PME Newsletter January 2015 by PME Administrative Manager

December 2014 / Jan nuary 2015

PM ME EN New wsllettterr Interrnationall Group for the P Psycholo ogy of Mathemat M atics Edu ucation

Messa M age frrom PME P E Presid dent Barba B ara Jaaworrski Deaar colleaguees in PME,, it is my plleasure to w wish you a very hap ppy new yeaar and all th he best of peace p and j oy in 2015. This issue of o our newsletter is broug ght to you ffrom a new w edittorial team.. Maike Vo ollstedt (Geermany) con ntinues as editor e and d she is join ned by Keith h Jones (UK), a curreent memberr of the IC. The retirin ng editor in Cynthia Nichol N (Can nada) to wh hom we express our wh holehearted d thanks for her wond derful work k in editting our new wsletter ovver the years – thank yyou Cynthiaa! I welcome the tw wo editors and thank them very much for this t issue of our o newslettter.

In nside th his issuee TPG G Report

Standding for thee IC

Younng Researchers’ Day at PME38

E38 reportss PME

PME E 39, 2015

27 7

Collloquium – a new w presentation form mat

0 30

Ourr PME con nference this year takes place in H Hobart at th he Uniiversity of T Tasmania, Australia. The themee is Mathem matics

Meessage from m the Editor E rs Welcome W to our Decem mber 2014 / January 22015 Newssletter. In th his issue wee bring repo orts from PM ME 38 and d the cordiaal invitation n to PME 339 in Hobarrt, Tasman nia. In addittion, there is in nformation about stan nding for thee IC and th he new PM ME handboo ok. Th here were cchanges in the PME Newsletter N tteam: We have h to say y farewell too Cynthia Nicol N and th hank her forr her great engagemen nt for the N Newsletter in i the last years y – and we welcom me Keith Jo ones of Enggland in thee Newsletteer team. En njoy readin ng the New wsletter! Tak ke care. Keith K Jones (dd.k.jones@sooton.ac.uk) and a Maike V Vollstedt (voollstedt@matth.uni-bremeen.de)

PME New wsletter Deecember 20 014 / Januaary 2015

Message M ffrom PM ME Preesident B Barbaraa Jaworrski (conntinuedd) Educcation: Clim mbing Mounttains, Buildiing Bridges aand refleects the parrticular land dscape at Hobart H as w well as ch hallenging us to see ou ur scientificc area throu ugh these metaphorrs. The Chaair of the Local L nd the Interrnational Orgaanising Committee an Proggramme Co ommittee iss Professor Kim Besw wick; we thank t Kim and her teaam for their ongoing work k on our beehalf. You can find th he First Ann nouncemen nt on the PM ME 39 web bsite at http://www.pm me39.com/ / and somee further information folllows below w.

th heir earlier experiencee, plus one further f edittor. Thus, T the ed ditors of thee new hand dbook are Angel A Gutieerrez (Spainn), Paolo Boero (Italy)) an nd Gilah Leder (Austrralia). A deecision has beeen taken to invite autthors who were not in nvited for th he 30 year hhandbook. It is expected th hat the hand dbook will again be published p by y SE ENSE.

PME E 40 will herald our 40th 4 year in n PME wheen our conferencee will take place p in Hu ungary, chaaired by Professor P Csaba CsĂko os of the Un niversity off Szegged. As parrt of celebraating 40 yea ars, the IC deciided to inviite a team of o editors to o prepare a new w PME hand dbook for 2016. 2 This should celeb brate anoth her decade of PME research from m the previous p haandbook which w was published p fo or 30 years y in 20006. We deciided to invite the edito ors of th he previouss handbook k, in order to t build on

Queries Q can go directlyy to our Ad dministrativve Manager, M Prrof. Dr. Betttina RĂśskeen-Winter via v th he email inffo@igpme. org. I look forward to o meeting m you u again in H Hobart.

You Y will find d more genneral matters about PM ME on n the IGPM ME web pag age at http:/ //igpme.orrg/.

Ba arbara Jawo orski January, 2015 2

PME New wsletter Deecember 20 014 / Januaary 2015

P PME Intern nation nal Co ommiittee R Reporrt Trreasurrer Po ortfoliio Gro oup (T TPG) Repo ort Subbmitted by O Olive Chapm man, PME Treasurer T Thee Treasurerr Portfolio Group G resp ponsibilitiess incllude: manaaging the fin nancial tran nsactions o of IGP PME (e.g., making paayments and d deposits, resp ponding to financial queries, q issu uing con nfirmations)), maintain ning recordss, advising on fiscal f questtions from present p and d future con nference orgganisers, an nd preparin ng annual fina ancial reporrts.

cu urrently working on moving m the acccounts to a ba ank in the UK. This wa as ap pproved by the IC C. This will also be neecessary if the t orrganisation co ontinues to pu ursue being g registered as a ch harity in thee UK.

Two o memberss of the TPG G ended th heir terms aat PM ME38 in Van ncouver. Beettina Dahll Soeendergaard (Denmark) completed d her term as trreasurer an nd member of the Inteernational Com mmittee (IC C), and Taii-Yih Tso (T Taiwan) ass mem mber of thee IC. We th hank them for f their valu uable work k for the TP PG over thee past year. A sp pecial than nks to Bettin na for her leadership of th he Group aand her com mmitted ma anagementt of th he financess of organizzation.

2. Financiall poolicies

We W are revieewing the hhistory of fin nancial po olicies of IG GPME to uupdate, reviise and co onsider new w ones as deeemed neceessary.

Durring PME 3 38, Olive Chapman C (C Canada) wass elected Trreasurer of IGPME by y the IC, with h Keith Jon nes (UK) co ontinuing as a member and d Stanislaw Schukajlow w-Wasjutin nski (Geermany) as a new mem mber to ma ake up the currrent TPG.

3.. Minimum emergeency fund d

IG GPME conttinues to bee financiallly strong. We W co ontinue to work w on a pproposal to o address th he minimum m am mount of m money the organization o n sh hould hold to t guarantyy covering its normal op perating exp penses in ccase of a situ uation thatt results in no income in a year and d covering co onference expenses in case of a â&#x20AC;&#x153;catastrophiic ev ventâ&#x20AC;? that leeads to an uuntimely cancellation n of a conference c .

Thee TPG has bbeen workiing on various projectts this year, whicch are of in nterest to the larger IGP PME comm munity. Som me items we w continuee to work w on:

1. New N bank

We currently h hold bank accounts a in n Finland but because off changes to o the Finnissh laws thaat requ uire us to conduct bussiness in Finland to con ntinue to ho old the acco ount there we w are

PME New wsletter Deecember 20 014 / Januaary 2015

Sttandin ng forr the PME P In nternaationaal Com mmitteee (IC C) Barb rbara Jaworsski, Presiden nt of PME 20013-2016 Eacch year, at tthe PME co onference, 4 memberss of th he PME IC C retire. This means in n general th hat they y have com mpleted a 4--year period d on the IC C. It means m also tthat we neeed to recruiit 4 new mem mbers. n the work o of It iss very rewarding to paarticipate in the IC and so ccontribute to PME an nd its role in n our Mathemattics Educattion Comm munity. In th his messsage, I sayy somethingg briefly abo out the worrk of th he IC and tthen tell yo ou how to stand to beco ome an IC member.

Th he IC has a special forrum as partt of the IG GPME web bsite where communiccations beetween IC members m taake place.

Beecoming a member of the IC

Yo ou need to contact thee Administtrative Manager M of PME, P who is currently Bettina Ro oesken-Win nter, via innfo@igpme.org. She will w seend you thee appropriatte applicatiion documeent to complete. You will bbe asked to write so omething ab bout your pparticipatio on in PME and wh hat you briing to PME E in terms of o your own n accademic and research experiencee. You will neeed two PM ME memberrs to nomin nate and seecond your applicationn. At any tiime there must m bee no more than three m members frrom any co ountry, including the P President, so s you need d to ch heck the currrent membbership of the t IC.

IC activity a

Eacch member of the IC contributes c to the overall work o of PME. Th his involvess: 1) Attendingg IC meetin ngs before (1.5 days) an nd after (1 daay) each con nference. 2) Becomingg a memberr of one of four f portfollio groups, wh hich are: a. The P PPG – Policcy Portfolio o Group – dealin ng with PM ME policy b. The SPG – Secreetary’s Porttfolio Grou up – dealling with co ommunicattion in PME E c. The T TPG – Treaasurer’s Porrtfolio Grou up – dealling with fin nance in PM ME d. The V VPPG – Vicce Presiden nt’s Portfoliio Group p – dealing with the accademic prograamme(s) in n PME 3) Playing a role in the work of thee chosen group, und dertaking taasks as agreed with other mem mbers.

al conferencce, all such h applicatio on Att the annua do ocuments will w be placeed on a cen ntral no oticeboard visible v to al all people atttending PM ME th hat year. Th hus PME m members can n see who is i staanding and d read the innformation n provided. Att the Annual General Meeting (A AGM) thosse staanding for election wiill be introd duced by th heir no ominator. There T will tthen be an election by y seecret ballot through whhich the req quired num mber off members will w be apppointed.

Thee IC has an Executive Committee consistingg of President, P V Vice Presideent, Secreta ary, Treasurer and Leader of the PPG. This T com mmittee meeets regularlly by Skypee during thee yearr between cconferencess.

Pllease consid der standinng for the PME P IC.

PME New wsletter Deecember 20 014 / Januaary 2015

PME E38 Reeportss PM ME Y Youngg Reseearcheersâ&#x20AC;&#x2122; Day D at PME E38 Subbmitted by N Nathalie Sincclair & Daviid Pimm (Sim imon Fraser University, Canada) Th here were two t plenaryy sessions and a two workshop w sessions (eacch participa ant chose tw wo fro om four, ass each workkshop was offered twiice). Th he first plen nary sessionn involved journal ed ditors from Educationaal Studies in Mathematiccs, Foor the learnin ng of mathem matics and Research R in Mathematics M Education, E eeach speak king for 10 minutes m abou ut the etho s of their jo ournal as well w ass one or two o issues conncerned wiith su ubmission, publishing and review wing of arrticles. Thiss was follow wed by a qu uestion and d an nswer sessio on with thee audience.. The secon nd pllenary sessiion, Life afte ter dissertatioon, consisteed off a panel off four speakkers, each of o whom haad fin nished their dissertatioon within the t past fivee yeears and weere pursuinng an acadeemic career, offfering cauttionary talees as well ass personal naarrative con ncerning thheir specificc early

Thee first PME E Young Reesearcher Day D (now Earrly Research her Day) to ook place in n Vancouvver on the t UBC caampus duriing the 24 hours h precceding the opening off PME 34. There T weree som me 80 particcipants from m twenty different d cou untries who o took part, each of wh hom was eith her in their final year of o their docctoral disssertation orr one year either e side of o this, whiich becamee a primary y selection criterion c from m the 120 iinitial appliicants. In addition, therre were 16 jjournal ediitors, worksshop leaderrs and d panellists who contriibuted to different parrts of th he program mme (Mich helle Cirillo, Jenny Craamer, Brentt Davis, Fraancesca Feerrara, Susaan Gerrofsky, And drew Hare,, Merrilyn Goos, G Harrpreet Kaurr, Ami Mam molo, Silkee Ladel, Dav vid Pimm, David Reid, Carlos Miguel M Rib beiro, Tim R Rowland, Nathalie N Sinclair, Keitth Weeber).

PME New wsletter Deecember 20 014 / Januaary 2015

PM ME You ung Ressearcherrs’ Day at PME ME38 (coontinuedd) On the first even ning of thee YRD, therre was a dinner at a pub on campus, wheree participan nts could meet in a much more inform mal setting and contin nue to conn nect other people at a simila ar stage in their t professsional educa ation. “Netw working” was w also a significantt desire expressed by the pa articipants prior to t arrival. The T YRD also enableed those particip pants who hhad not been to PME E beefore to atteend the firstt-timers meeeting, su urrounded by b people w with whom they had alrready spentt the past 224 hours. One O particip pant su ubsequently y wrote, “Iff I hadn't atttended YR RD, it would w havee been veryy easy for me m just to reely

trajeectory with hin academ mia. This wa as followed d by a very livelyy discussion n and interractive perio od whiich ran over the sched duled time. Thee four work kshops, who ose topics were w prim marily deterrmined by means of a questionnaire that was seent out to potentiaal partticipants, w were as follo ows: Workiing with langguage data aat the edges: Movving beyond word countss, Worrking with bbody data: from m theory to reesearch pracctice, Workin ng with teach cher as data aand Workin ng with h reasoning aand argu umentation ddata. Theree werre approxim mately twenty particip pants in eacch worrkshop, wh hich lasted for f two o hours.

PME New wsletter Deecember 20 014 / Januaary 2015

PM ME Young Reesearcheers’ Dayy at PM ME38 (coontinueed) on that set of connection ns througho out the con nference, an nd uninten ntionally av void meetin ng folk ks from outside that network. n Th hanks to YR RD, though h, I met maany folks in a similar acaademic possition to my yself that I'd d never meet beffore, and th hose relatio onships developed oveer thee conferencce.” We also o received the followiing com mment, “I met a ton of o folks from m outside m my norrmal netwo orks and made a numb ber of colllegiate friendships thaat I never otherwise o wo ould have. IIn my mind d, that was an incredibbly

po owerful parrt of YRD!”” With regard to workshop w pa articipant, ssomeone wrote, w “It was w on ne of the beest experiennces at PME that I haad. I haave always felt ‘on thee fringe’ wiith my in nterests. It was w exciting ng to think about a my own o work w differen ntly and thr hrilling to seee the otherr paarticipants interested iin things I like l too.” Th he evaluatiion of YRD D was carrieed out by th he members m of the IPC.

Discus D ssion Group G p 2: E Explorring Horizo H ons off Knowl K ledge for Teeachin ng Subbmitted by N Nicholas H. Wasserman, Ami Mam molo, C. Migu uel Ribeiro, and Arne Ja Jakobsen 

Du uring the firrst session of o this discu ussion grou up dev voted to Ho orizon Kno owledge forr Teaching (HCK) at PM ME, the prim mary activitties of the gro oup revolveed around three t presen ntations (giv ven by the four DG co oordinatorss) and sub bsequent diiscussion off the differeent perrspectives o on the math hematical horizon h rep presented. P Participantss were calleed to comppare and d contrast ssome of thee differencees and sim milarities off the discussed concep ptualization ns, and d ways thatt they may support orr enhance o one ano other in ressearch and practice. Su uch discussions an nd reflectio ons were theen com mplementeed with som me question ns to be con nsidered byy the particiipants to bee further deeepened in th he next disscussion gro oup session n:

 

Whatt primary im mpacts mig ght HCK have on the work of tteaching? What W are so ome exam mples / episoodes from your own classrroom experriences? methodolo Whatt are some m ogical appro oaches (pottentialities and a constraints) to sttudy HCK?? Whatt are some w ways to access and develop HCK inn and for teeaching?

Th he second session s hadd as a startiing point th he prreviously mentioned m qquestions and was deevoted, mainly, to disccussion and reflection ns am mongst parrticipants. T The beginniing and end d were w reserved for wholee group com mments. In n beetween, thee DG particcipants split into threee

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussioon Grou up 2: Exxploringg Horizoons of Knowled K dge for Teeachingg (contin nued) th he first issuee, some of tthe discussions were ceentered on task t concepptualization n, and in nterviews an nd observat ations that would w allow w fo ocusing on such probleems. How to understaand HCK’s H impa act, as well as ways lea ading to itss deevelopmentt, on and foor teaching g was noted d as a challenging g methodollogical queestion, with h co oncerns aro ound visibillity and selff-reporting. Questions Q were w raised aalso around d how to “m measure” th he manifesttation of HCK, H wheth her HCK H is som mething knoown or enaccted, and how we w might fra ame what cconstitutes HCK. H

groups, based on primary y interest in n one of thee queestions abovve, and discussed and d reflected upo on their (po ossible) answers for an nd reseervations abbout those questions. These werre shaared and reccorded with h the wholee group. Wh hat follows below triess to capturee some of th he group’s though hts concern ning each one o of the pro oposed refleection questtions:

Wh hat primarry impacts might HC CK have onn thee work of teeaching? What W are soome exaamples / eppisodes from m your ow wn classroom m expperiences?

What W are soome ways oof developing HCK in n an nd for teach hing?

Thiis question allowed diiscussion ab bout (possible) relationships between b currricular kno owledge an nd HCK, wh here particiipants noteed thatt HCK mayy provide a more proffound und derstandingg of the currriculum co ontent and relaationships aamongst th hose ideas. Some S of th he disccussed exam mples inclu uded studen nts’ confusiion oveer what is th he “biggestt number” and a what iss “infinity + 1”,, as well as unconventtional app proaches to o finding thee area of a rectangle usin ng geometrric means. The T examp ples were giveen as a meaans to explo ore the factt that HCK K allo owed teachers to draw w out imporrtant disttinctions (ee.g., betweeen arithmetic and geo ometric meaans) as welll as providee more than an a “n no, not corrrect” answ wer to studen nt confusio on.

Seeveral ideass emerged dduring the discussion fro om particip pants’ own experiencees and reeflections ass teacher edducators, mathematic m cs stu udents, and d mathemaatics teacheers. Attentio on was w paid to past-future p learning trrajectories, as well w as bread dth of connnections. On ne of the id deas th hat emerged d concernedd the fact th hat differen nt ho orizons exist for differrent subgro oups of mathematica m ally-engageed professio ons (e.g., en ngineers; teeachers; maathematicia ans; teacherr ed ducators). This T last quuestion led to several su uggestions on o possiblee ways on how h to ap pproach developing H HCK. Amon ngst those ways w we can n find: usingg exampless, ideas, or prroblems tha at can lead to the form mation of neew co oncepts, lin nked possibbly with a genetic ap pproach; acctivities thaat explicitly y link HCK K and cu urriculum related r know wledge (e.g g., divisibiliity in n the curricu ulum with numbers ex xtends to more m ch hallenging abstract a prooblems, wh hich could be b ex xplicitly linked to broaaden HCK)); concepts do no ot necessarily need to extend to other o moree ab bstract conccepts, but rrather HCK K can provide

Wh hat are som me methodo dological appproaches (pootentialitiess and consttraints) to study s HCK K? Meethodologiccal approacches were parsed out acccording to sstudies whicch might: access a and asseess HCK; d develop HC CK and exp plore its imp pact on teaccher effectiiveness; and d explore itts man nifestation in teachingg practice. Concerningg

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussioon Grou up 2: Exxploringg Horizoons of Knowled K dge for Teeachingg (contin nued) “bro oader view w” to see mo ore aspects of simple con ncepts, and broaden prrocesses to support morre in-depth mathemattical thinkin ng, also indiicating thatt the horizo on could bee more relaated to proccesses than content; teeacher prep paration at the second dary level could that inclludes practiicum at an elementary y school (th his is allready donee in some institutions i as a way to o accllimatize stu udents to th he professio onal settingg, as well w as give a sense of “past trajectories” so thatt methods ffor secondaary teaching g are info ormed by ex xperience and a knowleedge of whaat occu urs in previious schooll stages); an nd linking

s grraduate leveel mathemaatics with school mathematics m s, being onee of the posssible prresented ex xamples thee writing off a lesson pllan fo or elementa ary or seconndary schoo ol based on n th hese advancced conceptts (e.g., how w dynamiccal sy ystems can relate to thhe specific content c of a lesson in gra ade 7) – succh an appro oach was onveyed by y one of thee participan nts as an co im mportant op pportunity tto boil dow wn the conccept to o its roots, analyze a whhere it camee from, whaat it is connected to, and hoow it could connect to the co ontent those students w were suppo osed to deaal with. w

Book B A Announ ncemen nt Neetworkinng of Theeories as a Reseaarch Pracctice in Mathemat M tics Education An ngelika Bikkner-Ahsbah hs & Susan nne Predigeer (Eds.) 20 014 Th his book exxplains and d illustrates what it meeans to netw work th heories; it ppresents nettworking ass a challengging but fru uitful reesearch praactice and shows how the Netwoorking Theo ories Group p dealt d with th his challengge considerring five theeoretical ap pproaches, namely n the approach of o Action, Production P n, and Com mmunication n (APC), ( the Theory of Didactical Situations (TDS), thee Anthropolo A ogical Theo ory of the Didactic D (A ATD), the ap pproach off Abstraction A n in Contex xt (AiC), an nd the Theoory of Interrest-Dense Situations (IDS). ISBN: 9788-3-319-053 389-9 http://ww ww.springerr.com/education+%226+languag ge/ mathemattics+educattion/book/ /978-3-319--05388-2

PME New wsletter Deecember 20 014 / Januaary 2015

Discussio D on Grou up 2: Exploring Ex ng Horizzons of Knowle K edge forr Teaching T g (contiinued) ways w to furtther researcch about it challengin ng. As a communiity of matheematics edu ucators, thee DG has h served to t bring thiis matter to o the surface, with w the hope of furthe her propellin ng more wo ork to move m the field forwardd in terms of o the notio on of HCK. H Also as a resultt of the disccussions in and reflections r on o the DG,, the need for f next steeps, for f which we w intend too contribute, was appointed. a Amongst th those next steps s is the need for f bringing g together eexamples an nd vignettees of HCK H in pra actice to heelp the field d discern an nd ascertain a a more comm mon conception and understand u ing about H HCK as weell as its potentialitie p es and consstraints in and a for teaccher education e (and the rolle of teacheer educatorss’ knowledge k about suchh).

verall, from m the group’s discussio on emerged d Ov som me ideas th hat seem worth mentio oning. For exaample, in terms of som me of the ways w that HC CK might iimpact pracctice, particcipants seeemed to em mphasize tw wo aspects: 1) respond ding to unplanned d situations, including seeing the maathematicall significan nce of studeent ideeas/approaaches; 2) deesigning exa amples, pro oblems and d tasks – HC CK potentiially inffluences thee example choice c (perrhaps con nnecting to o future ideeas), the dessign approaach (em mphasizingg certain co oncepts), an nd the ease of exaample creaation (parsin ng out extraneous stufff). An nd while m much of the conversatio on about HC CK revolveed around more m advan nced maathematicall ideas, theere also seem med to be incclusion of a less advan nced horizo on as part o of thee notion, a sort of kno owing verticcal and ho orizontal co onnections across a the curriculum c m (bu ut outside o of what can n be considered currricular kno owledge); however, h itt was also no oted that inccluding too o much with hin the nottion of HCK migh ht cause it to t lose som me of its meeaning as a concept. mongst group members, there weere widely Am diffferent interrpretations of HCK, with w little mo ovement to owards agreeement as to what migght cou unt as the m mathematiccal landsca ape of the ho orizon, whaat might be evidence of o such kn nowledge, o or whether HCK H actua ally exists aas a useeful constru uct. Most people p in th he DG felt theere was som me value baased on theeir own exp perience reelated to kn nowing the mathematiical ho orizon, but w what precissely that wa as or how tto con nceptualizee the notion n more gen nerally felt elu usive. The llack of com mmon conception abou ut and understan nding of HCK H made discussing d

PME New wsletter Deecember 20 014 / Januaary 2015

Discuss D sion Group G p 3: M Mathem matical Disscoursse thaat Brreaks Barriiers an nd Creeates Spacees for Margin M nalized d Stud dents Subbmitted by R Roberta Hunter, Marta Civil, C Beth H Herbel-Eisenm mann, Daviid Wagner Thee aim of thee discussion n group wa as to consid der way ys in which h marginaliised studentts could bee provided with h space to eq quitably acccess matthematical discourse and a practicces. The goaal of the two sesssions was to create disscussion on n way ys to use, in nnovate on n, extend, and create a new w tool whicch could be used to bo oth scaffold d and d analyse m marginalised d studentsâ&#x20AC;&#x2122; access to disccourse and mathematical practicces.

inccrease studdent particip pation in th he ma athematicaal discoursee? In the first discussion grroup session the paarticipants were w providded with op pportunities to deescribe and make linkss with theirr own practtices rellated to how w barriers tto the disco ourse had been b ideentified and d removed for differen nt groups of o maarginalised students w within their own conteext. Th he rich disccussion provvided evideence of intternational interest annd activitiess the disscussion grroup particiipants are undertaking u g to ad ddress this topic. t

Thee discussion n group waas led by fou ur reseearchers an nd sixty-sevven particip pants atteended the fi first session. Many of the t same participants atttended thee second sesssion w lower) and there (although the numbers were on of eight new attend dees. wass an additio

Fo ollowing the initial disscussion a Co ommunicattion and Paarticipation n Framework (H Hunter & Anthony, 20011) designeed to suppo ort teaachers to sccaffold diveerse students to engagge in maathematica al practices was presen nted. This Frramework was w used ass a tool to critique c vid deo foo otage of thee mathemaatical intera actions of a gro oup of stud dents of Meexican origiin in the U.S. as they engag ged in probblem solving in their ho ome langua age (Spanishh) (Civil, 2012) . Ex xtensive participant diiscussion su upported intterrogation n of the pos sibilities th he Co ommunicattion and Paarticipation n Framework heeld to inform m actions teeachers miight take to provide spacee in reasonned mathem matical disscourse for marginalissed studentts.

or to the firrst discussio on group seession, Prio artiicles were p posted on th he PME co onference siite and d attendees were asked d to consider the following quesstions: 1.

Wh hat experien nces and reesearch can n you u share relaated to how w barriers to o the discourse have h been identified i and d removed for f differen nt groups off marrginalised students? s How w do these articles infform your worrk with increasing (or limiting) parrticipation of o marginallised stud dents? Wh hat ideas do o you have about inn novating on n and extend ding the fram meworks to o create a new n tool to

Th he focus of the secondd session off the discusssion gro oup was an nother Fram mework, which centreed on n authority structures w within the classroom (see

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussion n Groupp 3: Maathemaatical Diiscoursee that B Breaks Barriers B an nd Creattes Spacces for Margina M alized Students S s (contin inued) Herrbel-Eisenm mann & Waagner, 2010 0; Wagner & Herrbel-Eisenm mann, 2014 4). This Framework w was used d as an anaalytical tooll to revisit the t same videeo footage o of the first day. As thee participan nts view wed a sectio on of the viideo record d they explored the cllassroom discourse d fo or aspects o of ority, discou urse as auth hority, perssonal autho disccursive inevvitability, personal p latiitude and any y indicators of authoritty structurees or other auth hority strucctures. oncluded with w a rich discussion d Thee session co whiich examined the posssible purposes the fram meworks m might serve individuall i y and whatt they y might acccomplish to ogether thatt they do n ot acco omplish ind dividually to t support marginalise m ed stud dents to acccess space in i the math hematical disccourse.

Refferences

Civil, M. (20122). Opportu unities to Learn L in Mathem matics Edu ucation: Insights from

Researcch with “N Non-Domin nant” Commu unities. In T.Y. Tso (E Ed.), Proceedings of the 336th conferen nce of the Internattional Groupp for the Psyychology of Mathem matics Educaation (Vol.1, pp. 43-59 9). Taipei, Taiwan. Herrbel-Eisenm mann, B., & Wagner, D. D (2010). Appraising lexicall bundles in n matics classsroom disco ourse: mathem Obligattion and chhoice. Educaational Studies in Mathemaatics, 75(1), 43-63. Hun nter, R. K.,, & Anthonny, G. (2011). Forgingg mathem matical relaationships in i inquirybased classrooms c with Pasifiika students. Journall of Urban M Mathematics Education, 4(1), 98 8-119. Waagner, D., & Herbel-Eiisenmann, B. (2014). Identify ying authorrity structures in mathem matics classsrooms: a case c of a teacherr’s early expperience in a new contextt. ZDM.

Discus D ssion Group G p 4: N Negatiive Nu umbers: Brridgingg Contex C xts an nd Sym mbols Subbmitted by N Nicole Wessm man-Enzingger (Illinois S State Univerrsity) and La aura Bofferdi ding (Purduee University)) Thee discussion n group, Negative N Nu umbers: Briidging Conttexts and Symbols, S was coorgganized by L Laura Boffferding (Purdue Un niversity), N Nicole Wessman-Enziinger (Illino ois Staate Universiity), Aurorra Gallardo (Cinvestavv), Graaciela Salin nas (Cinvesstav), and Irit Peled

n group (Haaifa Univerrsity). This discussion em merged in response to a working group g on neggative numbers at PM ME-NA 35 (Lamb ( et all., 201 13) to exten nd the schoolarly discussion with an inteernational audience. A Across the two days of o sessions there were abouut 21 particiipants from ma

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussion n Group up 4: Neegative N Numbeers: Brid dging Co Contexts and Syymbols ((continu ued) variiety of coun ntries: Canada, Finlan nd, Gerrmany, Israael, Italy, Mexico, M Taiiwan, and the United Staates.

Eaach of the participants p s wrote a reesponse to wh hat they think it meanns to “undeerstand neegative num mbers.” Theese responsses were co ompiled and d placed onn an onlinee forum for paarticipants. Some of thhe participa ants highlighted the complexx use of thee minus siggn, wh hich is also o present inn the literatu ure. Drawin ng up pon the PM ME literaturre, the use of o the minu us siggn can be viewed v as: bbinary, una ary, and sy ymmetric.

Sesssion 1: Litterature Reeview of PM ME and PM ME-NA Prooceedings

To kick k off thee first meetiing, after grroup intrroductions, Laura Boffferding and d Nicole Wessman-Enzzinger jointtly prepared d and pressented a co omprehensiive literaturre review off neggative numbbers based on o past PM ME and PM ME-NA procceedings. Discussion D centered c on n the following tthemes present within n the PME and d PME-NA A proceedinggs: particip pant ages, theo oretical usee, integer co oncepts and d the meaaning of thee minus siggn, addition n and subttraction, m multiplicatio on and division, and histtorical high hlights. Onee of the main points off the literature rreview is th hat there is substantial s reseearch on stu udent think king about negative n num mbers in thee PME and d PME-NA A proceedin ngs over the past 338 years; ho owever, intteger researrch needs to makee better con nnections within w this dy of work. For examp ple, there iss a need forr bod inveestigations that conneect research h on additio on and d subtractio on to order or the meaning of thee min nus sign or multiplicattion and div vision. Keeeping with tthe title, “B Building bridges,” we disccussed the p possibilitiess of buildin ng con nnections beetween con ntent areas in future reseearch.

Taable 1: Mea aning of thee Minus Sig gn Biinary

The minuss sign is useed to indicate suubtraction between b a minuend aand subtrah hend. Un nary The minuss sign estab blishes a formal neggative num mber, relativve or directedd number, and a isolated d number orr result/solu ution. Sy ymmetric The minuss sign indiccates an opposite oor an action n to make opposite. Fu urther, the question, ““Does the developmen d nt of th he signed nu umber conccept depend d on the mo odel ussed? In wha at ways?” w was discussed and servved ass a transition for the prresentation ns in the second sessio on.

Seession 2: Modelling M P Perspectivess & the Meaning M off Negative N Numbers

Thee discussion n and lively y debates frrom the partticipants in n this sessio on were mo ostly focuse d arou und the folllowing two o questions:  

Iriit Peled started the seccond sessio on with a prresentation,, “Construccting Modeels for Teeaching Sig gned Numbbers.” In th his presentation sh he highlightted mathem matical mod dels, didacttical models, m and contextuall/real-life siituations fo or th he integers (Peled ( &C Carraher, 20 008). She illustrated a need n for futture researcch to addreess

does it meaan to undersstand What d negativve numbers? What iss a negativee number?

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussion n Groupp 4: Neg egative N Numberrs: Bridg dging Coontexts and Syymbols ((continu ued) the “grey area” between these modeels. She chaallenged parrticipants to o think abo out what it meaans to reaso on about neegative num mbers withiin thesse differentt models.

co ontexts, phy ysical conteexts, and ch hemical co ontexts. In this t presenttation, sign nificant discussion reevolved aroound Gallarrdo’s fraamework fo or understaanding the meaning m off neegative num mbers (Galllardo, 2002 2). We discussed thee frameworrk for negattive numbeer accceptance: subtractive s number, reelative nu umber, sign ned numberr, isolated number, n an nd formal negattive numberr. The pressentation co oncluded with w some di discussion of o their nextt directions off research, w which inclu udes in nvestigating g understannding of inteegers within n kin nemathematics (Mataais & Galla ardo, 2012) and ch hemical con ntexts.

Aurrora Gallarrdo and Graaciela Salin nas gave a pressentation, ““Negative Numbers N vs. v Sub btraction.” T This presen ntation serv ved as an inteeresting resp ponse to th he prior day y’s debates on the meaning o of the minuss sign and the t meanin ng of negative n nu umbers. Thee presentatiion took th he partticipants th hrough mucch of Gallarrdo’s con nsistent worrk within PME P and th he field, high hlighting diifferent con ntexts of thee negative num mbers: histo orical conteexts, geomeetrical

Figure 1. Participannts that disccussed the nnext steps fo or the discusssion group at the end of o the seconnd session..

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Discussio D on Grou up 4: Neegative Numbeers: Brid dging C Contextss and Syymbols (contin nued) (pp. 1358–13666). Chicago o, IL: Univeersity of Illlinois at Chhicago. Matias, M F., & Gallardoo, A. (2012 2). Distintoss nivelles de comppetencia formal en resollutores de pproblemas de d cinemáttica. En A. A Estepa, A Á. Contrera as, J. Deulo ofeu, M. C. C Penalva,, F. J. Garccía y L. Ordo óñez (Eds.)), Investigacción en Educcación Mateemática XVI VI (pp. 385–395). Jaén: SEIE EM Peled, P I., & Carraher, D. W. (200 08). Signed d num mbers and allgebraic thiinking. In J. J Kapu ut, D. Carrraher, & M. Blanton, Algebbra in the eaarly grades (p pp. 303–328). New w York, NY Y: Routledg ge.

Neext Directioons

Att the conclu usion of thee second sesssion, somee parrticipants sstayed to diiscuss the next n directio ons of the discusssion group (see Figuree 1). Two diffferent ideaas emerged from this conversatio c on. Th he first ideaa is to hold a monthly reading an nd disscussion gro oup. Differrent semina al pieces wiill be selected an nd shared with w participants thrroughout th he year. Discussion questions q fo or eacch of these pieces willl be posted in the onlin ne forrum, wheree participan nts can resp pond to the qu uestions. Th he second id dea is to sh hare com mmonly-ussed curricu ulum materiials and staandards for negative numbers n to initiate an intternational compariso on of these materials. m IIt waas decided tthat the con nference co o-chairs wo ould con ntact the paarticipants in the early y fall aboutt eacch of these areas of in nvestigation n. If anyonee waas not able to participaate in thesee sessions bu ut is interested i iin participaating in the reading gro oup or receiving eemail updattes please contact c Lau ura Bo offerding (lbbofferding@ @purdue.ed du) or Nico ole Wessman-En nzinger (nm menzinger@ @gmail.com m).

Ref eferences

Gaallardo, A. (2002). Th he extension n of the naturaal-number domain d to the t integerss in the traansition fro om arithmettic to algebbra. Educattional Studiies in Mathem matics, 49, 171–192. Laamb, L., Bisshop, J., Ph hilipp, R., Whitacre, W II., Stephaan, M., Boffferding, L, and otherrs. (2013)). Building on the emeerging knowlledge base for f teaching g and learn ning in relaation to inteegers. In M. M Martinezz & A. Castro Superffine (Eds.), Proceedingss of the 35tth annual meeting m of thee North Americcan Chapterr of the Intern national Grooup for the Psychology of Mathemaatics Educati tion

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Discuss D sion Group G p 5: N Numerracy across a the Curricu C ulum1 Subbmitted by M Merrilyn Gooos (The Univversity of Quueensland), Helen H Forgassz (Monash University), Vince Geigger (Au ustralian Cattholic Univerrsity) Thee aim of thiis Discussio on Group was w to exp plore international perrspectives on o embeddiing num meracy acro oss the scho ool curriculum, a new w field d of study iin which th here has beeen little reseearch to datte. Our purrpose was to t initiate disccussion on this theme with a view w to insttigating colllaborationss and, thereefore, provide the baasis for a prroposal for a Workingg Session at PM ME39.

urriculum” aapproachess is limited. “accross the cu Oth her concern ns includedd the impactt of inteernational testing t on cclassroom practices, p th he imp portance off pre-servicee education n to challen nge exissting notion ns of numeeracy and th he need to sup pport professsional deveelopment for f teacherss. Nin ne participa ants from siix countries attended the second 90 minute sesssion. We summarised s d the themes tha at emerged from the first fi session and d posed crittical questioons for furtther disccussion. Neew groups w were createed, each cen ntred on eith her a theore retical persp pective or app proach to reesearching w with teacheers, to form mulate questions to guuide future research. Wh hile the issu ue of numerracy is relev vant acrosss edu ucational co ontexts reprresented att the Disscussion Grroup, a majjor issue is the lack off a shaared language and a coommon deffinition of num meracy. Alsso of conceern was thee lack of imp pact that qu ualitative reesearch app pears to havve with h policyma akers; thereeby suggestiing a need to inveestigate alteernative forrms of asseessment forr num meracy.

Thee first 90 m minute sessio on was atteended by 133 partticipants fro om nine co ountries. We W introduc ed thiss session wiith a synop psis of our th heoretical model and currrent researrch. Small groups g werre form med to disccuss theorettical perspeectives thatt und derpin diffeerent concep ptualisation ns of num meracy and d how reseaarch on num meracy draaws on (or ( differs ffrom) reseaarch on worrkplace learrning, math hematical modelling m and a critical matthematics. Participantts were also o asked if “nu umeracy across the cu urriculum” was w an issu ue releevant to theeir own edu ucational co ontexts and d to consider c ho ow mathem matics educa ation reseearchers co ould work with w teachers to assist them m to embed d numeracy y across thee curriculum m. Thee small grou ups reporteed back to the t whole grou up the resp ponses to th he issues tha at they had d disccussed.

Mo ost participa ants indicatted interestt in con ntinuing disscussions byy providing g their con ntact detailss at the endd of the seco ond session n and d two particcipants indiicated an in nterest in con ntributing to o a forthcom ming issue of ZDM focu ussing on numeracy n thhat is being g edited by the Discussion n Group cooordinators. ___ __________ ____________________ _____________

Num meracy app pears to be a concern in all of thee edu ucational co ontexts reprresented at the Disscussion Grroup; howeever, there were w diversse inteerpretationss of the meaning of nu umeracy. Wh hile there seeems to be recognition r n that num meracy is crross-curricu ular in mosst of these edu ucational co ontexts, thee implemen ntation of

1 We W gratefully y acknowleddge the contrribution of Ann ne Bennison n to facilitatiing the Discu ussion Grou up and d preparing this report.

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Discuss D sion Group G p 6: O Observ ving Teache T ers Observ O ving Mathem M maticcs Teaaching g: Res earch hing th he Unobse U ervablle Submitted by D David A. Reidd (Universitäät Bremen, G Germany), Richard R Barrwell (Univerrsity of Ottawa, Canadaa), D Ma anuel (McGi Gill University ty, Canada) Lisaa Lunney Boorden (St. Frrancis Xavieer Universityy, Canada), Dominic Elaiine Simmt ((University of Alberta, Canada), Ca Chrristine Suurttamm (Univversity of Otta tawa, Canadda) Thee goal of thee discussion group wa as to addresss two o interrelateed methodo ological queestions:  

How caan we reseaarch the un nobservablee (e.g., reeasoning, beliefs, b etc.))? How caan our research ackno owledge an d make u use of the faact that “ev verything saaid is said bby an obserrver” (Matu urana, 1987)?

Partticipants in n the discusssion group p offered examples from m their own n experiencee of specificc reseearch studiees where th hese questio ons mattereed and d how they are/were addressed. a We then turneed to examp ples from th he Observin ng Teaachers studyy to provok ke deeper discussion d o of the key questio ons. The Observing O Teachers T stud dy seeks to explore ho ow middle school s matthematics p pedagogy differs d acrosss regions o of Can nada. “Pedagogy” is used u to refeer to the imp plicit culturral practicess of teacherrs, includin ng culttural beliefss about how w children learn l and how w teachers sshould teacch, that guide teachingg pracctice. Becau use pedago ogy is impliicit, it is nott direectly observvable, and so s the meth hodology o f the research prrogramme had to add dress the firsst key question: H How can we w research the uno observable? The researrch team ch hose to con nfront this cchallenge by y making use u of the faact thatt “everythin ng said is saaid by an observer”. o Ratther than atttempting to o observe and a analysee

teacchers’ practtice in the hhopes of rev vealing theeir ped dagogy, whiich would ssay as mucch about thee reseearchers as observers aas anything g, the reseearchers insstead obserrve and ana alyse teacchers’ obserrvations off their own teaching an nd otheers’ teachin ng. They reecognise tha at their observations of o the teachhers’ observ vations are also o observatio ons, and soo they comp pare analysses with hin the reseearch team in order to o observe them mselves as researcherss, as they observe o the teacchers’ obserrvations. Exaamples of tw wo analysees from the Observing Teaachers study y were sharred (see Ma anuel, Sav vard, & Reid, 2014, annd Reid, Sa avard, Manuel & Lin n, 2015). Thhe participa ants then enggaged in a comparative c e analysis of o two shorrt piecces of data from the sttudy. The second s day began with rep ports of thee observatio ons of the partticipants, which w serveed to illustra ate well thee observer depen ndence of ddata analyssis. Thee second session closeed with an extensive e an nd vigo orous discu ussion of thhe guiding questions. q Thee two sessio ons were atttended by over twenty y partticipants with consideerable overllap in the two o sessions. Maturana, H.R R. (1987). E Everything g said is said d by an observer. o Inn W. Thom mpson (Ed.), Gaia: A way of knoowing (pp. 65-82). 6 Hudson n, NY: Linndisfarne Prress.

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Diiscussioon Grou up 6: Obbservingg Teach hers Obsserving M Mathem matics Teeachingg: Reseaarching the Unoobservaable (con ntinued) d) Tob bin, J., Wu u, D., & Daavidson, D. (1989). Preschoool in three cultures: cu Japaan, China, aand the Uniited States. New N Haven n:Yale Univerrsity Press. Maanuel, D., S Savard, A. & Reid, D. (2014). Observving teacherrs: The ma athematics pedago ogy of Queb bec francop phone and anglop phone teach hers. Posterr. In Oesterrle, S., Niccol, C., Liljeedahl, P., & Allan, D . (Eds.) P Proceedings of the Joint Meeting of

PME 38 and PME ME-NA 36,Vo ol. 6, p. 360 0. Vancouver, Cannada: PME E. Reid, R D., Savard, A., M Manuel, D. & Lin, T. (2015 5). QueĚ bec anglophon ne teachersâ&#x20AC;&#x2122; pedag gogies: Obsservations from f an au utoethno ography. Suubmitted to o the Ninth annua al conferennce of the European E Socieety for Reseearch in Ma athematics Education. Praggue, Czech Republic. 4-8 Febru uary 2015.

Book A Annou uncemeent Algebra A Teachin ng aroun nd the W World Frederick F K Koon Shing g Leung (The Universsity of Hong Kong, Ho ong Kong), Kyyungmee Pa ark (Hongik k Universitty, Korea), Derek Hollton (Universityy of Melbourne, Australia) & Daavid Clarkee (Universitty of Melbou urne, Austraalia) (Eds.)) 2014 Utilizing tthe LPS daataset, this book b docum ments eighth grade algebra teeaching acrross a varietty of counttries that diiffer geograph hically and culturally. Different isssues in alg gebra teach hing are reporrted, and diifferent theories are ussed to charracterize algebra llessons or to t compare algebra teaaching in different d countriees. Many co ommonalities in algebbra teaching g around th he world arre identified, but theree are also st striking and d deep-rooteed differen nces. ISBN P aperback: 978946209 9 7056 ISBN H Hardcover: 978946209 9 7063 ISBN E E-Book: 978 8946209707 70 https:///www.senssepublisherss.com/cataalogs/book kseries/thelearneraas-perspectiive-study/a algebra-teacching-aroun nd-the-worrld/ 18

PME New wsletter Deecember 20 014 / Januaary 2015

Discus D sion Group G p 7: P Preparing an nd Su upportting Mathem M maticcs Teaacher E Educaators: Oppo ortuniities an nd Ch hallen nges Subbmitted by R Rachael M. Welder W Disscussion Grroup #7 waas organizeed by Rach ael M. Welder, H Hunter College of the City C Un niversity of New York (USA) and d Amanda Jan nsen, Univeersity of Deelaware (USA). Andreea MccCloskey, P Penn State University U (USA) also o con ntributed to o the planniing of the group, g but was unable to o attend thee conferencee.

enggaged in subgroup disccussions off the dileemmas they y perceive aas they work to becom me or prepare p MT TEs. Conceerns that were w raised ran nged from building b truust between n ePTs and MT TEs who la ack teachingg experiencce to con nsidering th he balance bbetween teaching and d sch holarship when w adjustiing to new roles as MT TEs.

Thee goals of o our discussiion group were w to: (a)) exp plore challeenges faced d by Mathem matics Teaacher Educcators (MTE Es) in theirr work with h pro ospective ellementary teachers t (eP PTs) and (bb) con nsider oppo ortunities to o address th hese chaallenges. Ou ur interest in this topic was driveen by the fact thaat most insttructors of mathemati m ics cou urses for eP PTs in the USA U do nott themselvees hav ve elementaary/primarry school teeaching exp perience (M Masingila, Olanoff, O & Kwaka, K 201 12). The atttendees of our o two gro oup meetin ngs included doctoral studen nts and facu ulty who arre MT TEs; many teach courrses for and d/or conducct reseearch surro ounding thee developm ment of ePT Ts and d MTEs. Th he followin ng countries were rep presented in n our group p: Australia a, Canada, Fraance, Germ many, Israell, Malawi, and the US SA.

o discussiions, the pa articipants Thrroughout our shaared inform mation regarrding the co ontexts surrrounding elementary e teacher preeparation in n theeir countriesâ&#x20AC;&#x201D;as this vvaried across nations, so did d the work of o MTEs. O Out of thesee discussion ns aro ose a valuab ble outcom me: an expan nded view of thee similarities and differrences occu urring in th he preeparation off elementarry teachers and MTEss spaanning five continentss. For instan nce, our parrticipant fro om Malawii shared an n example wh here new MTEs M withouut prior tea aching exp perience wiill teach in pprimary school settinggs beffore conduccting work as MTEs. Additional A ly, ourr participan nt from Fraance shared d a model wh here early MTEs M are chharged with h con nstructing their own pprofessionall dev velopment experiencees, some of which incclude observ ving or co-tteaching with w other MT TEs, workin ng in local primary scchools, or atteending conferences.

Thee first sessio on (7/16/1 14) opened with a sho ort intrroduction p providing th he rationale for the gro oup and thee goals of bo oth the gro oup organizzers and d the attend dees. Rachaael Welder presented find dings from her collabo orative reseearch with An ndrea McClloskey explloring the educational e l and d teaching eexperiencess of early-career MTE Es and d the challeenges they face f in prep paring ePT s. Thee participan nts reflected d on this work w and

The second seession (7/1 8/14) bega an with Rachael Weld der sharing additionall results from m herr work with h Andrea M McCloskey, highlightin ng pro ofessional leearning oppportunities reported to o con ntribute to the t work off early-careeer MTEs, in

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussioon Grou up 7: Prreparingg and Su upportiing Matthematiics Teeacher E Educatoors: Oppportuniities andd Challeenges (ccontinueed) lieu u of elemen ntary teachiing experien nce. This w was folllowing by a presentatiion by Ama anda Janseen, in which w she sshared her university’s u s model forr men ntoring docctoral studeents (futuree MTEs) ass they y begin teaaching courrses for ePT Ts. These ideaas served aas a springboard for th he participan nts to start s concep ptualizing strategies s fo or supportin ng MT TEs within the context of their ow wn countriies.

in nformation with the grroup organ nizers will be b co ontacted du uring the faall of 2014 to t discuss po otential research collaaborations and a the co ontinuation n of discusssions surrou unding the grroup’s goals at PME 339 and/or PME-NA P 3 36. References Masingila, M J., Olanoff, D., & Kwa aka, D. (20 012). Who teaches maathematics content coursses for prosppective elem mentary teach hers in the U United Stattes? Resultss of a natiional surveey. Journal of o Mathemattics Teach her Educatioon, 15(5), 34 47-358.

Thee second grroup meetin ng continueed to focus on identifyingg various diilemmas thee attendeess werre facing in n their own work and opportunitiies to address a them. Amand da Jansen in ntroduced a pro oposal to ad ddress the challenge c off balancingg teacching and sscholarship p: integratin ng teachingg resp ponsibilitiees with cond ducting research on teaccher educattion and teeachers’ learning. It is parrt of the cullture in Am manda’s workplace to inteegrate reseaarch and teaching. Ho owever, new w chaallenges were also exp plored as a consequenc c ce of considering c g this opporrtunity, succh as pottentially divverging from m one’s oriiginal reseearch area tto do so. Parrticipants reeflected on a collectivee challengee to identify i speecific learniing goals fo or the ePTs with whom th hey work. An A overarch hing concerrn wass how they, as currentt and futuree MTEs, deccide upon th he specific content an nd insttructional m methods to teach in th heir classes and d how to kn now if they y are making the best cho oices in term ms of what their ePTs need to learrn. This led d to an interesting exchange of ideaas about th he ways in which w goals for teacheer learrning are beeing and might m be deffined, withiin and d across nattions, and the t socio-political con ntext of estaablishing su uch goals fo or teachers’’ learrning. Thee participan nts who shaared contacct

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Discus D ssion Group G p 8: R Researrching g Thin nking Classro C ooms Subbmitted by G Gaye William ms (Deakin University, A Australia) and a Peter Lilj ljedahl (Sim mon Fraser University, U Can nada) stim mulation off discussionn through focus f on a Wo ordle <http p://www.w wordle.net> > that cap ptured frequ uently usedd words in the t parrticipant responses to the Buildin ng Thinking Claassrooms (DG5 PME377). Groups then forrmulated th heir own deefinitions off Thinking Claassrooms and d generatedd more reseearchable queestions abo out such claassrooms. A Gallery Waalk followed in whichh participan nts considerred researchable questions q ggenerated by y other gro oups and seelected the tthree questtions they wo ould most liike to exploore.

These organissers began their t conveersations abo out ‘Thinkinng Classroom ms’ in Moreelia, Mexicco in 2008 2 and o organised PME37 DG G5 Building Thinking Classsrooms as a result. Reseearching Thinking Classsrooms is a sequel s to th his previouss DG G. It was orrganised in response to participaant inteerest in exp ploring reseearch questions arisingg from the Buildding Thinkin ng Classroom ms DG. Research desiigns previou usly emplo oyed to stud dy Thinking Classsrooms fro om various theoreticaal perrspectives in nclude coggnitive, sociial, affectivve, em motional, pssychologicaal, and volittional perrspectives o on student learning, l an nd teacherss’ praactices and beliefs. Research desiigns em mployed havve included d: self-reporrts of affecttive exp periences (L Liljedahl, 2013), 2 video o analysis iin a ‘refform classrroom’ (Cob bb, Yackel, & Wood, 201 11), video o of own classsroom pracctice (Laampert, 20001), video-sstimulated student inteerviews (W Williams, 20 014), and sttudy of the “en ntanglemen nt of subjecct and objecct, mind an nd wo orld” (sensu uous cognittion, Radfo ord, 2014, pp. 352 2) which em mploys videeo as an an nalysis tool.. In add dition, netw working of theoreticall frameworkks hass been a pro oductive way w to find out o more abo out studentt and teacheer activity in i ‘thinkingg classrooms’ (H Hershkowittz, Tabach, Rasmusseen, & Dreyfus, D 20014). Teach her guidancce of studen nt thin nking has aalso been sttudied through video datta (Funahashi & Hino o, 2014).

m Daay 2 Includeed 35-45 paarticipants, many from Sesssion 1 and d some new w participan nts. Session n1 acttivity was reeviewed inn a Wordle representin ng thee frequency y of term ussage in definitions of Thinking Classsroom produuced in Sesssion 1 (seee nd Figgure 1). Diffferences beetween thiss Wordle an thee one displa ayed in Sesssion 1 (to captured c term ms used in discussingg Thinking Classrooms) C weere discussed. Caategories of research quuestions formulated in n Sesssion 1 werre then pressented. An example queestion from m each categgory is inclluded hereiin: “W What type of content (ee.g., tasks etc.) e promo ote a Thinking T Cla assroom?” “ How are mathematic m al stru uctures intrroduced intto discourse and does it maatter whether studentss or the teaccher bring theem in?” “W What are thee tools (inclluding com mpetencies) that enabble teachers to transitio on to a Thinking Classroom?”” “How do o teachers inittiate and su ustain Thinnking Classroooms?” “Ho ow doees thinking stop in a cclassroom and a why do oes it stop?” s “Wh hat techniquues give thee most

The two DG sessions weere focused d in the folllowing wayys: pants): Evid dence-based d Sesssion 1 (35--45 particip

PME New wsletter Deecember 20 014 / Januaary 2015

Diiscussioon Grou up 8: Reesearchiing Thin nking Classroo C oms (con ntinuedd) enggagement?”” “Given a Thinking Classroom, C wh hat are the o outcomes?”” Participan nts who waanted to foccus on the same s researrch questio on theen formed ggroups and d began to develop d a ressearch desiggn that wou uld help inv vestigate theeir question n. Various methodolog m gies were con nsidered an nd ideas weere shared. At least on ne gro oup decided d to continue to resea arch their question thro oughout thee subsequen nt year. Thee intterest stimu ulated throu ugh these questions q sugggests a Wo orking Gro oup on Desig igning Ressearch to Exxplore Thinkking Classroooms is waarranted at PME39. The T DG org ganisers inttend to subm mit this WG G.

math hematical iddeas towarrd meeting lesso on objectivees. ZDM, 466(3), 423-43 36. doi: 10.1007/s111858-014-0592-0 Lampert, L M. M (2001). T Teaching prooblems and thhe probllems of teachhing. New Haven, H CT:: Yalee Universityy Press. Liljedahl, L P. (2013). Illlumination n: an affectiive expeerience? ZD DM, 45(2), 253-265. 2 do oi: 10.10 007/s118588-012-0473 3-3. Hershkowit H z, R., Tabaach, M., Ra asmussen, C., C & Drreyfus, T. ((2014). Kno owledge sh hifts in a probability p y classroom m: a case stu udy coordinating tw wo methodo ologies. ZD DM, 46(3)), 363-387. doi: 10.100 07/s11858-014-0 0576-0. Radford, R L. (2014). Toowards an embodied, e cultu ural, and m material con nception of math hematics coognition. ZD DM, 46(3), 349-3 361. doi: 100.1007/s11 1858-014-05 5911. Williams, W G. G (2014). O Optimistic problemp , solviing activityy: enacting confidence c persiistence, andd persevera ance. ZDM,, 116. doi: d 10.10077/s11858-0 014-0586-y..

Co obb, P., Yacckel, E., & Wood, T. (2011). Youngg children’ss emotional acts whilee engageed in mathematical prroblem solvingg. In A. Sfaard, K. Gra avemeijer & E. Yacckel (Eds.),, A Journey in Mathem matics Educcation Reseaarch (Vol. 488 pp. 41-71 ). Neth herlands: Sp pringer. Fu unahashi, Y Y., & Hino, K. (2014). The teacheer’s role in guiding g chiildren’s

Figurre 1. Word dle of term ms used by pparticipantss in defining g Thinkingg Classroom ms 22

PME New wsletter Deecember 20 014 / Januaary 2015

Discus D sion Group G p 12: W Whatt is Qu uality Mathem M maticcs Teaachingg-Reseearch?? Subbmitted by B Bronislaw Czzarnocha andd Hannes Sttoppel Theere were tw wo meetingss of the DG G 12. Durin ng the first meetin ng there weere 8 particcipants, prim marily PhD D candidatees in Matheematics Edu ucation. Th he discussio on started by b reading exccerpts from reading maaterials to come c at thee adeequate ideaa about the quality of the t presenteed therre teachingg-research aims a and metthodologies. It centred d on the differences betw ween quan ntitative and d qualitativ ve approach hes to classroom c iinvestigatio ons. Raised questions werre:

ocused on th he detailedd discussion n of several fo teeaching exp periments, ppreferably one o designeed by y a teacher and one deesigned by a researcheer. Hannes H discu ussed the ddesign and conduct c off his multi-cyclica m al teaching experimen nt in Freshm man caalculus. Hiss descriptioon revealed large dose of th he intuitive teaching crraft knowleedge govern ning th he sequencee of refinem ments, whicch was appllied on n the “just-in-time” baasis. As com mpared witth teeaching exp periments ddesigned an nd conducteed by y researchers, his, teaccher’s desig gn was not fu ully conceiv ved apriori as the resea archer’s deesign. At th he same tim me, the desiign and co onduct wass successfull in terms of o finding th he efffective “rea al world” bbased pedag gogy in 3 cy ycles of the TE.

1. W When each of the meth hodologiess is aappropriate?? 2. C Can one creeate mixed approachess? 3. W What is the aim of teacching ex xperimentss? 4. H How should d the teachiing-research h reeports be written? w 5. W What exactlly is the diffference bbetween research and teachingt reesearch?

Th he discussions duringg both meettings of the DG 12 2 suggested d that the fuuture TR diiscussion grroups at PM ME’s shouldd be focuseed on the “n nuts an nd bolts” off designingg, conductin ng and asssessing cla assroom teaaching expeeriments.

Thee discussion n revealed that one off the most pertinent issuees was the design d and assessmen nt of a teaching eexperimentt. There weere significaant diffferences of opinion as to what is teaching exp periment; so ome of thesse differencces were motivated by tthe differen nces in expeeriences betw ween PhD candidatess and rank and a file teacchers (decissive minoriity amongsst parrticipants). T These diffeerences suggested thatt the second meeeting of th he DG 12 sh hould

PME New wsletter Deecember 20 014 / Januaary 2015

Workin W ng Sesssion 1: Teeacherr Notiicing: A Hidden Sk kill off Teacching Subbmitted by M Molly H. Fissher, Edna O. O Schack, Jeennifer Wilh helm, Jonath han Thomass, Cindy Jong ng, and Rebeecca McN Nall-Krall

This working group wass divided in nto three sub ubgro oups to con ntinue the work w of PM ME-NA 201 3 wo orking group p of the sam me name. The sub-grroups had d three maiin outcomees: A mono ograph on teaacher noticiing, a confeerence on teeacher notiicing, and d a strong iinternet preesence of th he topic of teaacher noticiing. The group forrmed to streengthen thee internet preesence discu ussed strateegies such as a a websitee on teaacher noticiing, social media m grou ups for discussion, blogs or wikiis to post in nformation , and mail listservss. At this point, p the em mail listserrv em hass been estabblished thatt combiness the 2013 wo orking group p attendeess with the 2014 2 workin ing gro oup attendeees and com mmunicatio on has begu un witth those gro oups. Additionally, a facebook ppage hass been creatted to bolstter teacher noticing discussions an nd this willl begin to bee used and adv vertised mo ore as the other o two su ub-groups h have mo ore informaation to shaare regardin ng the mo onograph an nd the confference. The sub-group p that discu ussed a teaccher noticin ng con nference deecided to ap pply for fun nding to sup pport a con nference cen ntered arou und teacherr notticing reseaarch. They plan to app ply for an AE ERA conferrence grantt to fund the conferencce and d the goal iis to hold th he conferen nce in the d days prior to the neext PME-N NA conferen nce, much like a pre-session p tto the main n conferencce. The ideea wo ould be thatt attendees could com mbine travell effo orts and atttend both conferences c s in one visiit.

on discussio ons to extennd mathem matical teach her noticing to science, s ultiimately lea ading to a monograph. m . The faciliitators of th he working seession had the good foortune to meet m with Jiinfa Cai, C co-edito or, with Jam mes Middleton, of thee Springer series, Researcch in Mathem matics Educcation. During D this meeting wee discussed d with Dr. Cai C th he purpose of the monnograph as defined by y the Working W Grroup particiipants. Fro om this we developed an outline too propose for f the monograph. m . The propposed mono ograph will build upon the wo ork of Sheriin, Jacobs, and Philip ppâ&#x20AC;&#x2122;s Mathematics M Teacher Nooticing: Seein ng Through Teachersâ&#x20AC;&#x2122; T Eyees (2011). At A the suggeestion of Drr. Cai, we will w strive to t in nclude both h seasoned and promising reesearchers/ /authors forr the chapters of the monograph. m . Additionaally, the ch hapter autho ors will w include internationnal mathem matics and sccience educcation reseaarchers. Th he sections of the book will include a com mmentary on o the chap pters within w that section. s W We have sen nt an initial in nvitation to o some of thhe top resea archers in this t fiield, receiviing a numbber of positiive responsses for ch hapter writting. Our O next steep will be too send a ca all for chaptter proposals fro om the part rticipants in n the workin ng seessions at both b the 20113 and 2014 conferen nces. Following F th his, a broadder call for chapters will w be reeleased. We W expect too submit th he monograaph proposal to the t Springeer series, Reesearch in Mathematics M Education bby early sum mmer 2015 5.

med is a group focusin ng The other subb-group form

PME New wsletter Deecember 20 014 / Januaary 2015

Working W Session n 1: Teaaching N Noticingg: A Hiidden S Skill of Teachin T ng Thee three sub--groups forrms will wo ork syn nergisticallyy as the auth hors in the monograpph willl be invited d to speak at a the confeerence and presenters from m the confeerence (tha at weren’t d to write for f the mon nograph) alreeady invited

could c be inv vited to subbmit for thee monograp ph. All A of these efforts willl be advertiised using social media a outlets, em mail listserrvs, and websites w tha at will be addministered d by the teechnology sub-group.

Workin W ng Sesssion 2: 2 Deevelop ping Preserv P vice Ellemen ntary Teach T hers’ M Matheematical Kn Knowleedge fo or Teaachingg Thiis working session waas organized d by Lynn C C. Harrt, Georgiaa State Univversity, USA, and Sussan Oesterlee, Douglas College, Canada. C Th he purrpose of thee working session s in Vancouver V wass to finish tthe work staarted in tw wo previous meeetings of th he working group in Chicago C and d Kallamazoo in n which meembers of th he group iden ntified and examined significantt cognitive and d non-cogniitive factorrs influencin ng the matthematics ccontent preeparation off preservicee teacchers from multiple, diverse d persspectives, inclluding thosse of matheematicians and matthematics eeducators.

developmen d nt using task sks originallly designed d for children. c Children’s C Thinking: T B Based upon n earlier reesearch from Cognitivvely Guided Instructio on and a more cu urrent studiies focused d on in nterpreting children’s mathematical thinkin ng, th his subgrou up shared hhow artefaccts of childrren’s th hinking can n promote m mathematiical understandi u ing with preeservice eleementary teeachers.

Identifying facctors that im mpact the developmen d nt of appropriate a e and adequ uate mathem matical con ntent knowlledge in preeservice eleementary teacchers is a complex issu ue in mathematics edu ucation reseearch. In th he first sessiion, each sub bgroup sharred their wo ork, as follo ows.

Mathematic M cal Habits oof Mind: Th his subgrou up unpacked u th he notion oof mathema atical habitss of mind. m Build ding from thhe literature, they provided p ex xample taskks for developing mathematic m cal habits off mind and d discussed how h these ta asks can bee used to bo oth raise awareness a of o and fosteer ways of thinking t in preservice p elementary teachers.

p’s work is Maathematicall Tasks: This subgroup grounded in a task development cyccle of designn, enact, reflect, an nd modify/rre-design ph hases. The group shared ttheir researrch on task

Affect: A Ano other influenntial factorr in the preparation p of preserviice teacherss is the affeect (ee.g., attitud des, beliefs,, emotions)) they bringg to and a acquire during uniiversity ma athematics

PME New wsletter Deecember 20 014 / Januaary 2015

Working W Session n 2: Devvelop Prreservice ce Elemeentary T Teacherrs’ Mathema M atical Knowled K dge for T Teachin ng con ntent coursees. This gro oup revieweed and sum mmarized th he state of research in n this area tto reveeal implicaations for prreservice ellementary teacchers’ learn ning of matthematics content. c

On O the secon nd day new wcomers an nd prior working w grou up memberrs were inv vited to reespond, first in plenaryy and then in themed su ubgroups. The T last houur was dedicated to diiscussing co ompilation of the fina al papers intto an n edited book to suppoort mathem matics in nstructors of preservicee elementary teacherss.

Thrree Internattional Persp pectives: This T subgrou up pro ovided exam mples from three coun ntries on matthematics ccontent cou urses for preservice teacchers.

Cheeck you ur PME E conttributio on befo ore subbmissio on! Su ubmitted by Stefan Uferr for the Vicee-President P Portfolio Grooup (Csaba Csikos, C Masaakazu Okazaaki, Wim vaan Dooren) E authors some s guida ance on thee details of preparing p a PME cont ntribution fo or To give PME ubmission, the Vice-President Po ortfolio Gro oup has preepared a Su ubmission Ch Check List. This su prrovides a co oncise list for f a “last-m minute-cheeck” of thin ngs like the length of suubmission,, the fo ormatting, tthe need fo or a blinded d version fo or review, details d of refferences, coopyright, an nd so on. The lists are on the IGP PME web site s (links ssee below) and a it is alsso planned to include them in thee co onference aannouncem ments and in n the Confttool system m for future conferencees. We hope that the lissts will be h helpful for PME P autho ors (since aall the form mal details can be founnd in one pllace without w searrching in lo ong documeents or webb sites) and will also reeduce the w workload of the co onference o organizers. So o, please taake some minutes m and d Check you ur Contribu ution with the t new listts! For Research h Reports: http://ww ww.igpme.o org/index.p php/annual-conferencce/sessionty ypes/researrch-report?iid=121 ations: http:://www.iggpme.org/in ndex.php/aannualFor Short Orral or Posteer Presenta co onference/session-typ pes/short-orral?id=122 For Poster P Presentation ns: http://w www.igpm me.org/indeex.php/ann nual-confereence/sessio onty ypes/posterr-presentatiion?id=122 2

PME New wsletter Deecember 20 014 / Januaary 2015

PM ME39,, 2015 5 Mathem M matics Edu ucation n: Cliimbing Mo ountain ns, Bu uildin ng Bridges Thee local orgaanising com mmittee is pleased p to inviite you to aattend the 39th 3 Meetingg of the Inteernational G Group for the Psychology of Matthematics Edducation (PM ME39) to be b held in Hob bart, Tasmaania, Austrralia from July J 13 to July y 18, 2015.

pro ovides oppo ortunities too highlight and exaamine resea arch that coonsiders thee challenges, obsstacles, link ks, and connnections in n mathematics education, e with a view w to pro ogressing what is know wn and valu uable in thee area. The them me Mathem matics Education: mbing Moun ntains, Buildding Bridgess reflects tw wo Clim of the t notable features off the landsccape of Hob bart: Moun nt Wellingtton (or kun nanyi, to givve it itts Aboriginal name), w which prov vides a

athematics E Education: Climbing g Ma Mo ountains, B Building Briidges has been b chosen n as th he theme o of the confeerence. Thiss theme

Figure 1. Aerial view w of Hobartt City, Tourrism Tasman nia and Alaastair Brett 27

PME New wsletter Deecember 20 014 / Januaary 2015

PM ME39, 2015: Mathem M matics E Educatioon: Clim mbing M Mountaains, Bu uilding Bridgess (contin nued) strik king backdrrop to the city, c and th he Tasman Brid dge, which crosses thee wide Derw went Riverr estu uary. ur plenary sspeakers wiill address aspects a of tthe Fou them me: Lyn En nglish (Ausstralia) who o has expertise in daata modellin ng and stattistical dhood; reassoning partticularly in early child Joh han Lithnerr (Sweden) who will brring a matthematician ns perspective to the problems p off teacching math hematics meeaningfully y; Oh Nam Kw won (Korea)) who speciialises in th he teaching and d learning o of senior maathematics;; and Martyy Sim mon (USA) whose reseearch focuses on the development o of conceptu ual understa anding in matthematics aand how mathematics teachers learrn to teach in ways thaat foster thiis. A plenarry pan nel will also o engage wiith the them me.

Figure 2. 2 Tasmaniaan devil (Sa arcophilus harrisii) - Bonorong Wildlife Pa ark, Tourism m Tasmania T annd Rob Burn nett

Th he main cam mpus of thee Universitty of Taasmania is located l in H Hobart and d as Au ustralia’s fo ourth oldestt university y will be cellebrating its 125th birtthday in 20 015.

We are proud to be hosting the confference in Hob bart, the caapital city of o the Austrralian island d state of Tasmaania (home to many animals and d plan nts found n nowhere elsse in the wo orld). Hobaart is sm mall city w with a popullation of around 200 000. Located aat 42°S, Ho obart is onee of the mosst beau utiful citiess in the worrld, situated d on the wid de Derwentt River, and d with the 1271m 1 high h kun nanyi/Mou unt Wellinggton as a dramatic back kdrop. It haas a cool bu ut not cold winter clim mate with avverage dailly maximum m July tem mperatures o of around 12°C. 1 Scen nic in its ow wn righ ht, Hobart iis also a sprringboard to many of Tasmania’s wo orld-renow wned natura al and derness areas. Anotheer of its fam mous wild attraactions is th he privately y owned Museum M of Old d and New A Art (MON NA), and theere are num merous gallleries, parkss, and an ex xcellent restaurant scen ne.

Th he venue for the confeerence is Ho obart Co ollege, one of the Tasm manian gov vernment sen nior second dary schoolls (Years 11 1 and 12 2), and is sittuated in thhe bush on Mt Neelson 6km from f Hobaart's CBD. It I offers a vaariety of con nference sppaces and has a commanding g view of M Mt Wellington. Trransport to and from th the CBD will be orgganised forr participannts. There iss a range of accommod dation optioons in the CBD C all wiithin easy walking w disttance of thee waaterfront wiith its manyy restauran nts. Fo or more info ormation aabout PME39, pleease visit th he website aat www.pm me39.com

PME New wsletter Deecember 20 014 / Januaary 2015

PME39, P , 2015: Mathem matics E Educatiion: Cliimbing Mounttains, Building B g Bridgees (conttinued)

Im mportant D Dates: Research Reeports (RR)) hort Oral (S SO) Sh Po oster Presen ntation (PP P)

Janua ary 15, 201 5 Marcch 6, 2015 Marcch 6, 2015

Figuree 3. Tasman nian seafoodd: Tourism Tasmania & Peter Aitcchison 29

PME New wsletter Deecember 20 014 / Januaary 2015

Colloqu uium – A new n prresenttation n form mat, piiloted att PME E 39! Subbmitted by W Wim Van Doooren, PME IC A colloquium m proposall consists off a set of (eexactly) threee researchh reports, to o be presentted by y members from at leaast two diffferent co ountries, an nd includess in addition n a one pagge su ummary by an organizzer, indicatting a specific prre-determin ned focus thhat is present in each reesearch repo ort.

Bessides the ex xchange of research r fin ndings, PM ME has the goal to o stimulate genuine co ollaboration n d incrementtal understaanding. For that reaso on, and PM ME will adopt a new prresentation n format in add dition to thee individuaal presentatiions in reseearch reporrts, short orrals, and po osters. ME 39 in Ho obart, it wiill be possibble Starrting at PM to submit s a gro oup activity y called “co olloquium””. A colloquium c consists off 3 related individual i reseearch reporrts, which will w be scheeduled togeether in one time slot in the prog gramme, an nd whiich are follo owed by a discussion initiated byy a disccussant who has prepaared his/heer con ntribution bbeforehand..

Th he deadlinee for propoosals of colloquia is thee saame as that of researchh reports. The T three seeparate reseearch report rts that com mprise the co olloquium have h to be ssubmitted via v the norm mal prrocedure, and a the orgaanizer addiitionally su ubmits a on ne page sum mmary of th he theme an nd th he goals of the t Colloquuium, inclu uding mentioning m the t person who agreed d to be diiscussant. The T research ch reports in ncluded in a co olloquium proposal p arre reviewed d in the usu ual way, w and if the t colloquiium is not accepted ass su uch, the ind dividual ressearch repo orts can stilll be acccepted in the t usual w way. Thus, there t is no risk in nvolved in trying t to suubmit in this new form mat ass compared d to submittting an indiividual reesearch repo ort.

Sev veral other o organizatio ons (e.g. EA ARLI, AERA, BERA A) have useed this preseentation form mat at theirr conferencce for a very y long timee, and d with greatt success. There T are seeveral possible advan ntages of th his new form mat. It enh hances collaaboration among a researchers, an nd may y stimulatee the inclusiion of new researcherrs in a particularr domain. For F conferen nce partticipants, th he program m becomes more m stru uctured and d coherent, and discusssion duringg sesssions can be enhanced d.

We W hope to see many ppositive rea actions, and d many m interessting colloqquia presen nted at PME E in Hobart. H

nk of colleaagues who could c subm mit If you can thin reseearch reporrts that in so ome way are a related tto each h other (e.gg. they depart from related or con ntrasting theeoretical stances, use identical insttruments orr methods or o investiga ate the sam me queestion usingg different methods, m fo ocus on clossely related d research questions, q etcetera), e w we wou uld like to iinvite you to t submit th hese researrch repo orts in the fform of a colloquium c for the nexxt PM ME conferen nce.