio. Eseigats, didacticies des mathématiques, mathématicies, chercheurs, parets et les élèves eux-mêmes, ot tous à coeur le développemet de la créativité mathématique et le succès e mathématiques. Ceci soulève quelques questios: Qu eted-o par créativité das l eseigemet et l appretissage des mathématiques? Est-ce que tous les élèves peuvet faire preuve de créativité ou est-ce réservé à seulemet quelques-us? Commet pouvos-ous développer ou stimuler la pesée créatrice das la classe de mathématiques? Commet harmoiser la créativité et le développemet d habiletés? Commet stimuler la créativité à l extérieur de la classe? Y a- t-il u lie etre la créativité et les compétitios mathématiques, etre la créativité et les tests stadardisés? Quel rôle joue la techologie das le développemet de la créativité das la classe de mathématiques et à l extérieur de la classe? Commet mesure-t-o la créativité mathématique? 3. Creativity i mathematical activities Mathematical creativity ad iovatio are ofte cited as critical to success i work ad i life i this twety-first cetury world. Teachers, mathematics educators, mathematicias, researchers, parets, ad studets themselves all have a stake i learig how best to urture ad support this developmet of mathematical creativity ad the realizatio of mathematical promise. Some of the questios to be ivestigated i this strad are : What does creativity mea i the process of teachig ad learig mathematics? Is this somethig for all studets or oly for a few? How might we develop or stimulate creative thikig i the mathematics classroom? How does this balace with skills traiig? How might mathematical creativity be stimulated outside the classroom? How is this related to mathematical competitios? What is the role of techology i the developmet of mathematical creativity withi ad outside the classroom? How might we measure mathematical creativity? How does this fit with high-stakes, stadardized or stadardsbased tests? Uiversité de Motréal 6 3 Juillet

2 Ivitig Creativity to Math Class: Ope-Eded Projects i a Middle School Classroom Heather Gramberg Carmody Purdue Uiversity ad Park Tudor School Itroductio Each studet seated i a math classroom has a uique way of viewig the world ad workig through mathematics. They come with differet isights, abilities, struggles ad eeds. Oe method of icorporatig this diversity is the developmet of ope-eded projects. Opeeded projects offer the opportuity for differetiatio ad high levels of egagemet that go beyod solvig a set of predetermied problems. Over the last several years, I have developed a structure for ope-eded math projects that helps me meet the eeds of a variety of middle school studets. The studets take great owership of the mathematics, ad their ethusiasm is cotagious. The iclusio of these projects has allowed me to ivite a great deal of creativity to math class. Opportuities for Excellece Studets have a variety of eeds whe learig mathematics. Gifted ad creative studets have a eed for depth of applicatio (Hirsch ad Weihold, 999). All studets eed to have opportuities to reflect upo their work ad refie it (Koshy, 00). To mirror true experts i the field, studets eed a opportuity to let ideas develop ad form (Hadamard, 945). Fially, may studets eed the opportuity to discuss their thikig with peers ad adults (Hirsch ad Weihold, 999). To egage studets i meaigful ways teachers eed to adopt may roles. The teacher becomes oe who discovers talet. High quality math experieces are ot reserved for a particular group of studets. Appropriate teachig ad activities ca allow may studets to demostrate advaced thikig. Teachers eed to provide opportuities for studets to reveal their abilities (Greees ad Mode, 999). It is oly through itetioal classroom activities that studets creativity ad potetial ca emerge. Pupils who are give a diet of repetitive exercises ad closed problem-solvig exercises from text books are ulikely to show their true potetial. A child who is capable of detectig patters ad geeralisig will oly do so if suitable activities are provided. (Koshy, 00, pg. ). Oce talet is oticed, teachers eed to provide support so studets move beyod what is familiar ito areas of challege ad creativity (Ma, 006). This type of teachig requires a high level of flexibility. Teachers eed to appreciate creativity ad to ejoy the upredictability of workig with diverget thikers (Greees ad Mode, 999, p. ). The eed for flexibility also applies to the classroom settig. Classrooms that ecourage mathematical creativity are learer cetered, have a emphasis o idepedece, favor flexibility over structure or chaos, ad are ope to ew iovatios (Wheatley, 999). The studets work idepedetly towards solutios ad ew uderstadigs of mathematics (Hirsch ad Weihold, 999). Zemelma, Daiels ad Hyde made a umber of suggestios for classroom structure icludig that assessmet be a itegral part of teachig, that class time is spet developig problem situatios that require applicatios of a umber of mathematical 5

3 ideas, ad that teachers focus o usig multiple assessmet techiques, icludig writte, oral, ad demostratio formats (998, pg. 05). Project Settig The followig project structure has bee applied repeatedly i seveth grade mathematics classes. A series of eight projects were developed as a attempt to address the variety of eeds ad to restructure learig. They address the Natioal Associatio for Gifted Childre (NAGC) curriculum stadards ad the Natioal Coucil for Teachers of Mathematics (NCTM) stadards. They have prove effective i large urba districts as well as small private schools. Sometimes my studets have bee i completely heterogeeous groups. However, i other settigs they were i distict groups after a extesive placemet process. These math projects fit the eeds of a wide array of childre. Studets who were classified as gifted have completed these assigmets. Some were labeled twice exceptioal, exhibitig gifted characteristics as well as a variety of special eeds. Studets with learig disabilities, commuicatio disorders, those with autism, ad those with behavioral disorders have completed these projects. Project Purposes ad Goals Project Purposes Movig beyod traditioal curriculum takes time ad thought. It is importat to kow the purpose of the time ad activities. These projects first seek to provide egagig cotet i order to icrease studet motivatio ad learig. Challegig tasks yield higher cogitio, metacogitio ad motivatio (Diezma ad Watters, 005). Secodly, the projects are desiged to give opportuity for studets to move beyod beig passive subjects of math curriculum to creative thikers who view the world aroud them through the les of mathematics. Sheffield suggests that there is a cotiuum i mathematics of iumerators, doers, computers, cosumers, problem solvers, problem posers, [or] creators (999, p. 43). As studets work through the log term activities, they move towards the more creative ed of the spectrum. Fially, the projects serve to provide opportuity for studet-drive differetiatio. Curret research suggests the importace of differetiatig studets work i terms of cotet, process, product or epistemology (VaTassel-Baska ad Stambaugh, 006). Project Goals Ay ew studet project should be developed with several goals i mid. The first is to ecourage studets to apply mathematics to real life (NCTM, 000). Secodly, these projects should ecourage studets to move beyod the basics of mathematics ad to allow them to explore meaigful applicatios. These projects are ot simply to have a fu activity. They must move past uimportat kowledge to a deeper exploratio of the world aroud them (Rezulli, 98). Next, these projects should ecourage studets to have idepedece ad autoomy i their mathematical learig. The proposal stage ad the opportuities for peer collaboratio allow studets to experimet ad move beyod what is comfortable without the fear of failig. The hope is that these projects allow for resposible risk takig (Costa, 00). The fial goal of the projects is to require studets to reflect upo their work, ad to refie their projects. Authetic learig ad complex mathematics are very rarely achieved o the first try. Just as studets are taught to refie their writig through a sequece of editig differet drafts, they eed 53

4 to lear how to revise their work i mathematics. The goal is to build i time for plaig, testig ad creatig quality projects. Format of Projects Although the specific mathematical cotet chages betwee projects, the format remais the same. The followig factors are always icluded: Extesive mathematical computatio Algebraic otatio Writte expressio Visual or graphic represetatio Project proposal Peer collaboratio Evaluatio of complexity The extesive mathematical computatio is the achor for the projects. Studets should get as much, if ot more, practice o essetial skills whe completig these projects as they would whe workig through traditioal curricula. The algebraic otatio is importat i developig abstract thikig ad reasoig. The writte expressio i projects allows studets to articulate their discoveries. Mathematics becomes much more tha a completed list of problems. The iclusio of visual represetatio i each project allows studets to itegrate multiple ways of viewig mathematics. The artistic stregths, creativity with words ad strog iterpersoal abilities that some studets have ca lead to a deeper uderstadig ad explaatio of mathematics. As metioed before, each project cotais a madatory proposal well before the deadlie. Studets are give time to daydream ad ask some of the what if questios at the begiig phases of a project. I additio to collaboratio with the teacher, studets also work with each other. May of the projects ivolve days where studets brig i their works i progress to get reactios ad suggestios from peers. Fially, a compoet of the gradig evaluates the complexity of studets work. Studets reflect upo ad discuss the level of challege ad complexity i their ow work. I my experiece, studets will push themselves to heights far beyod the accepted curriculum or stadards. Questios to ask whe developig a project With all of these goals ad compoets i mid the projects begi to evolve. Whe I am cosiderig a ew project, the followig questios help to provide the structure: What topic is essetial to the curriculum? Or, what topic is a corerstoe of the course? What are some meaigful ad real world applicatios of this topic? What applicatios are appropriate for studets of this age ad level of mastery to explore? How ca I iclude mathematical computatio, algebraic otatio, writte expressio, visual represetatio, peer collaboratio, evaluatio of complexity, ad opportuities for reflectio ad revisio? What choices ca I offer i terms of cotet or product? What resources do I eed to fid for the project? Am I esurig that all studets have equal opportuities for success i terms of required resources? 54

5 What amout of time will studets eed to produce quality work? How ca I structure a timelie that allows for creativity ad rigor? Are there adequate opportuities for differetiatio to accommodate various studet eeds? Does this project ecourage a deeper uderstadig of mathematics tha studets would otherwise have? Coclusio The icrease i the diversity of our studets ad the complexity of their eeds ca be a rich additio to a mathematics classroom. The challege for teachers is to fid a way to iclude studets backgrouds ad creativity i a way that allows for rigorous mathematics. The idea of ope-eded projects has allowed me to look beyod textbooks ad fu activities whe I pla for my studets. This structure provides studets the opportuity to excel ad flourish. The elemets of collaboratio ad reflectio erich the culture i a classroom. These log term projects ivite creativity to math class, ad the result is well worth the risk of movig beyod what is familiar. 55

6 Refereces Costa, A. L. (00). Habits of Mid. I A. L. Costa, ed., Developig Mids: A Resource Book for Teachig Thikig (pp ). Alexadria, VA: Associatio for Supervisio ad Curriculum Developmet. Diezma, C.M ad Watters, J.J. (005). Caterig for mathematically gifted elemetary studets: learig from challegig tasks. I S.K. Johso ad J. Kedrick (Eds.) Math educatio for gifted studets (pp ). Waco, TX: Prufrock Press, Ic. Greees, C. ad Mode, M. (999). Empowerig teachers to discover, challege, ad support studets with mathematical promise. I L.J. Sheffield (Ed.), Developig mathematically promisig studets (pp. -3). Resto, VA: Natioal Coucil of Teachers of Mathematics. Hadamard, J. (945). The psychology of ivetio i the mathematical field. Mieola, NY: Dover. Hirsch, C.R. ad Weihold, M. (999). Everybody couts- icludig the mathematically promisig. I L.J. Sheffield (Ed.), Developig mathematically promisig studets (pp. 33-4). Resto, VA: Natioal Coucil of Teachers of Mathematics. Koshy, V. (00). Teachig mathematics to able childre. Lodo: David Fulto Publishers. Ma, E. L. (006). Creativity: the essece of mathematics, Joural for the Educatio of the Gifted. 30 () p Natioal Coucil of Teachers of Mathematics (000). Priciples ad Stadards for School Mathematics. Resto, VA: Author. Rezulli, J. S. (98). What Makes a Problem Real: Stalkig the Illusive Meaig of Qualitative Differeces i Gifted Educatio. Gifted Child Quarterly, Sheffield, L.J. (999). Servig the eeds of the mathematically promisig. I L.J. Sheffield (Ed.), Developig mathematically promisig studets (pp ). Resto, VA: Natioal Coucil of Teachers of Mathematics. VaTassel-Baska, J., & Stambaugh, T. (006). Comprehesive Curriculum for Gifted Learers, 3 rd Editio. Bosto, MA: Ally ad Baco. Wheatley, G.H. (999). Effective learig eviromets for promisig elemetary ad middle school studets. I L.J. Sheffield (Ed.), Developig mathematically promisig studets (pp. 7-80). Resto, VA: Natioal Coucil of Teachers of Mathematics. Zemelma, S., Daiels, H. ad Hyde, A. (998) Best Practice. Portsmouth, NH: Heiema. 56

7 I classroom Mathematical activities for families Díez-Palomar, J. Departmet of Mathematics Educatio ad Scieces Uiversitat Autòoma de Barceloa (Spai) Molia Roldá, S. Departmet of Mathematics Educatio ad Scieces Uiversitat Autòoma de Barceloa (Spai) Rué Rosell, L. Departmet of Mathematics Educatio ad Scieces Uiversitat de Barceloa (Spai) Abstract Previous research shows that studets who their families are gettig ivolved i the school practices ad activities are gettig better performaces i mathematics tha other studets. The curret research lies i mathematics educatio (CEMELA, Harvard, Iclud-ed) suggest that to create opportuities ad spaces for families to participate ito the dyamics of the schools are a way to fight agaist situatios of iequity that, sometimes, are barriers for learig of mathematics. I this paper some evideces about how to desig ad how to implemet mathematical activities with the families are discussed. The paper offers a look o the role that families play i the process of the creatio of the activities, how they approach this process, what are the difficulties (ad how they are maagig them), ad the impact that their prior experieces have i the process of learig mathematics. Key words: dialogic learig, family ivolvemet, mathematical activities, ad the other mathematics Itroductio Family traiig by itself is ot eough to have success i learig mathematics. However, prior research claims that studets with difficulties i mathematics have better performaces i this matter as far as their families start to get ivolved i the school dyamics (Klima & Mokros, 00). As Sheldo & Epstei (005) declare: ecourage parets to support their childre s mathematics learig at home was associated with higher percetages of studets who scored at or above proficiecy o stadardized mathematics achievemet tests. This paper addresses the aalysis of the outcomes achieved i a research project focused o teachers mathematics traiig from the les of the work with families. The objective is to aalyze the kid of practices ad activities that families are carryig out to help their childre to lear mathematics. We explore this questio from a didactical poit of view. The a reflectio o the impact of the mathematics teachig ad learig is provided. Fially, the paper cocludes with some cotributios drawig o the discussio of our data. Family traiig ad mathematics educatio There are several experieces about how families get ivolved i the mathematics educatio. Hoover-Dempsey & Sadler (005, 997) distiguish three differet ways to categorize how families participate i the mathematics educatio of their childre: modellig (that iclude Examples may be visited at two special issues focused o this topic: Mathematics Thikig ad Learig, ad Adult Learig Mathematics: Iteratioal Joural (special issue). 57

8 their ifluece o studets attitudes ad routies of study); reiforcemet (embracig all the families behaviours that ecourage the school success of their childre); ad direct istructio (which meas all the situatio i which a member of the family take courses to lear mathematics ad be able to help their childre at home with the homework). There are differet experieces aroud the world about this kid of practices. For example, i the Uited States cetres of research such as CEMELA, or MetroMath are very well kow. The Harvard Family Project is also a famous program i this field. I Europe there is a icipiet iterest for this type of work. Although there are ot may experieces workig with families, is importat to poit out examples such as the Learig Commuities developed by CREA, or Parets Nights, which is a case from Swede. This field has a huge importace because as Klima & Mokros (00) claim: parets have great potetial to ifluece childre s mathematical learig ad paretal support is ecessary for successful implemetatio of reform mathematics programs (p. ). Methodology This paper discusses dada comig from a research titled: Teacher traiig for a family mathematics educatio i multicultural cotexts. It is a research project fuded by the Research Departmet of the Catala Govermet (AGAUR, Agècia Catalaa de Gestió d Ajuts Uiversitaris per a la Recerca). The goal of this project is to improve the quality of the teachig practices carried out i Cataloia, through the itervetio o family traiig. The cocrete objectives iclude: () to idetify the elemets ad educative strategies i the work with adult people i the field of mathematics educatio, from a multicultural les; () to create traiig resources addressed to teachers of mathematics i adult educatio, i order to promote equity ad opportuities for all to learig mathematics; ad (3) to offer resources for a teacher traiig of quality, coected to real classroom-situatios, i order to promote a iclusive family traiig i mathematics educatio. I order to achieve all these objectives, we carried out a case study (Stake, 995), that icludes several workshops of mathematics addressed to the families, carried out i two schools, accordig to the research work developed by CEMELA i the Uited States, ad CREA i Spai ad Cataloia. A total of 4 workshops of mathematics for families are carried out durig 008. The workshops have bee coducted i two differet schools: a elemetary school placed i a city south of Barceloa, ad a middle/high school located i a workig-class eighbourhood i Barceloa. Aroud sixty persos were ivolved i these 4 workshops, from differet coutries of origi, icludig Cataloia, Morocco, Equator, Romaia, Czech Republic ad Armeia. Table : Steps to create the workshops of mathematics. Step : To meet with a perso from the mai office of the school (like the pricipal), to explai the project. Step : To ask the teachers staff which curriculum of mathematics they are implemetig i the school, how they orgaize the cotets, ad what kid of difficulties they otice amog the studets. 58

9 Step 3: To talk with the families that have childre i the school i order to figure out what are the troubles that their childre eed to face learig mathematics, i order to reach a agreemet about what to iclude i cotets of the workshop. Step 4: To desig a body of activities icludig all the voices ad demads from both, teachers ad parets. Step 5: To implemet the activities i the workshops. Amog the topics icluded there are fractios, equatios, umber sese, basic operatios (additio, subtractio, multiplicatio ad divisio), ad problem solvig. Data collected is qualitative. Videotapes, field otes, discussio groups, ad i-depth iterviews were collected. The methodological approach used has bee the critical commuicative methodology (Gómez, Sáchez, Latorre, Flecha, 006). Discussio Activity : The Cady Jar This is a ice breaker activity. This kid of activities are useful because facilitate to build a commuity sese. Whe a ew workshop starts, i may occasios people do t kow each other, or if so, it is ot i a educative milieu, where variables such as motivatio, safety, role played iside the group, etc., are really importat (Goffma, 959). Lave & Weger (99) talk about commuity of practices to aalyze this kid of situatios. We further study this issue i aother paper (Díez-Palomar, Prat Moratoas, 009). The evideces that we have collected durig this project show that as soo as this commuity sese is already built, the results i terms of mathematics learig are better. The activity about The Cady Jar is a very well-kow activity, already used i other cotexts (MAPPS project Math for Parets-, Tucso, October 004). It works with a jar of glass full of cadies. Participats i the workshop may take the jar i their hads, they ca look at it, aalyze it, but they caot ope the jar ad cout the cadies. They must figure out the total umber of cadies by estimatio. The perso who provides the closest estimatio to the result, the real umber, gets the jar as a prize. Table : Strategies used by families to solve the activity. Coutig guessig ad Approximatio estimatig the volume of the jar Two families took the jar i their hads; they tur it aroud several times, coutig the cadies through the glass. The they preseted a tetative umber. Most of the families took the jar, they tured it up ad dow several times, ad they idetified layers of cadies. The they couted the umber of cadies that they saw through the glass (a perfect cylider), ad the they multiplied that umber for the total of layers that they already couted from the bot- 59

10 tom to the top of the jar. Prior experiece A family just came up with a radom umber, drawig o their prior experiece related to the capacity of this kid of jars. Activity : Solvig axb = c equatios This activity was about how to solve several equatios with oe ukow factor. The families directly proposed the topic because it is oe of the difficulties that most of them have i order to help their childre to solve their homework. Next quote shows the kid of dyamics that occurred i the group: (Cotext: We are i a classroom placed i a high school. Parets are workig with first grade equatios with oe ukow. Now the topic how to solve a equatio is showig up. The facilitator solves the problem usig a method, ad oe mother claim that her daughter uses other way to do it. At this poit the facilitator explais the method used by the daughter. She has dived the chalkboard ito two colums: o the first oe there is the method used by the facilitator which is the oe kow by the mother- ; o the secod oe the facilitator wrote the daughter method which is the oe used by teachers ad childre at the school-). Facilitator: How it is goig? Good? Mothers: yes... very good (the mom who asked the questio is the oe who speaks louder). Mother: We did t uderstad it at home. Facilitator: eh? Mother: I did t uderstad it like this at home; this that you have explaied to us my daughter used to say mom, we wrote this here, ad I say where do you put this? because I kow it i the other w... i the old way (a oise i the backgroud is heard, like admittig she is right) ad I was ot able to uderstad it because there is o explaatio o the text book. Facilitator: But, ow did you get it? Mother: (Some mothers admittig o the backgroud are heard) Kid of, but what happes is that here is so easy... but to me... (She starts to laugh ad makes gestures with her hads to say that sometimes the activities are difficult). Facilitator:... well... this is the same... but you have to go to... Mother: (At the same time) ow you re gettig it, because, because... Facilitator: (At the same time) to everybody. Mother: she explais that she does it that way, but I do t kow how to explai it... I this activity appears as aspect that is really commo while workig with families: the coflict that sometimes emerges betwee parets ad childre, because they use differet strategies 60

11 to solve the same problem. The use of activities like this oe is useful to discuss all this differeces. Through the dialogue the families lear how teachers solve the equatios i the school owadays, ad also they share their differet ways to solve this kid of activities. Coclusios Data collected durig the research shows that families have differet ways to solve mathematical problems. To solve the coflicts that sometimes arise betwee parets ad childre, a possibility is to build what we call spaces of dialogue where everybody could iclude their voice (parets, childre ad teachers as well). Doig that it is possible to reach a agreemet that has as beefit the improvemet of the quality of the help that families are able to offer their childre i solvig mathematics (homework). I additio, the existece of these spaces also allow the appearace of differet ways to solve the activities, so they become learig spaces where differet approaches are valued. As Hoover-Dempsey & Sadler (005) claim: the opportuities for family participatio coditio their ivolvemet i the educatio of their childre. 6

12 Refereces Díez-Palomar, J., Prat Moratoas, M. (009). Discussig a case study of family traiig i terms of commuities of practice ad adult educatio. Paper preseted i CERME 6, Lyo (Frace). Goffma, E. (959). The Presetatio of Self i Everyday Life. Lodo: Pegui. Gómez J., Latorre A., Flecha R., Sáchez M (006). Metodología comuicativa crítica. Barceloa: El Roure. Hoover-Dempsey, K. V., & Sadler, H. M. (997). Why do parets become ivolved i their childre's educatio? Review of Educatioal Research, 67(), 3-4. Hoover-Dempsey, K. V., Walker, J. M. T., & Sadler, H. M., (005). Parets' motivatios for ivolvemet i their childre's educatio. I E. N. Patrikakou, R. P. Weisberg, S. Reddig, ad H. J. Walberg, (Eds.), School-Family Parterships for Childre's Success (pp ). NY: Teachers College Press. Klima, M., & Mokros, J. (00). Parets as iformal mathematics teachers of their elemetary grades childre. Coferece Report. Lave, J., & Weger, E. (99). Situated learig: Legitimate peripheral participatio. Cambridge: Cambridge Uiversity Press. Stake, R. E. (995). The Art of the Case Study Research. Thousad Oaks, CA: Sage Publicatios. 6

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22 Posig ad Icorporatig Ethomathematical Problems for School Mathematics: Providig Teachers with Required Prospective Tools Katsap Ada Kaye Academic College of Educatio Itroductio The educatio system challeges for st cetury committed the teacher to uderstad how the societal surroudigs ad commuicatio affect his pupil's perceptios ad forekowledge. Cosequetly, the orietatio i teachers' educatio, ad so aturally, i the math teachers' educatio, would be developig teacher's comprehesio of day's tedecy. These iclude, iter alia, teacher's capability to bridge betwee the academic commo math kowledge ad the kowledge roots i the culture ad society his pupils come from. Beig the case, teachig, classroom activity, cotets ad tasks i mathematics lesso should be adapted to the sociocultural backgroud of the classroom populatio. The questio is therefore this: how this purpose ca be realized? Ethomathematical Program (D'Ambrosio, 006), icorporated as a part of syllabus i the 'History of Mathematics' Educatioal College course, allowed us to give a umber of aswers to aforemetioed questio. D Ambrosio lauched Ethomathematic at the NCTM, 985, while he explaied the coceptio of socio-cultural basis of mathematics educatio, ad iterpreted ethomathematics as corpora of kowledge derived from practices, such as coutig, weighig, measurig, ad etc. Ethomathematics represets practical mathematics i both wester ad developig societies, ad has bee idetified withi the areas of architecture, orametatio, weavig, sewig, agriculture, kiship' relatios, spiritual ad religious practices. Prospective teachers (PT), represetatives of two ulike cultures backgroud, Jewish ad Bedoui, that comprise the college course populatio, lear about ad experiece the mathematical practices i their ow culture. Previous study i the subject, coducted i the course, demostrated that Ethomathematics i mathematics teacher educatio could ifluece the teachers' professioal developmet so that they icorporate humaistic ad social aspects of mathematics, ad coceive the perceptio of mathematical cocepts as a mix of abstract iterpretatio i academic mathematics ad comprehesio of practical math rooted i their culture. As a recommedatio it was writte that the educators' missio is to lead the PT's experieces so that they are likely to lik classroom ecouters with mathematics cotet ad pedagogy (Katsap & Silverma, 008). The ogoig study i the course focused o examiatio of PT's traiig i posig ethomathematical problems for school mathematics, ad clarificatio of this problem's ature. The program was carried out by qualitative-orieted study, while the collectio ad aalysis of the data (observatios, feedback-questioaire, discussios ad iterviews) follows the method of the Grouded Theory. Ethomathematical problem posig: the process ad how PT have doe with Developmet of the ability to uderstad mathematical activity i the cultural mathematical practicum ad acquiremet of learig-teachig skills i posig problems i ethomathematical cotext, are amog the goals of the ethomathematics program i the course. The traiig proc- 7

23 ess model was costructed based o the uio of three theoretical frameworks: ethomathematical approach (D'Ambrosio, 006), method of problem posig (Brow ad Walter, 005), ad coceptualizatio of the model, 'Preparig Teachers for a Chagig World' (Darlig-Hammod ad Brasford, 005). The term problem posig was itroduced by Friere. This method of educatio is based o creativity ad ecourages geuie thikig effort ad actio that ifluece reality. Later, Brow ad Walter (ibid) gave the cocept a ew meaig ad iterpreted it as a 5 stages process which guided the learer to uderstad ad solve the math problem through ivestigatio. Whe we discuss the disciplie of mathematics, our mai attetio focuses o the developmet of the learers' ability to solve problems. The problems these pupils become exposed to durig the lessos are of dual origi: teachers ad books. Sice there are ot books cotaiig ethomathematical problems, the teachers should be traied to pose these problems by themselves. As a part of the program duties i the course the PT's had to coduct a field ivestigatio ivolvig screeig ad idetificatio of the compoets embedded i mathematical practice foud i texts ad tales of their ow society ad culture. Followig this they have posed ethomathematical problems. Accordigly to these theoretical frameworks 5 stages for ethomathematical problem posig process were costructed as follows: Stages i PT learig process towards ethomathematical problem posig a) Study ad idetificatio of the practical mathematics used i the socio-cultural commuity the teacher belogs to. b) Examiatio of the cotext ad bridgig betwee formal mathematical kowledge ad the commo mathematical kowledge i the commuity practice. c) Plaig of origial math lessos combiig ethomathematical cotets. d) Posig ethomathematical problems. e) Stregtheig professioal resposibility ad commitmet to the commuity at a time of deliverig the problems for commo use. Searchig i the sources ad posig of ethomathematical problems, whose ethic cotet matches the practical math existig i the commuity, customs ad traditio, was the method that iduced the PT's to itegrate the subjects metioed durig the course already i the forthcomig lessos i their schools. The motive behid the desire to itegrate these problems i the Math teachig i the school was showig their care for the pupils, as expressed by oe of the PT's: " I thik that solvig ethomathematical problems costitutes much greater cotributio for the pupils tha solvig math problems they're familiar with at math lessos. First of all, with these problems the pupils erich their vocabulary, sice the vocabulary i these problems is ot used routiely by them [the pupils]. Beside that, the cotributio of ethomathematical problems stems from the possibility to fid essece i the problem's substace ad uderstad that othig is obvious The pupil caot say: "what do I eed it for" regardig this problem. I thik that these problems would cause the pupils to iteralize mathematical cocepts ad forge the cotexts. It tempts me to thik ad lear about my cultural lives ad borrow examples from life ad eviromet. durig the course I was able to 'come full circle' for thigs I did ot kow so well, ad did ot kow how do they happe, ad what are their sources. I thik that experiecig ethomathematical problem posig would chage the way pupils perceive the subject of mathematics." 73

24 Orey & Rosa (004) raised perceptios about multi-cultural educatio ad multi-cultural mathematics that ca be useful as iterpretatio i the discussio about ethomathematics ad professioal developmet of teachers i Brazil ad USA. They summarize various researches showig that multi cultural perspective ecourages the itellectual developmet of the teacher ad eables him to lear mathematical cotets through umerous experieces referrig to cultural, historic ad scietific developmet of mathematics. PT's expressios eable to idetify how exploratio ad scrutiy of the ethomathematics' subject helped them to comprehed the academic ad pedagogic potetial offered by it. Followig are 5 th grade PT quotatios: " Already from the glace at the subject of Jewish rugs it could be see how much kowledge ca be produced, for mathematics lesso purposes, from the study of this subject. This is somethig that previously escaped our attetio ad was cosidered as uimportat Eve whe the subject of 'rugs' did surface durig a math lesso, we did ot lik the rug to culture ad mathematics. It was just a 'example'. Now however, if I'll brig rug pictures to the class, I would ask the pupils to begi with the study of the geometric figures i them. The I would sed them to look for matchless o the cultures that made the rugs, which is already a differet iterestig approach i teachig, to idetify the differeces ad variety i their geometric figures. At the ext stage, I'll ask them to apply their coclusios o the study of the differeces betwee our ad other cultures, as reflected i the geometric figures o the rugs " This is the place to metio that PT's approached the itegratio of ethomethematical cotets with great ethusiasm, ad may of them thaked the course for its cotributio i brigig them closer to the roots ad sources of the society they grew i. I oe case, a Bedoui PT told how chagig of the word 'first perform', appearig i order of operatios (calculatios iside paretheses first perform) with the word ruler, i a story expressig hierarchy of rulers i the Bedoui family, helped him to "reach a state of 00% memorizig by the pupils the rules of order of operatios". This case study took place i a school after this PT leared the method of ethomathematical problem posig. Accordig to him, his studies i the college ispired him to have a idea he would ot have thought about before. He otes: "I was surprised ad very satisfied with myself ad the class that was so coected to my story. To pose a problem that did ot exist before, a problem so tagible ad pretty, that helped the pupils to remember the rules." The Bedoui PT experimetig permits him to supplemet a additioal way for teachig/learig Mathematics i the school that looked the Mathematics as spiritual edowmet belogig to a particular society's, traditios ad customs, ad opes to the pupils a widow to a mathematical world with a cultural way of expressio. What did the PT lear durig the program? The process uderwet by the PT's i the Ethomathematics program ca be diagosed as fosterig of mathematical literacy. As said above Ethomathematical problems are uavailable i cotrast with mathematical oes, ad the mathematical situatio take from mathematical practice i the PT's ow culture was ot preseted as a formal give of the mathematical problem. The field ivestigatio icluded clarificatio ad idetificatio of otios, which are mathematical cocepts preset i the texts ad/or deeds, ad stories of the PT's ow people. While filterig the collected data ad choosig the subject matter from mathematical practice 74

25 the PT's bega adaptig the raw material: rewritig, recostructio ad decodig. The followig citatio by oe PT describes the way she wet through: " I bega searchig the verses ad could't uderstad aythig, there were words with mathematics, but how does it all come together to establish a lik? I saw there was a ocea of kowledge - the problem was how to look for it. There are verses which ca be tured ito a problem with equatio or addig ad deductio problems it was like a verbal problem that had to be traslated ito mathematical laguage, ad these problems ca be posed easy. There were texts however, that required really scrupulous diggig for words to fid the math i them. Usually, these texts could be coverted ito geometry problems. I other problems the difficulty was that there was ot a sigle comprehesive setece ad I had to reivet it myself. I texts cocerig iheritace for example, which ca be coverted ito "percetage" problem, there is a certai formula how to calculate the percetages ad the text had to be iveted " Wrap thigs up expose to view that i order to pose a ethomathematical problem the PT's had to acquire ew skills, which they did ot eed previously, ad put ito practice both mathematical ad laguage literacy. Moreover, the PT's provided direct ad idirect cofirmatio of the aforemetioed assumptio ad stated that they have ideed felt the professioal chage they've wet through followig the program. The cases described above, ad others, which occurred durig the program, testify how the ivestigatio, deeper delvig ito the ethomathematical subject - matter ad experiece of ethomathematical problem posig brought the PT's toward the comprehesio of the academic ad pedagogic potetial it offers. I the post course feedback-questioaire at the ed of the course all PT's poited out that they do iterweave ethomathematical discussio i math school teachig i this way or aother. The citatio below reflects these feeligs: "As a teacher I have acquired a ew method for lesso structurig. Now I have aother optio for plaig a lesso ad teachig-learig methods i the mathematical class I ca give them (childre) iformatio about the culture ad traditio as well as mathematical kowledge This is a meetig with sorts of problems that eable to uderstad ad lik betwee mathematics ad ways of life of the pupils i their culture It tempts me to thik, lear about my cultural life, import examples from life ad eviromet - I am creatig a iterdiscipliary lesso." Ethomathematical problems ad the outcomes While the plaig the ethomathematics program was chose a 'ethomathematical problem' (EMP) as a major outcome of it. Furthermore, was fix o that the EMP hold uique structure, ad permeated with the awareess that mathematics assimilated its essece from huma developmet ad socio-cultural ifluece of the eviromet. Ethomathematical problem is a mathematical problem i which the verbal text uses a arrative to describe mathematical practices preset i the customs, traditios ad daily experieces of differet socio-cultural groups. The umerical value of the problem solutio must be examied i social cotext (Katsap, 008). The text itself ad phrasig of the tasks i it i EMP's differ from those appearig routiely i math-study books, ad revolves ot oly o mathematical iformatio, but also cotais philosophical-social oe. The umber of words, i which the mathematical essece hides i umbers, symbols, geometric forms, etc, is sigificatly lesser tha that of other problems, thaks to 75

26 the descriptive abudat iformatio the problem should iclude. The EMP has a permaet structure, where the problem's text is built from two parts. The first part cotais a prelude, which is a iformatio segmet of arrative ature ad is associated with culture, traditio or customs of the teacher's ow culture. The secod part of the text cotais two types of questios i the task. Oe of them is mathematical questios whose solutio addresses mathematical objects ad structures ecessitatig ivestigatio ad/or proof. The other type refers to o-mathematical questios, such as those referrig to social issues ad whose aswers satisfies the eeds of society or facilitates the commo mathematical practise i this society. The text of the problem preserves two priciples. Firstly, the questios of both types ad the prelude would have commo cotet subject, ad secodly, the EMP's text would have a arrative form. Narrative is explaied i the literature as a "process or actio durig which the story is relayed i writig or orally". The choice of the 'arrative form' of expressio is based o the ature of ethomathematics itself, where the arrative eables actual bridgig betwee the "etho" ad the "mathematics" i the text of the problem. The works doe by ma ad requirig mathematical kowledge ecompass a variety of huma activities occurrig o daily basis i various cultures ad societies dwellig i the same area. This variety eables PT's to fid a topic they hold dear, ad to create i a college class a multiplex of teachig uits with ethomathematics cotets, all of which deservig itegratio i the teachig of mathematics i the school. The study aalyzed the ature of ethomathematical problems posed by the PT's i the course. The areas ad the cotets put forward by the PT's Family ad Commuity Area: Ma & Place Religio Area: Area: Bedouis Jews Bedouis oly Jews ad Bedouis. Legacy ad iheritace X. Settlemet ad its physical geography. Shapes of structures, portals ad items particularly idetified. Kiship. Kiship. Residetial employmet: me, wome 3. Justice-makig X X X X X with syagogues/mosques.. Numbers i the Holy Scriptures desigated to assist i the decipherig of the meaig ad/or symbolise religious values. 3. Verses describig problem solutio with mathematical tools origiatig i the Holy Scriptures. 4. Symbols which are shapes ad/or saliet religious features, becomig recogizable sociocultural sigs. The coclusios suggested by observatio of topics selected by PT's for their EMP's (see table above), reveal that all cotets of the Religio Area ad Kiship cotets i the Family ad Commuity Area were commo to both Jews ad Bedouis sectors. Whereas the remaiig cotets i the Family ad Commuity Area, as well as Ma & Place Area cotets were idetified oly amog the PT's from the Bedoui sector. The puzzlig fact is the absece of the followig area: A perso as a idividual ad his pri- 76

27 vate dealigs. Last ca attest to that the PT's from both sectors perceive the field of ethomathematics ad EMP's as a subjective mutual-commuity substace i which ma does ot exist by his ow. The research fidigs suggest that the EMP's posed by PT's ca be characterised by seve statemets: The ethomathematical problem: a) Reflects reality ad the eviromet where the teacher who posed problem lives i. b) Imbued i words which are symbols ad images belogig to the teacher's ow culture. c) Iterlaces questios from mathematical ad socio-cultural areas ito oe set of tasks. d) Is liked to a variety of mathematical subjects. e) Trasfers the problem solver from the world of abstract mathematics ito the mathematical daily practices acquired by his people via various chaels: occupatio i a specific area of iterest, fulfillig of rites, preservatio of commuity customs, etc. f) Forces the problem solver to idetify with the matter of the text ad through this to play a active role i the text completio by solvig the problem ad providig the aswers it eeds. g) Bridges betwee the cotemporary world of the problem solver ad his roots ad people. EMP's costitute a uique positio i the teachig of verbal problems i mathematics, which ivites the solver of mathematical problems to feel ad experieces himself as a itegral part of his commuity trough laguage whose mathematical meaig is a part of the traditio of his ow culture. Summary The fidigs idicate that PT's learig/teachig occurred durig the process of posig ad icorporatig ethomathematical problems i school mathematics, cotributed to them acquisitio of ew learig-teachig-pedagogic skills, fosterig of mathematical literacy ad reestablishmet uderstadig the purpose of mathematical problems. The mai maifestatio of the PT's successful fuctioig were the awakeig setimet of kiship to the cultural mathematical heritage foud i the commuity the PT belog to. Recommedatios I the istructios to PT's embarkig upo a collectio of materials ecessary to compile EMP's it is desired to iclude a list of questios each of them should ask him toward the problem posig. "I the EMP I've posed:. Does the topic I've selected serve the pupils from my ow culture?. How the prelude assists uderstadig of the mathematical ad socio-cultural questios appearig i the ext stages of the problem? 3. Is there a idetifiable a mutual lik ad commo laguage betwee socio-cultural questios ad mathematical questios of the problem? 4. It there a solutio of the problem limited by a set of values, ad if so, what are the values, ad why they must be limited? 5. Do i the text of the problem cotais mathematical cocepts that are uique to the mathematical practice i my ow culture, ad if so, what are the parallel mathematical cocepts i the wester culture?" 77

28 Bibliography Brow, S. & Walter, M. (005. 3rd Editio). The Art of Problem Posig. Third Editio, Lawrece Erlbaum Associates, Mahwah, New Jersey. D Ambrosio, U. (006). The Program Ethomathematics: A Theoretical Basis of the Dyamics of Itra Cultural Ecouters. The Joural of Mathematics ad Culture. May 006, V.() ISSN Darlig-Hammod, L. & Brasford, J,. (Eds) (005). Preparig Teachers for a Chagig World: What Teachers Should Lear ad Be Able to Do. Sa Fracisco. Jossey-Bass. Katsap, A. (008). Ethomathematical Problems for School Mathematics: Teachers Posed Math Problems i Cotext o Their Ow Culture. A Paper preseted at HPM 008, July 4-8, 008, Mexico. Katsap, A., Silverma R. (008). A Case Study of the Role of Ethomathematics amog Teacher Educatio Studets from Highly Diverse Cultural Backgrouds. The Joural of Mathematics ad Culture. August 008, V.3() ISSN Orey, D.C., Rosa, M. (004). Ethomathematics ad The Teachig & Learig Mathematics from a Multicultural Perspective, Califoria State Uiversity, Sacrameto IV Festival Iteracioal de Matemática, Sa José Costa Rica. Appedixes Ethomathematical Problems Problem : The rural settig of water ad pasture Read the prelude before you ad aswer the followig questios: Prelude The Bedoui settlemet ca exist i a place that has water sources ad sheep pasture. This is the reaso the ew settlemet is erected at reasoable walkig distace from these resources. The settlemet also caot be located too close to the well, which is a source of water to all passers-by, who ca stop ad drik water or give it to their horses, camels or sheep. Therefore, the erectio of the tets i the ew settlemet takes these limitatios ito cosideratio ad erects them at a distace that leaves sufficiet space to movemet betwee the settlemet dwelligs ad the water source. Moreover, it is take care that the pasture would be withi the boudaries of the settlemet. The task a) The distace betwee the water source ad Bedoui settlemet is km. It is kow that a wome leavig the settlemet i the directio of the water source with oe jar requires oe hour to reach the water source ad retur ad that every additioal jar prologs the walkig time.5-folds. The jar cotais 30 litres. Calculate the time eeded for wome that left her home i the settlemet to go forth ad back if she carries 3 jars of water. b) Describe what you've leared about the rural settig of the water source ad pasture from the prelude iformatio. Problem : Omer Cout Read the prelude before you ad aswer the followig questios: 78

29 Prelude I the Jewish traditio the Omer cout is a 49-day coutig from Passover to Petecost. It has seve cycles of 7 days each, ad o the fiftieth day the Petecost (Shavu'ot) is celebrated, amed after the seve weeks (shavu'a) of the cout. For religious persos, this period is characterised by relative absece of happiess ad joy avoidace of shavig ad haircuttig, o weddigs, avoidace of listeig to music (the leiet approach edorses music without musical istrumets) or dacig. The origi of the commadmet is Leviticus 3: Ad you shall cout from the ext day after Sabbath, from the day I've brought you, the Omer: seve Sabbaths shall thee cout, util the ext day after Sabbath you shall cout fifty days ad made a ew sacrifice to the Lord". The task a) The beard growth rate of a certai perso is.5 mm per week. What shall the legth of his beard be at the ed of the Omer Cout? b) The hair legth of a 0 year-old child is 0 cm. Durig the moth (4 weeks) his hair reached the legth of 6 cm. What would his hair legth be at the ed of the Omer Cout? c) A perso is used to go to the sea o daily basis. Durig the Omer Cout days he avoids goig to the sea. What is the percetage of the days' amout i relatio to the all year he does ot go to the sea? d) Thik how mathematical kowledge helped religious Jews to observe the commadmets. Problem 3: Mosque - a place for worship ad mathematical arts Read the prelude before you ad aswer the followig questios: Prelude A Mosque is characterized by multiple usages of arts characterized by a variety of geometric patters. Each of the mosque's structure parts: the dome, turret, widows ad the iche are pletiful with decoratios. Typically, these decoratios feature Arab writig or plat motives. Accordig to Islamic laws o huma or aimal images are allowed. The door is the etrace ito the mosque. Moslem art creativity foud its expressio i wood etchig ad irowork, as well as i decoratio ad orametatio of the mosque's doors, characterized by various colorful geometric patters. The task a) Describe what you've leared about the rural settig of the water source ad pasture from the prelude iformatio. b) Which geometric patters ca be idetified o the doors ad decoratios of the mosques i the picture below? c) Copy the geometric forms ad patters you've idetified i the picture to your otebook. If you ca idetify symmetry i the copied doors ad patters, ad if so, list the symmetry types. d) Thik about the artists that embellished the mosques, ad coceive i the mid did they have to lear geometry? Discuss this questio with your frieds. Problem 4: Wome's Sectio Read the prelude before you ad aswer the followig questios: Prelude A old Jewish custom says that me ad wome should ever sit together i order to avoid distractio. The custom triggered a separate sectio for wome i each syagogue, kow as the wome's sectio. This ame stems from the Jewish Temple's, ad the Gemara expouds what was the great repair istalled: 79

30 that it was surrouded by a balcoy ad the wome see from above ad me from below. The porch wome sit i has a high parapet ad usually a curtai, through which they peep o what happes i the me's sectio. The task a) What do you thik is felt by the wome attedig a syagogue? There are three rows i the wome's sectio ad each oe has 5 seats. b) I how may ways it is possible to arrage: Five wome i a row? The five wome, oe mother ad four daughters, whe the mother has a permaet seat - first o the right? c) I how may ways it is possible to arrage i the wome's sectio mothers ad 5 daughters i oe row: Whe the mothers sit ext to each other? Whe the mothers do't sit ext to each other? The five daughters sit ext to each other? d) Pose two additioal problems o the subject ad write dow their solutios. Problem 5: Rape or a Rape Attempt Read the prelude before you ad aswer the followig questios: Prelude I the case of crimes agaist wome, such as abductio, rape or attempted rape - the puishmet is very severe. These offeces are see as violatio of family hoor. Bedouis compare rape to a situatio of breakig a clay pottery, which caot be repaired. A girl cooperatig with the abductor ad does ot resistig loses her rights i the trial. As for wome that was take by force or raped, she must brig at least three witesses to support her testimoy, those that heard her cries ad rushed to her aid. I this case, ay perso rushig to the aid of the assaulted girl receives a reward, grated by the judge. The sum of the reward is seteced accordig to the umber of steps the perso made back ad forth. Except this, there is the fixed paymet to the girl's father, at the sum of 000 diars. The task a) Do you thik the traditioal puishmet, is ideed, just? Explai. b) If you were the judge, would you setece the rapist to the same puishmet ad adjudicate the same sum as compesatio? c) The judge awarded 0 diars for every step made by the ma who came to aid the girl. What is the sum to be paid to the ma who wet 0 meters i each directio, if it is kow that every meters are equal to 3 steps? d) Pose two additioal problems o the subject ad write dow their solutios. Problem 6: Haukkiyya I have. Read the prelude before you ad aswer the followig questios: Prelude The Haukkah Lamp (Haukkiyya) is costructed from ie braches: eight braches for the burig cadles ad the ith for the 'sexto' (shamash), a cadle servig to light the other cadles. I the Jewish traditio, the eight cadle braches symbolize the eight days of the Haukkah miracle. I compariso, the Meorah (Cadelabrum), oe of the Jewish symbols sice the days of the Temple, appears seve braches. As argued i the Talmud, it might be a result of the cotroversy cocerig the prohibitio of usig a lamp resemblig the Cadelabrum of the Temple. I a 'kosher' Haukkah lamp the eight braches must be i a straight lie, at uiform height ad at equal spacig, while the brach for 'sexto' should differ from the others i some way. 80

31 The task a) A DIY store sells 5 cm iro poles at the price of 0 NIS each. Members of the Yehuda Ha- Maccabi family came to store to buy iro poles for a Haukkiyya. Help them to calculate the budget required to build the lamp. b) Draw a Haukkiyya ad calculate the umber of iro poles required, ad the price to be paid for them. c) Propose two additioal Haukkiyya desigs. d) The Ha-Maccabi Family also decided to costruct a Cadelabrum with braches of similar height to those i the Haukkiyya. Kowig that the Cadelabrum ad Haukkiyya differ from each other, poit out the differece ad the reaso behid the appearace of each oe of these Jewish symbols? e) Calculate the umber of iro poles required to build the Cadelabrum you've desiged, ad the purchasig expeses of the Ha-Maccabi Family. 8

32 La commuauté virtuelle CASMI ecourage-t-elle plus de créativité das les solutios d élèves? Domiic Mauel Viktor Freima Uiversité de Mocto Abstract May researchers suggest that the use of rich ad complex mathematical tasks to schoolchildre may give them opportuities to develop problem solvig abilities such as geeratig multiple ad possibly origial solutios ad strategies, to reaso mathematically, to makig coclusios, justifyig them ad to commuicate mathematical ideas (Sheffield, 008). This visio does ot however seem to be makig its way ito the classroom. I fact, may authors say that the pedagogy used i mathematical classrooms ted to focus more o what studets reproduce istead of how they thik, which would foster studets atural curiosity (Meisser, 005). Could the rapidly grow virtual olie learig commuities help brig this visio with teachig practices givig ew opportuities to develop mathematical creativity amog studets? I our exploratory study, we will aalyze a case of the CASMI sciece ad mathematics iteractive learig iteret-based commuity ( created by a team of researchers from the Uiversité de Mocto (Freima, Mauel & Lirette-Pitre, 007). We will first look how rich are the problems posted o a bi-weekly basis o the website accordig to the list of characteristics that we foud i the literature o what could be defied as a good mathematical problem. Also, we will aalyze a collective solutio space of selected problems as it has bee suggested by Leiki (007) lookig at its potetial i terms of mathematical creativity by examiig the fluecy, the flexibility ad the origiality of each solutio. Further, we will explore if there exists a relatioship betwee the richess of the problems posted ad the potetial mathematical creativity foud i the collective solutio space. We assume that solutios will be more creative whe problems are rich. We will preset some prelimiary results of our study durig the coferece, followed by recommedatio o the use of potetially rich olie tasks by teachers i order to develop more creative approach to mathematics i their studets.. La résolutio de problèmes e salle de classe : cotexte actuel Plusieurs chercheurs metioet qu e proposat aux élèves de problèmes riches et complexes e mathématiques, o leur permet de géérer des solutios multiples ou de stratégies origiales et aisi appredre à raisoer, justifier des coclusios et commuiquer mathématiquemet des résultats (Sheffield, 008). Ces diverses habiletés e sot pas iées chez ue persoe et e se développet pas automatiquemet ; pour u eseigat, il deviet 8

33 alors importat d apporter de chagemets réels das leurs pratiques afi de cultiver et de ourrir la créativité mathématique chez les élèves Les auteurs soutieet que la pédagogie utilisée e salle de classe devrait permettre aux élèves de développer leurs propres stratégies de résolutio de problèmes et de chercher de solutios origiales au lieu de mettre l accet sur l habileté des élèves de reproduire les procédures eseigées. Selo Meisser (005), cette pratique courate risque de uire à la curiosité aturelle des appreats et leur ethousiasme evers les mathématiques à mesure qu ils gradisset. De plus, état habitués à chercher ue répose e appliquat ue stratégie apprise, les élèves s adaptet difficilemet aux eseigats qui tetet de développer la créativité mathématique das leur salle de classe (Messier,000). Le développemet des espaces virtuels au cours de dix derières aées a permis d erichir les pratiques d eseigemet de mathématiques e proposat ue vaste gamme de problèmes e lige (Klotz, 003). La commuauté virtuelle CASMI (Commuauté d appretissages scietifiques et mathématiques iteractifs, cotiet des cetaies de problèmes potetiellemet riches et complexes qui ot été résolus par de milliers d élèves membres de la commuauté. Repreat la défiitio de l espace collectif de solutios multiples à u problème mathématique (Leiki, 007) das le cotexte de la commuauté virtuelle CASMI, ous allos étudier l origialité, la flexibilité et la fluece das les solutios des élèves e lie avec la richesse des problèmes mathématiques. Afi de costruire u cadre de référece permettat cette étude de cas, ous allos tout d abord ous pecher sur les caractéristiques de la créativité mathématique. Esuite, ous allos aalyser e profodeur le cocept d u problème riche e mathématiques. Nous allos égalemet décrire l espace virtuel collectif de solutios qui est la commuauté CASMI tout e précisat os choix théoriques et méthodologiques.. Le développemet de la créativité mathématique et les problèmes riches Les chercheurs ot différetes visios par rapport au cocept de créativité mathématique, ce que reflète ue multitude de défiitios qui se trouvet das la littérature recesée. Tadis que Ruco (993) tiet compte de la relatio etre la pesée covergete et divergete, d autres auteurs comme Haylock (997) et Krutetskii (976) tieet compte de la fluece (trouver plusieurs réposes à u problème), la flexibilité (utiliser plusieurs stratégies pour résoudre u problème) et l origialité (produire des solutios uiques) des solutios aux problèmes mathématiques. Deux élémets semblet toutefois se trouver das la majorité des défiitios : - la créativité et l iovatio sot souvet citées avec la résolutio de problèmes, la pesée critique, la commuicatio et la coopératio comme état des coaissaces et des habiletés cruciales et des expertises que les élèves ot besoi pour avoir du succès das leurs vies das la société de os jours ; - il faut u eviroemet particulier pour que la créativité apparaisse (se développe). 83

34 E effet, les chercheurs comme Clie (999), Sheffield (003), Freima (006) recoaisset que la créativité mathématique se développe lors de la résolutio des tâches riches e mathématiques preat souvet la forme de problèmes ouverts et de situatios-problèmes. Les problèmes ouverts semblet permettre de se décoecter du stéréotype que chaque problème possède seulemet ue répose e icitat les élèves e trouver différetes e utilisat leurs propres stratégies (Klavir & Hershkovitz, 008). E procédat aisi, les élèves peuvet trouver différetes stratégies pour résoudre u problème, trouver différetes possibilités de réposes, développer des stratégies origiales, et predre des risques, toutes des caractéristiques de la créativité mathématique ressorties par des chercheurs (Ma, 005). L auteur affirme que l utilisatio de problèmes mathématiques ouverts permet aux élèves d atteidre automatiquemet les premiers stades de la créativité mathématique. État doé que la résolutio de problèmes ouverts semble favoriser la créativité mathématique, le site CASMI peut-il doer l occasio aux élèves de développer leur créativité mathématique? Das la sectio suivate, ous allos préseter brièvemet cette commuauté virtuelle. 3. Le CASMI : ue commuauté virtuelle à potetiel créatif? Créé e 006, le site Iteret CASMI est ue commuauté d appretissage virtuelle pour les mathématiques, les scieces, les échecs et la lecture. Ayat pour but d augmeter chez les élèves les expérieces positives e mathématiques, d améliorer les habiletés de résolutio de problèmes chez les jeues à travers des défis de grade taille e utilisat les techologies et de favoriser l appretissage par la collaboratio, la commuauté virtuelle CASMI deviet u exemple d eviroemet favorisat des appretissages plus riches (Freima, Mauel, & Lirette-Pitre, 009). Selo ue étude meée auprès les élèves participats au projet (Freima & Mauel, 007), ces deriers aisi que leurs eseigats semblet apprécier les défis que leur poset les problèmes affichés sur le CASMI e leur permettet d améliorer leurs habiletés de résolutio de problèmes. De plus, les élèves semblet être motivés à résoudre les problèmes et appréciet le déroulemet du site et les rétroactios qu ils reçoivet. Pourtat, les raisos de cette boe appréciatio e sot pas ecore coues. Etre autres, comme bie de chercheurs cités ci haut, ceci pourrait être attribué à la richesse de problèmes proposés e lige et aux possibilités qui s offret aux élèves d être plus créatifs. Qu est-ce qu u problème riche? Les problèmes posés das la commuauté virtuelle CASMI sot-ils riches? Mais quelles sot les caractéristiques de problèmes riches? Les solutios sot-elles plus créatives lorsque les problèmes sot les plus riches? Ces questios émerget aisi e ous permettat de mieux cibler les objectifs de otre recherche. Les participats vot-ils toujours utiliser les mêmes stratégies lors de la résolutio des problèmes sur le CASMI? Ou vot-ils plutôt développer leurs propres stratégies? Trouveros-ous des stratégies origiales? U problème potetiellemet plus riche, crée-t-il u espace virtuel collectif de solutios et de stratégies plus origiaux? E étudiat ces élémets plus e détail, ous pourros détermier s il existe des lies etre la richesse des problèmes posés das le CASMI et la créativité 84

35 mathématique retrouvée das les solutios soumises par les membres. Les retombées de cette recherche pourrot servir à doer des pistes aux eseigats des types de problèmes qui ot le potetiel de développer davatage la créativité mathématique chez les élèves. 4. Choix méthodologiques Notre recherche se déroule e deux étapes. Tout d abord, ous aalysos la richesse des problèmes qui sot posés das la commuauté virtuelle CASMI. Suite à otre aalyse approfodie de la littérature, ous avos ressorti plusieurs critères mesurables et observables qui caractériset la richesse d u problème. Ue grille d aalyse coteat ces critères a été créée et validée. Par la suite, otre deuxième étape est d étudier le potetiel de la créativité mathématique de l espace virtuel collectif de solutios de ces problèmes. Les composates de la créativité mathématique que ous utilisos das cette aalyse sot la fluece (ombre de boes réposes trouvées au problème), la flexibilité (ombre de stratégies différetes utilisé pour résoudre le problème) et l origialité (la fréquece miime des réposes et stratégies et de la commuicatio mathématique des solutios). État doé la ature exploratoire de otre recherche et l espace limité de cet article, ous allos illustrer otre aalyse prélimiaire à l aide d u exemple. 4. Exemple d u problème Depuis la semaie où ous vous avios proposé de jouer avec les palidromes, otre mascotte CASMI a composé ue éigme avec des lettres de so om. Il a tout d abord pris u ombre à ciq chiffres et il l'avait additioé avec u autre ombre qui cotiet les mêmes chiffres, mais das l ordre iverse. Esuite, il a calculé la somme. Fialemet, il a remplacé tous les chiffres sauf u par les lettres de so om (ue lettre remplace u chiffre; les mêmes lettres remplacet les mêmes chiffres). CASMI te lace u défi de trouver quel chiffre se cache derrière la seule lettre qui e se trouve pas das so om. 4. Richesse du problème Selo os critères de la richesse des problèmes, ous pouvos établir que ce problème possède plusieurs réposes possibles. De plus, il peut être résolu e utilisat plus d ue stratégie. Par exemple, ous pouvos toujours procéder par essai erreur pour résoudre ce problème. Mais ous pouvos aussi étudier toutes les possibilités d obteir des sommes de deux ombres de faço systématique. Ce problème est complexe, car l élève qui le résout doit passer par plusieurs étapes afi d arriver à la boe répose. Cepedat, selo os critères de richesse, ce problème aurait pu être plus riche e présetat u cotexte réel. De plus, il e demade pas à l élève de faire des choix etre différetes optios (comme, par exemple, choisir le meilleur pla d achat). E somme, ous cosidéros le problème moyeemet riche par rapport à otre modèle. Nous pouvos doc s attedre que l espace collectif de solutios soumises par les élèves à ce problème sur le site CASMI ous doe quelques solutios créatives. Nous avos doc examié cet espace e appliquat les critères d origialité, de fluece et de flexibilité. 85

36 4.3 Potetiel de créativité mathématique de l espace collectif de solutio U total de 37 solutios a été soumis pour ce problème. Nous e avos elevé 8 car elles étaiet icorrectes, ou e coteaiet pas suffisammet d iformatios pour les évaluer. Fluece : Le problème a demadé de trouver la valeur de la lettre R, la seule qui e se trouvait pas das le mot CASMI. U total de 5 réposes différetes a été trouvé, soit les valeurs de R égales à 3, 4, 7, 8, et 9. La plupart des jeues (85,7%) ot trouvé ue seule répose. 7,% d élèves ot trouvé réposes et 7,% ot trouvé 3 réposes. Flexibilité : Les stratégies employées étaiet surtout essai et erreurs. Aucue solutio e coteait plus d ue stratégie. Das quelques cas, les élèves se sot servis des propriétés des ombres (pair et impair) et d ue recherche systématique de toutes les faços d obteir des sommes requises. Origialité : Le tableau ci bas représete la fréquece des réposes obteues. Nous pouvos remarquer que deux réposes (R = 3 et R = 4) étaiet les plus rares, doc, elles sot cosidérées comme origiales, selo otre défiitio. Répose Pourcetage (%) trouvée 3 3, 4 3, 7, 8 39,3 9 33,3 E somme, le potetiel créatif pour ce problème était surtout au iveau de la fluece et de l origialité, car u total de 5 différetes réposes ot été trouvées pour ce problème et que parmi les 5 étaiet origiales puisque très peu d élèves les ot trouvées. Pour ce qui est de la flexibilité, celle-ci est pas exploitée du tout. Les élèves semblet se coteter d ue seule stratégie pour résoudre le problème. Nous pesos que le potetiel créatif serait plus élevé pour des problèmes plus riches. Compte teu la ature exploratoire de otre étude et du ombre limité de solutios aalysées, ous e pouvos pas tirer de coclusios plus profodes. Par cotre, e faisat ue aalyse avec plusieurs problèmes et avec u plus grad échatillo de solutios, o pourra mieux répodre à os questios de recherche. Toutefois, os aalyses démotret que les eseigats voulat développer la créativité mathématique chez leurs élèves peuvet se servir de ce problème et de l espace collectif de solutios pour orgaiser des discussios à l itérieur de la commuauté virtuelle ou e salle de classe afi de permettre aux élèves d exploiter leurs différetes solutios, les iciter à chercher esemble différetes stratégies et de chercher d autre solutios origiales ou de prouver qu ils e existet plus. 86

37 Référeces Cha, C. M. E. (008). The use of mathematical modelig tasks to develop creativity. Paper preseted at the th Iteratioal Cogress o Mathematical Educatio: Discussio group 9, Moterrey, Mexico (July ). Clie, S. (999). Giftedess Has May Faces: Multiple Talets ad Abilities i the Classroom: Distributed by Wislow Press for The Foudatio for Cocepts i Educatio, 770 East Atlatic Ave., Delray Beach, FL ($34.95). Freima, V. (006). Problems to discover ad to boost mathematical talet i early grades: a challegig situatios approach. The Motaa Mathematics Ethusiast, 3(), Freima, V., Mauel, D., & Lirette-Pitre, N. (007). CASMI Virtual Learig Collaborative Eviromet for Mathematical Erichmet.. Uderstadig our Gifted, Summer 007, pp Haylock, D. (997). Recogizig mathematical creativity i school childre. Iteratioal Reviews o Mathematical Educatio, 9(3), Klavir, R., & Hershkovitz, S. (008). Teachig ad evaluatig "Ope-Eded" Problems [Electroic Versio]. Iteratioal Joural for Mathematics Teachig ad Learig. May 0th. Krutetskii, V. A. (976). The psychology of mathematical abilities i school childre. Chicago: Uiversity of Chicago Press. Leiki, R. (007). Habits of mid associated with advaced mathematical thikig ad solutio spaces of mathematical tasks. Paper preseted at the The fifth coferece of the Europees society for ressearch i mathematics educatio-cerme-5. Ma, E. (005). Mathematical Creativity ad School Mathematics: Idicators of Mathematical Creativity i Middle School Studets. Uiversity of Coecticut, 005, Meisser, H. ( 005). Creativity ad mathematics educatio.. Paper preseted at the The 3rd East Asia Regioal Coferece o Mathematics Educatio from Ruco, M. A. (993). Diverget thikig, creativity ad giftedess. Gifted child Quarterly, 37(), 6 -. Sheffield, L. (003). Extedig the challege i Mathematics. Califoria: TAGT & Corwi Press. Sheffield, L. (008). Promotig creativity for all studets i mathematics educatio: A overview. Paper preseted at the th Iteratioal Cogress o Mathematical Educatio: Discussio group 9, Moterrey, Mexico (July ). 87

38 PATTERNS OF CHANGE IN SOLVING DYNAMIC AND STATIC PROBLEMS Ildikó Pelczer Florece Mihaela Siger Cristia Voica Natioal Autoomous Uiv. Mexico Uiversity of Ploieşti Romaia Uiversity of Bucharest Romaia Subject: Problem solvig ad istitutioalizatio of kowledge. Das cet article, ous aalysos l évolutio das le temps du ombre de réposes correctes et icorrectes aisi que du pourcetage de l absece de réposes à des problèmes proveat d u test iteratioal à choix.. Le pricipal itérêt à ce travail est de relier certaies caractéristiques de problèmes à certais chagemets. Pour catégoriser les problèmes, ous avos reteu les catégories «dyamique» et «statique». Les problèmes dyamiques impliquet u chagemet das leur cofiguratio, ue trasformatio du cotexte ou ue aalyse des variates ce que impliquet pas les problèmes statiques. Ces deux catégories de problèmes semblet défiir différets patters de réposes et ce, à différets iveaux scolaires. La coaissace de tels comportemets pourrait aider les eseigats à mieux structurer les activités de résolutio de problèmes tout au log de la scolarité. INTRODUCTION We have chose for the classificatio of math problems a kiematical poit of view. A kiematical view is sigificat for a couple of reasos, which go from the curriculum to the idividual. I geeral, the curriculum provides a successio of topics ad activities across ages from exploratios to formal kowledge, which supposes a evolutio from dyamic searches to more static descriptios (Siger, 007). This is relevat for may coutries because the structure of cotemporary curriculum i mathematics throughout the world reflects uifyig tedecies that show its globalizatio (e.g. Atweh & Clarkso, 00). Whe referrig to studets, while i the first years of schoolig childre perceptio seems to be mostly dyamic-rhythmic, while progressig to high school, studets explaatios ivolve more formal argumets of a static ature, iduced by the ew acquired kowledge (Siger & Voica, 008). If we look at the idividual s developmet through learig, the dyamic ifrastructure of mid (Siger, 008) allows successive phases i the process of cogitive growth. Therefore, a aalysis of the dyamics of problems could offer ew data o the problem solvig process to both: the teachig practice, ad the didactical research. We further defie two geeral categories of problems. A dyamic problem supposes a movemet of a cofiguratio, or a trasformatio of the cotext, or some degrees of freedom which lead to more possible results. Coversely, a static problem does ot suppose motio or variability of data. These defiitios equally refers to the text of the problem as well as to the cogitive processes ivolved i problem solvig activities ad allow classifyig problems i two categories that do ot overlap: dyamic or static. Because the classificatio emphasizes some complex characteristics of problems, sometimes it is ot obvious to which category a specific problem belogs to. For istace, problems with similar text ca be situated i differet classes, sice the essetial criteria is the dyamics of the cotet (Siger & Voica, 006). It has to be metioed that, for some problems, differet persos might have differet opiios about 88

39 to which category a problem belog. This ca happe because each oe has its ow way to formalize the problem ad to iterpret it. The importace of a characteristic depeds o the possibility to highlight sigificat differeces i some descriptors. I the preset paper, we are iterested to see how the dyamic-static characteristic affects o-aswers ad the ratio of correct aswers per distracter with highest percetage (descriptors). To exemplify our method, we ext look at two problems i detail. The two problems (oe dyamic ad oe static) were chose such to belog to the same domai (geometry) ad to make referece to the same type of cofiguratio. Problem (static) I a rectagular triagle, a ad b are the legs. If d is the diameter of the iscribed circle ad D of the circumscribed oe, the d D is equal to: A) ab B) (ab) C) 0.5(ab) D) ab E) a b Problem (dyamic) I the figure there are two half circles of radius cm ad ceters E ad F ad a circle of diameter AB (A ad B are o the two half circles so that AE EF ad BF EF). The area of the grey regio is: The first problem is a static oe because, i order to solve it, we eed to visualize the iscribed circle, as determied by the taget poits. The fiishig of the drawig with these segmets does t chage the iitial cofiguratio, but rather puts i evidece the data give i the problem. For solvig it, it is eeded to idetify equal segmets, but this process will ot iduce chages i the cofiguratio. O the cotrary, the secod problem is dyamic, because a fast solutio (it has to be remided that these problems are give at a competitio) supposes to trasform the figure by traslatig parts of the grey area (two quarters of the disc of diameter AB) over two white quarters (circles with radius AE ad BF). By this traslatio, the area asked i the problem ca be easily computed as the area of the rectagle ABFE. We cosider as give that problems ca be categorized i a disjoit way ito static ad dyamic, ad there are several types of static ad dyamic problems (Siger & Voica, 006). However, we have degrees of dyamic ad static characteristics, especially for problems that allow a dyamic ad static approach. I this article we argue i favor of the followig claims: ) o-aswers are i geeral higher for static problems; ) dyamic problems have a lower rate of growth whe we look at the ratio betwee correct aswers ad the percetage of the maximum distracter tha static oes; 3) the rate of chage i case of the dyamic problems depeds o the type of the dyamic problem; 4) static problems icrease the ratio withi the advaces of the grades due to icrease i hadlig mathematical kowledge or due to alteratives i solvig the problem; 5) there is a chage i the ratio betwee static ad dyamic problems used i school i favor of the static oes as progressig i schoolig; 6) these aalyses iform both classroom practice ad didactics o better approaches o problem solvig. METHODOLOGY Sample ad tools. We applied the dyamic-static classificatio to the problems give at the iteratioal applied mathematics cotest Kagaroo i order to allow the aalysis of the relatio 89

40 betwee problems ad studets resposes. Kagaroo is a multiple choice cotest where the studets have to aswer, i 75 miutes, 30 questios of 3, 4 or 5 poits worth. For each questio five aswers are give, but oly oe is correct. Wrog aswers are pealized by decreasig the accumulated poits by the quarter of the problem s worth, meawhile o-aswer or multiple aswers are ot take ito accout (but are still stored). The tests are commo for two cosecutive grades (from grade 3 up). I Romaia, each year, there are approximately 50,000 participats at the cotest. The umber of participats decreases betwee grades, from a iitial 40,000 i grade (7-8 years old) to aroud,00 i grade (8-9 years old). Give the high umber of participats, the aalysis of resposes ca give statistically sigificat results. Method. I order to sustai the claims, we aalyzed several aspects by lookig at the dyamic ad static problems give across several grades at the Romaia editio of the Kagaroo competitio. Problems were selected i pairs (dyamic-static), such to show certai similarities i cotet or i cocept ad to have bee give at several grades. We examied the: evolutio of average o-aswers percetage, variatio i the ratio of correct ad maximum valued distracter or, i some cases, the sum of distracters with highest percetage of selectio. RESULTS AND DISCUSSION I the geeral aalysis we looked at the evolutio of the average of o-aswers (table). It ca be observed that there is a cotiuous icrease for both types of problems, but up to grade 6 dyamic problems have a higher rate of o-aswers average tha static oes. From grade 7, there is a iversio that remais stable i time. We argue that this is due to the relatio that static problems have with formal kowledge. As participats advace i their studies, problems are more focused o the use of formal kowledge, but also to hadle them becomes heavier for studets, so they avoid aswerig to those they cosider too difficult at the first sight. Grade Static Dyamic Table. Average of o-aswers (i %) per problem type ad grades The differeces betwee static ad dyamic problems are maitaied at particular aalysis. I order to see whether the differece i these characteristics of the problems triggers differet behaviors from participats, we aalyzed the ratio betwee the percetage of correct aswers ad the maximum percetage from the distracters. Table cotais the results. Problem Classificatio Grade Static Dyamic Table. Patter i time of the correct/maximum distracter ratio for a dyamic ad a static problem It ca be observed directly that the ratio is growig faster for the static problem tha for the dyamic oe. I the case of the static problem, this ca be due to the fact that studets ca reaso more effectively about the segmets as they advace i their studies. The icrease i case of 90

41 the dyamic problem is slower, but still gets to oe at the highest grade (that is, oly i grade the correct aswers overcome the wrog oes). We cosider that this particular result is ot suggestig that i grade studets solved the problem dyamically; o the cotrary, we cosider it as a cofirmatio of the fact that static thikig becomes domiat i time. The reaso for such iterpretatio comes from cosiderig the aswers to distracters, ot oly the ratio. I case of problem, i grade 7 ad 8, the distracter with maximum percetage was B, that is (ab); meawhile from grade 9 it is the distracter E, that is a b. But, distracter E is a aswer that should be elimiated at oce, sice it represet the diagoal of the triagle (D) ad it is obviously ot the correct oe. Still, choosig this aswer reflects that studets retrieved, almost i a automatic way, the kowledge that was liked to a right agle triagle with the give legs (Pythagoras theorem) ad do ot estimate the plausibility of the aswer. The same argumet goes for the dyamic problem. The most frequetly chose distracter (from grade 0 up) was C, which expresses the area of a half circle (so, obviously less tha the grey area). Oly i grade 9, the distracter with maximum percetage was E, at least ot so clearly wrog as C. Based o what kid of thikig is required by the most frequet error, we suppose that the same way of thikig was applied i case of correct solutios. This is why we argue that, i case of the dyamic problem, the.0 ratio at grade could be still the expressio of a static thikig. It has to be remided that this is also possible because the problem allows a static way of solvig (by drawig the triagles BFO, AEO where O is the itersectio of the three circles), eve if that oe is ot the most efficiet (at least as the eeded time cosidered). I coclusio, beside beig a argumet i favor of hypotheses ad 3, the above situatio illustrates that the ratios for dyamic problems that also have a alterative static way of solvig icrease slower tha for completely static oes (hypothesis 4), but probably faster tha for pure dyamic problems. CONCLUSIONS We looked at two categories of problems: dyamic ad static, ad argued that they defie differet patters of behaviors. This hypothesis was verified by a geeral overview of a multiple choice cotest problems ad particular aalysis of a pair of static-dyamic problems. The first aalysis focused o the evolutio of average percet of o-aswers from grade 3 to grade ad cocluded that after grade 7 (whe i most of the curricula there is a complete tur ito teachig formal mathematics: provig, formal reasoig, various formulas, etc.) static problems have costatly higher percet of o-aswers, meawhile durig all grades there is a icreasig tedecy for both types of problems. The particular aalysis uderlied that static ad dyamic problems show differet patters of ratio (correct aswers per distracter) chage betwee grades: with static oes growig faster. The rate of growig depeds o the type of static or dyamic problem, cocretely whether there are alterative solutios to the problems. We cosider these results as highly relevat to teachig ad assessmet, sice they treat complemetary aspects of problem solvig. 9

42 REFERENCES Atweh, B. & Clarkso, P. (00). Iteratioalisatio ad globalisatio of mathematics educatio: Towards a ageda for research/actio. I B.Atweh, H.Forgasz & B.Nebres (Eds.). Sociocultural research o mathematics educatio: A iteratioal perspective (67 84). New York: Erlbaum. Siger, F.M., Voica, C. (006). A kiematical classificatio of mathematics problems (i Romaia). ROMAI Educatioal Joural,, pp Siger, F.M., Voica, C. (008). Betwee perceptio ad ituitio: thikig about ifiity. Joural of Mathematical Behavior, 7(3), Siger, F.M. (007). Balacig globalisatio ad local idetity i the reform of educatio i Romaia. I B. Atweh, M. Borba, A. Barto, D. Clark, N. Gough, C. Keitel, C. Vistro-Yu, & R. Vithal (Eds.), Iteralisatio ad globalisatio i mathematics ad sciece educatio (pp ). NewYork: Spriger Sciece. Siger, F.M. (008). The Dyamic Ifrastructure of Mid - a Hypothesis ad Some of its Applicatios. New Ideas i Psychology, DOI:0.06/j.ewideapsych

43 RELATIONSHIP BETWEEN ITEM DIFFICULTY RATIO AND SUBJECTIVE PERCEPTIONS OF DIFFICULTY IN MATHEMATICAL PROBLEM SOLVING Ildikó Pelczer*, Viktor Freima** *Cetre for Techical Developmet, UNAM, Mexico ** Uiversity of Mocto, New Bruswick, Caada Subject: Problem solvig ad istitutioalizatio of kowledge. Das cet article, ous présetos les résultats d ue evaluatio prélimiaire de la relatio etre la performace e résolutio de problèmes mathématiques, la perceptio des solutioeurs quat à la difficulté des probèmes et les caractéristiques de ces problèmes. Les problèmes proveat de l eviroemet virtuel CASMI sot caractérisés par leur cotexte familier et par le processusaticipé de résolutio. Ue aalyse exploratoire ous amèe aux coclusios prélimiaires suivates : il existe des patters que l o peut idetifier relatifs à la difficulté perçue des problèmes et chacu de ces patters est relié aux caractéristiques du problème aisi qu à la difficulté relative de l item. Nous poitos commet ces résultats peuvet être utiles aux eseigats.. AIMS OF THE STUDY I their lesso preparatio, mathematics teachers have to evaluate the difficulty of problems chose for studets. There are beefits of doig so: it allows selectig the problem that is appropriate to the studet s kowledge ad, lets icrease gradually difficulty levels i order to keep studet motivated (Newma et al., 998). Research related to difficulty of mathematical problems ca be grouped as follows. First, we have studies that search for domai-idepedet measures of difficulty. Grodlud (98) defied difficulty as the ratio betwee the umber of correct ad total umber of solutios. He itroduced the term item difficulty ratio (ITR) for this cocept ad, today, is still oe of the most used measures. Aother commo measure from the same category is respose time: time passed betwee the presetatio ad resolutio of a item (Maso et al., 99). Secod, there are the studies that search for factors that affect success i solvig a problem ad thus ifluece its difficulty level. For example, Lae (99) foud that difficulty of algebraic word problems depeds o such factors as umber of itermediate results to be obtaied, a ecessity to reformulate result i oe short setece ad familiarity of problem cotext. Horke & Habo (986) defie these factors as cogitive. O the other had, Jerma (983) used the term complexity factor to deote elemets that relate to the umber of steps required to achieve results, umber of partial results ad computatioal complexity. I these cases, difficulty was measured as ITR. A third lie of studies focuses o problem categories ad their difficulty. Researchers i this lie, defie frameworks to classify problems ad the try to associate difficulty with categories. The difficulty i questio ca be the subjectively perceived difficulty (Craig, 00), ITR (Caldwell & Goldwi, 987) or a researcher established scale based o the degree of solutio correctess (Galbraith & Haies, 000). I this approach, the umber of difficulty categories is aother aspect. I case of ITR we have a cotiuous measure; i the other cases we have crisp values. I Craig (00), difficulty measures were established by compariso of problems from differet categories tryig to set up a oe to oe correspodece betwee problem category ad diffi- 93

44 culty. The same is doe i Galbraith & Haies (000) where the authors propose a three-level taxoomy accordig to the icreasig mathematical demads. Our exploratory study belogs to the latest lie of research. However, we aalyze the problems i a differet way. First, we aim to establish a set of problem characteristics that would reflect several stages of the solvig process. Rather the workig with a priori defied problem categories (as i the above metioed studies) we search for combiatios of these characteristics related to ITR. O the other had, we look at subjective perceptios of difficulty o a threelevel scale (from easy to difficult). Istead of usig these levels of idividually perceived difficulty, we look for patters of perceived difficulty (PPD) i order to relate them to sets of problem characteristics. There are some beefits i doig so. First, relatig problem characteristics to ITR, we obtai a more geeral way to assess difficulty. Secod, by usig PPD, we allow to iclude variatios of perceived difficulty (it is importat sice the appreciatio of difficulty will vary from studet to studet) ad redefie difficulty categories based o these patters (istead of a priori defiig them). Third, relatig PPD with a set of problem characteristics, we have, agai, the advatage to idetify aspects that make the problem more difficult. Last, studyig the relatio betwee ITR ad PPD we ca have a isight as from where differeces arise ad which are the probable error sources (sice, ITR is related to correct solutio). To kow these characteristics may be useful for classroom teachig sice it helps to do problem selectio by teacher by balacig betwee difficulty perceptios ad maitaiig studets iterest i solvig problems. Therefore, we are tryig to fid a aswer to the followig questios: - how ca we properly characterize a problem; - how PPD ca be idetified ad what they mea; - what is the relatio betwee PPD ad characteristics; ITR ad characteristics; ITR ad PPD.. METHODOLOGY The problems were selected from the CASMI website ( a iteret commuity providig studets with challegig mathematical ad sciece problems at various levels of difficulty (Freima & Lirett-Pitre, 009). New problems are posted every two weeks ad participatio is ope to everybody. After submittig a solutio electroically, participats, who are schoolchildre ad pre-service teachers, have bee asked to assess the problem s difficulty o a three level scale: ot difficult at all, a bit difficult ad very difficult. For each problem, the total umber of submitted solutios ad the umber of correct solutios are also available. I order to characterize these problems, we focused o the cotext of the problem (familiarity) ad o the way to solve it. The familiarity meas that the situatio described i the problem ca be associated to some kid of everyday life experiece. The secod aspect refers to the solutio process, amely to the possibility to fid the solutio by a step-by-step approach. This meas that the solutio ca be reached by tryig out a limited umber of cofiguratios, combiatios ad the elemets ivolved i these variats are all give i the problem. The cotrary way is to idetify a geeral procedure to treat the problem ad the apply this to the particular case of the problem. Examples of such type of problems are combiatorial problems. Each problem - for the purpose of our aalysis - is described as a vector of 6 compoets: familiarity (- preset, 0 - ot), solutio process, ITR, percets of ot at all, a bit ad very diffi- 94

45 cult. For the exploratory study, we selected 5 problems i the way to have differet combiatios of characteristics (table ad appedix). Maximum percet is i bold. Pb. Familiarity Solutio process ITR (%) Not at all difficult (%) Somehow difficult Very difficult PPD Iterpretatio of PPD Very easy Itermediate Table. Characteristics ad differet idicators for the aalyzed problems 3. PRELIMINARY CONCLUSIONS Patters of perceived difficulty: A patter is defied by the distributio of percetages o the three difficulty levels; i such way we ca idetify three patters (PPD colum). The iterpretatio of a patter is based o the percet of the ot at all difficult level (last colum). For example, for values higher tha 75 we have very easy problems, for values betwee 50 ad 75 we have easy oes. Relatio betwee ITR ad PPD: For the patters ad 3 (PPD colum) we foud a correlatio betwee ITR ad maximum values of difficulty levels (relatio highlighted i italic i table ). However, for the patter, ITR values are split. We explai this case thru problem 4 (lie 4, i table). Meawhile ITR is low, the percet o ot difficult at all is i average rage (50-75) which suggests that studets kow how to solve the problem (thus, they cosider it easy), but make mistakes i the solvig it (so, ITR is low). Further research is eeded to cofirm this fidig. Relatio betwee problem characteristics ad patters of difficulty: As it ca be observed from the table (values i italic ad uderlied), very easy problems are familiar ad allow a step by step approach. As metioed, familiarity makes easier the uderstadig of the problem sice there is o eed for iterpretatio of mathematical terms. The step by step approach, allows easy solutio: oe tries out combiatios give i the problem s text or iterprets data. O the other had, easy problems have either familiarity either solvig process, but ot both. It might be that participats perceive problem as difficult due to the lack of familiar cotext (ad that lack maifests itself i difficulties to uderstad / iterpret the problem) or because the problem eeds a geeral procedure i order to be solved. Itermediate problems are either familiar or step by step. I this case, difficulties ca arise from iterpretatio /uderstadig of the problem ad solvig it. This seems to be the reaso why the value o the very difficult level is above 0%. More geerally, it seems that the lack of familiar cotext characterize easy ad itermediate problems (lies 3, 5 i table ). Still, this does t mea that problems i familiar settig will be ecessarily easy oes (see the example of problem 4, table). Further aalysis is required to properly asses the role of each characteristic over PPD. Easy 95

46 Relatio betwee problem characteristics ad ITR: We foud that ITR was higher for stepby-step problems (grey cells i table ). Agai, these problems ca be solved by a combiatio of give data ad do ot (ecessarily) require the use of formula or complex operatios that could lead to mistakes ad perhaps, to icorrect solutios (so, lower item difficulty ratio). However, the aalyzed data are ot eough to draw coclusios o the role of the familiar characteristic over ITR. 4. SUMMARY I the paper, we treated the problem of ITR, PPD ad their relatio to problem characteristics. Five problems, from the CASMI site, were aalyzed, each with a high umber of solvers. We report three prelimiary coclusios. First, three patters of subjective difficulty have bee idetified. These patters show differet percet distributios o the difficulty evaluatio levels. Secod, there is a relatioship betwee the two problem characteristics, familiarity ad step by step approach, ad the idetified patters. Familiar ad step by step problems are i the very easy category; meawhile, whe oly oe of the characteristics is preset, problems are easy. Problems with oe of the characteristics preset are itermediate, with more tha 0% of the participats judgig them as very difficult. Third, ITR is directly related to the step by step characteristic: its presece defies high ITR. O other had, the role of familiarity i ITR is still ot clear. The experimet suggested several lies for further research. First, these prelimiary coclusios should be cofirmed by a larger scale evaluatio. Secod, the lik betwee PPD ad problem characteristics should be further aalyzed by usig sequeces of problems i which the problem characteristics are modified ad each problem is assessed separately. Third, it is ecessary to thik of special settigs that would allow to properly assessig the role of the familiarity characteristic i gettig childre ito mathematics: icitig to solve problems, but also to pass betwee problems with ad without this characteristic. 96

47 REFERENCES Caldwell, J.H. & Goldi, G.A. Variables affectig word problem difficulty i secodary school mathematics, Joural for research i Mathematics educatio, 8(3), pp , 987. Craig, T. Factors affectig studets' perceptios of difficulty i calculus word problems, Proc. of the Secod Iteratioal Coferece o the Teachig of Mathematics, Crete, Greece, 00. Freima, V.& Lirette-Pitre, N.. Buildig a virtual learig commuity of problem solvers: example of CASMI commuity, ZDM- The Iteratioal Joural i Mathematics Educatio, 4(-), pp , 009. Galbraith, C. & Haies, C. Coceptual mis(uderstadigs) of begiig udergraduates, It. Joural of Mathematical Educatio i Sciece ad Techology, 3(5), pp , 000. Grolud, N. E. Measuremet ad evaluatio i teachig. Macmilla, New York, 98. Horke, L. & Habo, M. Rule-based bak costructio ad evaluatio withi the liear logistic framework. Applied Psychological Measuremet, 0(4), pp , 986. Jerma, M.E. Problem legth as structural variable i verbal arithmetic problems. Educatioal Studies i Mathematics, 5, pp. 09-3, 983. Lae, S. Use of restricted item respose models for examiig item difficulty orderig ad slope uiformity. Joural of Educatioal Measuremet, 8(4), pp , 99. Maso, E. et al. Respose time ad item difficulty i a computer-based high school mathematics course. Focus o Learig Problems i Mathematics, 4(3), pp. 4-5, 99. Newma, D. et al. Effect of varyig item order o multiple-choice test scores: Importace of statistical ad cogitive difficulty. Applied Measuremet i educatio, (), pp , 998. Appedix: problems used i the aalysis. Paul goes to the shop, where they sell videogames ad sweets. Paul goes to the sales sectio i order to check for ew discouts. After a momet, his fried, Marc arrives. Sice he has o moey, Paul decides to share his moey with him. Paul has 60$ ad gives the half to Marc. Marc decides to buy a videogame Guerre civile ($), a copy of Le chevalier (7$) ad chocolate bars (50c per bar). Paul, who ca t eat sweet because of his diabetes, oly buys videogames: a copy of Supercourse (3$) ad oe of Le chevalier (7$). At that momet, Paul s father arrives ad offers to pay the rest of moey if eeded. How much Paul s father should pay?. Marc puts five apples o the table. He cuts three ito halves. How may apples are still o the table? 3. Here is a umber game: a. thik of a umber; b. multiply by ; c. add 8 to the result; d. divide the ew result by ; e. subtract the umber you iitially thought of; f. your aswer is 4. Try out several umbers. What do you get as fial aswer? Ca you explai i mathematical terms how you got the aswer? 4. The Doucette family ivites their frieds to celebrate Thaksgivig i a restaurat. All tables there are square ad oly oe perso ca be seated o oe side. We ca arrage tables as we wish, but tables must touch at east o oe side each other. Beside the 4 family members, there are guests. What is the miimum umber of tables eeded? 5. The admiistratio of the park Hiberville decided to trasform their 8 km side triagular track ito a rectagular with keepig the same legth. Which are the differet possibilities for the legths of the sides whe all of them are iteger umbers? Could you fid all solutios? 97

48 DIFFERENT WAYS TO SOLVING OF ONE PROBLEM AS A MEANS FOR STIMULATING THE INTEREST AND CREATIVE THINKING OF COLLEGE STUDENTS IN MATHEMATICS Miriam Daga, Pavel Satiaov Sami Shamoo College of Egieerig, Beer Sheva, Israel pavel@sce.ac.il; daga@sce.ac.il Résumé L'objet de cet article réside das de ouvelles approches das l'eseigemet des preuves mathématiques, e ce qui cocere les séries. Ces méthodes sot basées sur deux vues alteratives. Soit, e cosidérat ue formule doée comme le cas particulier d'u résultat plus global, soit e la cosidérat elle même comme la règle propre à ue variété de cas particuliers, à priori sas lies commus. As it was writte i the Secod Aoucemet of CIEAEM 6 the oe of the mai teacher's tasks is i motivatig studets by puttig mathematical cocepts ito cotext ad likig them to other academic subjects ad everyday life. Mathematics teachers look for attractive topics for their lessos, which truly draw attetio to studies of mathematics ad provoke studet's creative thikig, rather tha merely provide them techical skills. Such topics may be directed to differet mathematical problems, but they may also itegrate differet mathematical themes by applyig differet methods to solvig of a sigle problem. The latter is i the spirit of the Poicare s defiitio of mathematics as "the art of givig the same ame to differet thigs" (i Rose, N., 988). By solvig a sigle problem i various ways studets are likely to gai a deeper uderstadig of the problem. Moreover, by beig exposed to differet kids of mathematical ideas, studets are provided with a rage of effective patters for their ow mathematical thikig. Thus, our presetatio takes as it motto ad guidig priciple Polya s advice that It is better to ivestigate oe problem from may poits of view, tha to solve may problems from oe poit of view (Polya, G., 96). We will preset a exteded example, based o this priciple, related to the formula for computatio of the sum of powers of two: I our presetatio, we shall give seve differet approaches to discovery ad proof of the formula for this sum based, as a rule, o fidig coectios of it with some practical processes. We are sure that diversity of approaches to solutio of a sigle problem stimulate creative thikig ad arose iterest of the studets i mathematics. No-formal Iterest-Evokig Itroductio for studets: Accordig to oe well-kow aciet leged the emperor of Chia so loved the game of Chess that he offered the ivetor of the game aythig he wished as a gift. The ivetor asked that a grai of rice be placed o the first square of a chess board, two grais o the secod square, four o the ext, the eight, 6, 3, 64, 8, 56 ad so o util the etire chess board was covered. After the ivetor had left, members of the court were calculatig a log time the umber of grais of rice to pay the ivetor:

49 They were astoished to fid that this was more quatity of rice the existed i the etire world. This much rice would cover the etire earth several iches deep. (Pedoe, D., 973). Here we preset some differet approaches to the formula for this sum.. Multiplicatio by S = S =... S = S S =. (This summatio is demostrated ad used as far back as Euclid's Elemets (300 BCE). It is the most widespread school approach to this sum because i this way the commo formula for 3 computatio of the sum S ( q) = q q d... q may be easily proved. But from the affective poit of view it is formal ad ot very impressive method. We check may times the first year studets (of egieerig college) competece about this sum ad it was foud that oly quarter of them were able to give the precise formula for its computatio ad o more tha 5% of the asked studets were able to prove it. This state says about the little attetio of the secodary school teachers to this sum ad givig it as some formal mathematical equatio without uderstadig its itrigue philosophical essece which my be doe as you ca see i some of the ext approaches.. Decay Iteger Model [Expasio approach] = = - - = = = This solutio may be described as mathematical model of uclear decay of some material cosists from atoms (with use of half-decay period). Here frame used for umber of atoms left at the proper stage of decay. 3. Fractio decay model 99

50 ;... 3 = = = = = = = = = = = = = = F S F S F F S 4. Touramet Model K players play had-to-had ad the loser leaves the field. How may games will be played till the fial game? Logical Solutio: After oe game, oe player leaves. I the ed K - of K players leave the field ad so K - games were played ad it is the aswer. Special case = K direct solutio: If there were = K players i the begiig as i the first roud games will be played; i the secod roud games will be played,, i the last roud game will be played. So the total umber of the games is:.... So, combiig two metio above solutios, we have:... = = K. 5. Probability Model What is the probability that i a family with childre at least oe is a daughter [assumed that p male = p female =0.5] Divide all elemetary evets ito subsets accordig to the umber of births before the first girl was bor: 3. the solutio stage of the fial see ad... 4 ) ( ) ( ; ) (... 4 ) ( A p A p A P A p = = = = = 6. Distace computatio Model

51 7. Biary Arithmetic Method I biary otatio: = ; = 0; = 00; 3 = 000, ad so o S = = ( times ) S =... = = times times 0 All or part of the approaches give i this paper was preseted, at various times, to lecturers, teachers ad college studets. Cosistetly, we were told that the lesso was a very iterestig oe, that it revealed ew ways of thikig, ad that it icluded momets of surprise. It was evidet that the approach demostrated to the studets the stregth ad the beauty of mathematical thikig. For some of the studets it iitiated or icreased their iterest i mathematics. Coclusios The approach preseted i this paper promotes a positive attitude towards mathematics i which mathematics is regarded as a powerful ad a beautiful istrumet for problem solvig. It combies otios ad methods from differet fields of sciece ad real life ad highlights the uiversality of mathematics - how oe simple formula may describe differet problems from differet fields of kowledge. Cosidered i light of research ito the value of multiple represetatios, for example, it provides a opportuity for two-way commuicatio betwee theory ad practice i mathematics teachig. Refereces Rose, N. (988). Mathematical Maxims ad Miims. Rome Press Ic. Polya,G. (96). Mathematical Discovery. O uderstadig, learig ad teachig problem solvig. Joh Wiley & Sos, Ic., New York-Lodo, Vol.. Pedoe, D. (973). The Getle Art of Mathematics. Dover Publicatios, Ic. N.Y. 30

52 Patters i the city: a mathematics project Isabel Vale, Aa Barbosa, Elisabete Cuha, Isabel Cabrita, Lia Foseca, Teresa Pimetel School of Educatio of the Polytechic Istitute of Viaa do Castelo, Portugal Oe aspect of mathematics educatio, i this cetury, is the lack of scietific culture ad curiosity of youg people as well as the low mathematics kowledge of studets. These studets low mathematics skills traslate ito lack of motivatio for learig this subject, traditioally idetified as a hard task. It is our belief that it is importat to show the ivisible face of mathematics aroud us to get studets ad teachers more motivated. I this presetatio we itroduce a overview of this project whose first goal is to promote the mathematics culture of elemetary pre-service teachers ad studets through the observatio ad exploratio of the urba eviromet while desigig mathematics materials for elemetary educatio. We will preset the descriptio of some of the tasks, i particular those coected with patters. U aspect de l'eseigemet des mathématiques, das ce siècle, c est le maque de culture scietifique et de curiosité des jeues et leur faible coaissace des mathématiques. Les faibles compéteces des élèves e mathématiques se traduiset par l'absece de motivatio pour l'appretissage de cette disciplie, traditioellemet recoue comme difficile. C'est otre forte covictio qu'il est importat de motrer la face ivisible des mathématiques autour de ous pour que plus d'étudiats et d eseigats e soiet motivés. Das cette commuicatio, ous préseteros brièvemet ce projet dot le premier objectif c est de devélopper la culture des mathématiques élémetaires chez les eseigats et les étudiats par le biais de l'observatio et l'exploratio de l'eviroemet urbai tat bie que la coceptio de matériaux pour l'eseigemet des mathématiques. Nous préseteros aussi la descriptio de certaies tâches, e particulier celles liées aux padros. Betwee we developed a project that was orieted by the followig reasos: As teachers educators we have bee oticig the lack of scietific culture ad curiosity of youg people that is oe of the most importat features i mathematics educatio i this cetury. This happes i short aywhere aroud the world, but i particular i Portugal, sice the recet iteratioal studies (e.g. TIMSS, PISA) have show that our studets have weak performace i Mathematics tests. These studets low mathematics skills traslate ito a lack of motivatio for learig this subject, traditioally idetified as a hard task. Whe studets eroll i school they carry, with them, a egative attitude towards mathematics, subject about which they have already heard, most of the times i a iexact way, but about which they have a limited kowledge yet. This situatio provokes i the teachers despodecy, lack of motivatio ad of alterative strategies to break this vicious cycle i which mathematics teachig have bee trasformed. This happes ot oly at schools but it is a socially istalled attitude towards mathematics. It is our strog belief that mathematics is accessible to everybody ad is preset everywhere aroud us. Oly actios that show these features of mathematics ad coduct to a great aware- This project, MatCid, refª CV/33 006, was supported by Ciêcia Viva Agecy 30

53 ess of people i geeral ad studets i particular, will allow us to recover the delay ad ivert geeral umotivatio. With this project, we iteded to promote the cotact with a cotextualized mathematics, startig from the daily life features, walkig through ad aalyzig the city where we live i, coectig some of its details with exploratio ad ivestigatio tasks i school mathematics. Its aim is to promote the mathematics culture of elemetary (pre-service ad i-service) teachers ad studets through the observatio ad exploratio of the urba eviromet, while desigig mathematics curriculum for elemetary educatio. At the same time they lear social ad historical evets related with what they saw, cotributig for a more civic, historical ad cultural kowledge of the place where they live. Backgroud Teachig a cotextualized, applied, experieced, visual, ituitive mathematics to all studets, promotig peer ad teacher/studets iteractio, usig didactical materials, ad commuicatig their ow ideas ca cotribute to a more sigificat learig of mathematics. I this perspective, learig requires a active ad reflexive studet egagemet i sigificat ad diversified tasks. I additio, most of mathematics studets failures derive from the affective classroom eviromet, oce it ca seriously compromise their iitial expectatios ad motivatios. After all we must develop ad stimulate creative thikig i the mathematics classroom. Teachers have a determiat role i the teachig process, so accordig to that perspective, teacher educatio should promote a ew visio about mathematics kowledge ad its teachig. Teacher educatio must create opportuities either for pre-service or for i-service teachers to explore their world ad discover that mathematics is everywhere, coectig mathematical ideas to real world iterests, experieces, ad empowermet. Teacher educatio must as well develop teachers competeces such us to be aware, critic ad more cofidet i their mathematical abilities, but most of all teachers have to be alert to kow ad to discover the ivisible mathematics that surrouds us. Amog the differet mathematical tasks that we use i mathematics classes, those ivolvig problem solvig play a importat role i the learers lifes. To look for a patter is a powerful problem solvig strategy ad mathematics may be defied as the sciece of patters (e.g. Devli, 999; Orto, 999). Patter study has a growig importace i particular o algebra, sice algebraic thikig has become a catch-all phrase for the mathematics teachig ad learig that will prepare studets with the critical thikig skills eeded to fully participate i society ad for successful experieces i algebra. Algebra as the study ad geeralizatio of patters meas that studets should be able to observe a patter, form a geeral algebraic rule ad the be able to justify that rule (e.g. Lai et al., 006; Maso, 996; Rivera & Becker, 005). Moreover patters are everywhere; we see them i ature, i architecture ad i art. I this presetatio we will give a special attetio to differet types of patters: those related to umbers ad algebra ad those related to tessellatios ad friezes, where studets ca geeralize i umerical or geometric cotexts. The project methodology ad procedures 303

54 The aim of this project is to awake elemetary pre-service teachers, studets ad populatio i geeral to the beauty ad utility of mathematics i daily life, discoverig it, uveilig it, ad explorig the multiples features of the city where we live. Our mai goals are: to promote mathematics culture of populatio; to cotribute to the architectural, atural ad historic kowledge of the city; to cotribute to a more positive view of mathematics; to promote the mathematics culture of elemetary teachers ad studets through the observatio ad exploratio of the urba eviromet while desigig mathematics curriculum for elemetary educatio; to promote mathematic ad trasversal competeces of studets; ad to cotribute to the professioal developmet of teachers. Accordig to the aims of this project we adopted a exploratory methodology where the participats were pre-service elemetary teachers of mathematics; i-service teachers; ad elemetary studets of grades -6. The project had differet phases, but it started i the classes of teacher educatio at a School of Educatio. Durig the classes of Didactics of Mathematics some themes were selected of differet features of the city, such as: documets; traffic ad sigs; gardes; moumets; buildigs; widows; forged iro; tiles; tessellatios; ad regioal embroideries, clothes, palmitos, pottery, gold. The those studets, future teachers, had to walk aroud the city of Viaa do Castelo lookig for differet elemets, accordig to a previous theme ad work it mathematically at a elemetary level proposig adequate tasks for childre (grades -6). These tasks were used, with some adaptatios, i the posters preseted i the exhibitio. The differet data was collected ad aalyzed by the project team accordig to the previous objectives ad the expected products, supported by literature o teachig ad learig mathematics educatio for elemetary levels ad teacher traiig. The products The mai products/tasks created withi this project were: () Oe booklet of tasks explorig mathematic details of particular itieraries of the city of Viaa do Castelo. This booklet is for usig by commo citizes that ejoy mathematics while visitig our city, i teacher traiig (pre-service ad i-service) ad i elemetary school mathematics; () Oe CD-ROM/DVD ad a Website with iformatio, materials ad selected resources for studets ad teachers of elemetary educatio; (3) A exhibitio ExpoMatCid. This exhibitio was ad ca be preseted i schools ad places of geeral public access. It icludes posters with mathematical tasks grouded o several aspects of the city of Viaa do Castelo. These tasks are to be solved by people with elemetary mathematics kowledge ad also ited to motivate youg studets to mathematics; 304

55 ad (4) A mathematical walk through the city, where studets of grades -6 solve mathematics tasks durig a arrative path through some streets of Viaa do Castelo ad was itegrated i the Summer Courses of our istitutio. This task was replicated with i-service teachers ad after with their ow studets. Some coclusios At the ed of this project we hope to have cotributed to: the promotio of sciece; a more positive attitude towards mathematics; ad to broade the visio of the possible coectios that ca be established betwee mathematics ad the world aroud us. To desig all the tasks i a adequate way to geeral populatio with basic mathematics kowledge was t always easy. A project that ivolved studets till the secodary level would more ad more provide to deepe mathematics subjects ad would allow the exploratio of differet tasks. Patters are ideed a powerful resource to develop mathematics cocepts ad to establish several coectios amog differet subjects ad school levels. I geeral the iterveiet i this project fid the more obvious coectios of patters those related to geometry (geometric motios - flips, slides ad turs friezes, tessellatios) because they are more visual ad familiar. The ew coectio for these iterveiet was that related with the developmet of the algebraic thikig through geeralizatio of patters. This approach it was less familiar because it was t i the Portuguese mathematics curriculum of school. To coclude we ca say that the implemetatio of a project such as MatCid-Mathematics i the City has edless chaces of exploratio. 305