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ICMɉ Studɹ 16 ഍਀਀㰀䠀㄀㸀㰀䤀䴀䜀 匀刀䌀 㴀 ∀椀挀洀椀猀㄀㘀氀漀最漀⸀最椀昀∀ 䄀䰀吀㴀∀嬀䤀䌀䴀䤀匀㄀㘀 䰀伀䜀伀崀∀㸀㰀⼀䠀㄀㸀ഀഀ ഍਀ഀ਀਀㰀䠀刀㸀ഀഀ
  • Currenɴ Context

    ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀⠀愀⤀ 倀爀愀挀琀椀挀攀猀 愀渀搀 䔀砀愀洀瀀氀攀猀㰀⼀䈀㸀㰀⼀瀀㸀ഀഀ

    Therɥ arɥ manɹ wayɳ thaɴ studentɳ arɥ currentlɹ beinɧ challengedȮ Thesɥ challengeɳ occuɲ botɨ withiɮ anɤ outsidɥ oɦ schooɬ anɤ includɥ studentɳ aɳ welɬ aɳ generaɬ memberɳ oɦ thɥ publicȮ Theɹ caɮ alsɯ bɥ classifieɤ iɮ severaɬ categorieɳ sucɨ aɳ competitionsȬ probleɭ solvingȬ exhibitionsȬ publicationsȬ anɤ whaɴ maɹ bɥ roughlɹ calleɤ 'mathematicɳ assemblies'Ȯ Beloɷ wɥ refeɲ tɯ somɥ particulaɲ caseɳ wherɥ challengɥ iɳ organisedȮ Tɯ illustratɥ thiɳ wɥ havɥ useɤ exampleɳ whicɨ arɥ familiaɲ tɯ memberɳ oɦ thɥ Internationaɬ Programmɥ Committee.

    ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀䌀伀䴀倀䔀吀䤀吀䤀伀一匀㰀⼀戀㸀㰀⼀倀㸀ഀഀ

    Exclusive and Inclusive Competitions

    Therɥ arɥ manɹ welɬ knowɮ competitionɳ sucɨ aɳ thɥ ഍਀㰀愀 栀爀攀昀 㴀 ∀愀洀琀猀琀愀昀昀猀琀漀爀漀稀栀攀⸀栀琀洀氀∀㸀㰀椀㸀䤀渀琀攀爀渀愀琀椀漀渀愀氀 䴀愀琀栀攀洀愀琀椀挀愀氀 伀氀礀洀瀀椀愀搀㰀⼀䤀㸀㰀⼀愀㸀 ⠀䤀䴀伀⤀ 愀渀搀 ഀ਀਀㰀愀 栀爀攀昀 㴀 ∀眀眀眀⸀洀愀琀栀欀愀渀最⸀漀爀最⼀∀㸀㰀椀㸀䰀攀 䬀愀渀最漀甀爀漀甀 搀攀猀 䴀愀琀栀洀愀琀椀焀甀攀猀㰀⼀䤀㸀㰀⼀愀㸀⸀ ഀഀ The former involves small groups of students from many countries (an example of an exclusive competition) while the latter involves thousands of students in France and Europe (an example of an inclusive competition). Details of these and many other competitions can be found on their web sites as well as in the World Federation of National Mathematics Competition's journal Mathematics Competitions.

    ഍਀㰀倀㸀吀栀攀 眀漀爀搀 ✀挀漀洀瀀攀琀椀琀椀漀渀猀✀ 洀愀礀 椀渀椀琀椀愀氀氀礀 挀漀渀樀甀爀攀 甀瀀 愀渀 椀洀愀最攀 漀昀 爀椀瘀愀氀爀礀 戀攀琀眀攀攀渀 椀渀搀椀瘀椀搀甀愀氀 猀琀甀搀攀渀琀猀 眀椀琀栀 ✀眀椀渀渀攀爀猀✀ 愀渀搀 ✀氀漀猀攀爀猀✀⸀ 圀栀椀氀攀 琀栀椀猀 洀愀礀 戀攀 猀漀 椀渀 挀攀爀琀愀椀渀 猀椀琀甀愀琀椀漀渀猀 椀琀 椀猀 渀漀琀 愀氀眀愀礀猀 琀栀攀 挀愀猀攀⸀ 䔀瘀攀渀 椀渀 琀栀攀 䤀䴀伀Ⰰ 眀栀攀爀攀 洀攀搀愀氀猀 愀渀搀 瀀爀攀猀琀椀最攀 愀爀攀 愀琀 猀琀愀欀攀Ⰰ 琀栀攀爀攀 椀猀 洀漀爀攀 挀漀漀瀀攀爀愀琀椀漀渀 琀栀愀渀 爀椀瘀愀氀爀礀 漀甀琀猀椀搀攀 琀栀攀 挀漀洀瀀攀琀椀琀椀漀渀 爀漀漀洀⸀ 䤀渀 愀氀氀 挀漀洀瀀攀琀椀琀椀漀渀猀Ⰰ 琀栀漀甀最栀Ⰰ 猀琀甀搀攀渀琀猀 眀漀爀欀 ✀愀最愀椀渀猀琀 琀栀攀 瀀爀漀戀氀攀洀✀ 愀猀 洀甀挀栀 愀猀 琀栀攀礀 眀漀爀欀 ✀愀最愀椀渀猀琀 攀愀挀栀 漀琀栀攀爀✀ 愀渀搀 琀栀攀爀攀 愀爀攀 猀椀琀甀愀琀椀漀渀猀 眀栀攀爀攀 挀漀洀瀀氀攀琀椀渀最 琀栀攀 焀甀攀猀琀椀漀渀猀 椀猀 琀栀攀 洀愀椀渀 愀椀洀 爀愀琀栀攀爀 琀栀愀渀 ✀眀椀渀渀椀渀最✀⸀ 䄀渀搀 琀栀攀爀攀 愀爀攀 挀漀洀瀀攀琀椀琀椀漀渀猀 眀栀攀爀攀 琀栀攀 猀琀甀搀攀渀琀猀 栀愀瘀攀 琀漀 挀漀洀瀀漀猀攀 焀甀攀猀琀椀漀渀猀 昀漀爀 漀琀栀攀爀 猀琀甀搀攀渀琀猀 琀漀 猀漀氀瘀攀 爀愀琀栀攀爀 琀栀愀渀 栀愀瘀椀渀最 琀栀攀 焀甀攀猀琀椀漀渀猀 椀洀瀀漀猀攀搀 戀礀 琀栀攀 挀漀洀瀀攀琀椀琀椀漀渀 漀爀最愀渀椀稀攀爀猀⸀ 䈀攀氀漀眀 眀攀 最椀瘀攀 攀砀愀洀瀀氀攀猀 漀昀 琀眀漀 挀漀洀瀀攀琀椀琀椀漀渀猀 琀栀愀琀 愀爀攀 搀椀昀昀攀爀攀渀琀 椀渀 猀漀洀攀 眀愀礀 昀爀漀洀 琀栀攀 琀爀愀搀椀琀椀漀渀愀氀 挀漀洀瀀攀琀椀琀椀漀渀 眀栀攀爀攀 猀琀甀搀攀渀琀猀 愀爀攀 攀猀猀攀渀琀椀愀氀氀礀 猀甀戀洀椀琀琀攀搀 琀漀 愀渀 攀砀愀洀椀渀愀琀椀漀渀⸀ ഀ਀਀ഀഀ

    An exclusive competition of interactive style

    Thɥ competitioɮ EuromathKappAbel is a Nordic competition for 14 year olds in which whole classes participate as a group. The first two rounds consist of problems distributed on the Internet and downloaded by the teacher. Within a 90 minute time limit, the class discusses the problems and decides how to answer each problem. The third round is divided into two parts: a class project with a given theme (that ends with a report, a presentation and an exhibition), and a problem solving session run as a relay where two boys and two girls represent the class. Recent themes have been Mathematics and local handicraft traditions (2000), Mathematics in games and play (2001), Mathematics and sports (2002), Mathematics and technology (2003) and Mathematics and music (2004). The three best teams from the third round meet on the following day for the final, which is a problem solving session with the teams that did not make it to the final as audience.

    ഍਀㰀瀀㸀㰀戀㸀䌀䰀䄀匀匀刀伀伀䴀 唀匀䔀 伀䘀 䌀䠀䄀䰀䰀䔀一䜀䔀㰀⼀戀㸀㰀⼀倀㸀ഀ਀਀㰀倀㸀㰀䈀㸀倀爀漀戀氀攀洀 匀漀氀瘀椀渀最㰀⼀䈀㸀㰀⼀瀀㸀ഀഀ

    The words `problem solving' have been used to cover a variety of experiences but by the words here we mean allowing students to work on closed questions that they are not immediately able to solve. Hence they need to apply their mathematical content knowledge as well as ingenuity, intuition and a range of metacognitive skills in order to obtain an answer.

    ഍਀㰀倀㸀倀爀漀戀氀攀洀 猀漀氀瘀椀渀最 椀猀 漀昀琀攀渀 甀猀攀搀 椀渀 挀氀愀猀猀爀漀漀洀猀 愀猀 愀 漀渀攀ⴀ漀昀昀 攀砀攀爀挀椀猀攀 琀栀愀琀 洀愀礀 漀爀 洀愀礀 渀漀琀 戀攀 挀漀渀渀攀挀琀攀搀 琀漀 琀栀攀 洀愀椀渀 洀愀琀栀攀洀愀琀椀挀愀氀 挀甀爀爀椀挀甀氀甀洀⸀ 䤀琀 挀愀渀 戀攀 猀攀攀渀 愀猀 愀 ✀昀椀氀氀攀爀✀ 琀栀愀琀 洀愀渀礀 猀琀甀搀攀渀琀猀 攀渀樀漀礀 戀甀琀 椀琀 椀猀 渀漀琀 愀氀眀愀礀猀 瘀椀攀眀攀搀 愀猀 挀攀渀琀爀愀氀 琀漀 琀栀攀 洀愀琀栀攀洀愀琀椀挀猀 挀氀愀猀猀爀漀漀洀⸀㰀⼀瀀㸀ഀ਀਀ഀഀ

    Investigations and projects may be extended problem solving exercises where students look into more difficult problems over more than one period of class time. They frequently involve a written report.

    ഍਀㰀倀㸀吀攀愀挀栀攀爀猀 眀栀漀 甀猀攀 瀀爀漀戀氀攀洀猀 琀漀 搀攀瘀攀氀漀瀀 猀琀甀搀攀渀琀猀✀ 椀搀攀愀猀Ⰰ 欀渀漀眀氀攀搀最攀 愀渀搀 甀渀搀攀爀猀琀愀渀搀椀渀最 漀昀 挀甀爀爀椀挀甀氀甀洀 洀愀琀攀爀椀愀氀 挀愀渀 戀攀 挀漀渀猀椀搀攀爀攀搀 愀猀 琀愀欀椀渀最 愀 怀瀀爀漀戀氀攀洀 猀漀氀瘀椀渀最 愀瀀瀀爀漀愀挀栀✀ 琀漀 琀栀攀 琀漀瀀椀挀⸀ 吀栀椀猀 愀瀀瀀爀漀愀挀栀 挀愀渀 爀攀昀氀攀挀琀 琀栀攀 挀爀攀愀琀椀瘀攀 渀愀琀甀爀攀 漀昀 洀愀琀栀攀洀愀琀椀挀猀 愀渀搀 最椀瘀攀 猀琀甀搀攀渀琀猀 猀漀洀攀 昀攀攀氀 昀漀爀 琀栀攀 眀愀礀 琀栀愀琀 洀愀琀栀攀洀愀琀椀挀猀 椀猀 搀攀瘀攀氀漀瀀攀搀 戀礀 爀攀猀攀愀爀挀栀 洀愀琀栀攀洀愀琀椀挀椀愀渀猀⸀ 䔀砀愀洀瀀氀攀猀 漀昀 戀漀琀栀 瀀爀漀戀氀攀洀 猀漀氀瘀椀渀最 氀攀猀猀漀渀猀 愀渀搀 氀攀猀猀漀渀猀 眀栀椀挀栀 琀愀欀攀 愀 瀀爀漀戀氀攀洀 猀漀氀瘀椀渀最 愀瀀瀀爀漀愀挀栀 挀愀渀 戀攀 昀漀甀渀搀 漀渀 琀栀攀 眀攀戀 猀椀琀攀 㰀愀 栀爀攀昀 㴀 ∀眀眀眀⸀渀稀洀愀琀栀猀⸀挀漀⸀渀稀∀㸀眀眀眀⸀渀稀洀愀琀栀猀⸀挀漀⸀渀稀㰀⼀愀㸀⸀㰀⼀瀀㸀ഀ਀਀ഀഀ

    Challenge in traditional education: An example

    Ɂ traditionaɬ methoɤ iɮ Japanesɥ elementarɹ schooɬ iɳ tɯ solvɥ ɡ probleɭ througɨ full-clasɳ discussionȮ Witɨ ɡ skilfuɬ teacherȬ thɥ childreɮ caɮ learɮ morɥ thaɮ thɥ curriculuɭ intendsȮ Foɲ exampleȬ supposɥ thaɴ theɹ arɥ giveɮ thɥ probleɭ oɦ dividinɧ 4/ȵ bɹ 2/3Ȯ Onɥ studenɴ mighɴ observɥ thaɴ ȶ iɳ thɥ leasɴ commoɮ multiplɥ oɦ Ȳ anɤ 3Ȭ anɤ write

    ഍਀㰀倀㸀⠀㐀⼀㔀⤀⼀⠀㈀⼀㌀⤀㴀⠀㐀砀⠀㘀⼀㈀⤀⤀⼀⠀㔀砀⠀㘀⼀㌀⤀⤀㴀⠀㐀砀㌀⤀⼀⠀㔀砀㈀⤀㴀㄀㈀⼀㄀ ⸀㰀⼀倀㸀ഀ਀਀㰀倀㸀吀栀攀 挀栀椀氀搀爀攀渀 挀愀渀 挀漀洀攀 琀漀 爀攀愀氀椀猀攀 琀栀愀琀 琀栀椀猀 洀攀琀栀漀搀 椀猀 攀焀甀椀瘀愀氀攀渀琀 琀漀 琀栀攀 猀琀愀渀搀愀爀搀 愀氀最漀爀椀琀栀洀 愀渀搀 挀愀渀 戀攀 甀猀攀搀 眀椀琀栀 漀琀栀攀爀 挀栀漀椀挀攀猀 漀昀 昀爀愀挀琀椀漀渀猀⸀ 䘀爀漀洀 琀栀攀 琀攀愀挀栀攀爀✀猀 瀀漀椀渀琀 漀昀 瘀椀攀眀Ⰰ 琀栀椀猀 搀礀渀愀洀椀挀 椀猀 甀渀瀀爀攀搀椀挀琀愀戀氀攀Ⰰ 愀渀搀 猀漀 琀栀攀 琀攀愀挀栀攀爀 爀攀焀甀椀爀攀猀 搀攀攀瀀 洀愀琀栀攀洀愀琀椀挀愀氀 甀渀搀攀爀猀琀愀渀搀椀渀最 愀渀搀 猀甀爀攀 猀欀椀氀氀猀 椀渀 漀爀搀攀爀 琀漀 栀愀渀搀氀攀 琀栀攀 猀椀琀甀愀琀椀漀渀⸀ 䈀甀琀 眀栀攀渀 琀栀攀 愀瀀瀀爀漀愀挀栀 猀甀挀挀攀攀搀猀Ⰰ 琀栀攀 挀栀椀氀搀爀攀渀 搀攀攀瀀攀渀 琀栀攀椀爀 洀愀琀栀攀洀愀琀椀挀愀氀 攀砀瀀攀爀椀攀渀挀攀⸀㰀⼀倀㸀ഀഀ

    EXHIBITIONS

    ഍਀㰀倀㸀䔀砀栀椀戀椀琀椀漀渀猀Ⰰ 椀渀 琀栀攀 猀攀渀猀攀 漀昀 最愀琀栀攀爀椀渀最 洀愀琀攀爀椀愀氀 琀漀最攀琀栀攀爀 昀漀爀 瀀攀漀瀀氀攀 琀漀 瘀椀攀眀 漀爀 椀渀琀攀爀愀挀琀 眀椀琀栀Ⰰ 愀爀攀 戀攀挀漀洀椀渀最 椀渀挀爀攀愀猀椀渀最氀礀 挀漀洀洀漀渀⸀ 吀栀攀猀攀 愀爀攀 最攀渀攀爀愀氀氀礀 漀甀琀猀椀搀攀 漀昀 琀栀攀 挀氀愀猀猀爀漀漀洀 愀渀搀 洀愀礀 戀攀 愀椀洀攀搀 愀猀 洀甀挀栀 愀琀 琀栀攀 最攀渀攀爀愀氀 瀀甀戀氀椀挀 愀猀 琀栀攀礀 愀爀攀 愀琀 猀琀甀搀攀渀琀猀⸀ 吀栀攀礀 挀愀渀 愀氀猀漀 琀愀欀攀 瀀氀愀挀攀 椀渀 愀 瘀愀爀椀攀琀礀 漀昀 猀攀琀琀椀渀最猀 昀爀漀洀 猀挀栀漀漀氀猀 琀漀 洀甀猀攀甀洀猀 琀漀 猀栀漀瀀瀀椀渀最 洀愀氀氀猀 琀漀 琀栀攀 漀瀀攀渀 愀椀爀⸀ 圀攀 洀攀渀琀椀漀渀 猀攀瘀攀爀愀氀 攀砀愀洀瀀氀攀猀 漀昀 琀栀攀猀攀⸀㰀⼀倀㸀ഀ਀਀ഀഀ

    The idea of a science centre is to present scientific phenomena in a hands-on way. This means that the visitors are challenged by a real experiment and then try to understand it. Some science centres have mathematical experiments, but there are also science centres devoted exclusively to mathematics, for instance the <ɡ hreɦ Ƚ "http://www.mathematikum.de">MathematikumThere are also annual exhibitions, varying in content from year to year. An example of this which attracts tens of thousands of visitors per day is <ɡ hreɦ Ƚ "http://www.cijm.org">Lɥ Saloɮ dɥ lɡ Culturɥ Mathématiqueɳ eɴ deɳ JeuxExperiencinɧ Mathematics ഍਀ഀ਀਀㰀倀㸀䔀砀栀椀戀椀琀椀漀渀猀 挀愀渀 栀愀瘀攀 愀 猀瀀攀挀椀愀氀 琀栀攀洀攀Ⰰ 猀甀挀栀 愀猀 琀栀攀 漀渀攀 愀琀 琀栀攀 唀渀椀瘀攀爀猀椀琀礀 漀昀 䴀漀搀攀渀愀 愀渀搀 刀攀最最椀漀 䔀洀椀氀椀愀 昀攀愀琀甀爀椀渀最 㰀愀 栀爀攀昀 㴀 ∀栀琀琀瀀㨀⼀⼀眀眀眀⸀洀洀氀愀戀⸀甀渀椀洀漀⸀椀琀∀㸀洀愀琀栀攀洀愀琀椀挀愀氀 洀愀挀栀椀渀攀猀㰀⼀愀㸀⸀ 吀栀攀猀攀 洀愀挀栀椀渀攀猀 愀爀攀 挀漀瀀椀攀猀 漀昀 栀椀猀琀漀爀椀挀愀氀 椀渀猀琀爀甀洀攀渀琀猀 琀栀愀琀 椀渀挀氀甀搀攀 挀甀爀瘀攀 搀爀愀眀椀渀最 搀攀瘀椀挀攀猀Ⰰ 椀渀猀琀爀甀洀攀渀琀猀 昀漀爀 瀀攀爀猀瀀攀挀琀椀瘀攀 搀爀愀眀椀渀最 愀渀搀 椀渀猀琀爀甀洀攀渀琀猀 昀漀爀 猀漀氀瘀椀渀最 瀀爀漀戀氀攀洀猀⸀㰀⼀倀㸀ഀഀ

    Instrumentɳ foɲ museumsȬ laboratorieɳ oɲ mathematicɳ centreɳ maɹ bɥ verɹ expensiveȮ Foɲ classrooɭ usɥ smalɬ cheapeɲ kitɳ maɹ bɥ availablɥ witɨ informatioɮ abouɴ possiblɥ classrooɭ use.

    ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀倀唀䈀䰀䤀䌀䄀吀䤀伀一匀Ⰰ 䤀一䌀䰀唀䐀䤀一䜀 䤀一吀䔀刀一䔀吀㰀⼀䈀㸀㰀⼀倀㸀ഀഀ

    Publications cover at least books, journals, web sites, CDs, games and software. They are generally accessible to a wide audience.

    ഍਀㰀倀㸀㰀䈀㸀匀挀栀漀漀氀 䴀愀琀栀攀洀愀琀椀挀猀 䨀漀甀爀渀愀氀猀㰀⼀䈀㸀㰀⼀倀㸀ഀ਀਀㰀倀㸀吀栀攀爀攀 愀爀攀 洀愀渀礀 攀砀愀洀瀀氀攀猀 愀爀漀甀渀搀 琀栀攀 眀漀爀氀搀 漀昀 樀漀甀爀渀愀氀猀 搀攀猀椀最渀攀搀 琀漀 猀琀椀洀甀氀愀琀攀 猀琀甀搀攀渀琀 椀渀琀攀爀攀猀琀 椀渀 洀愀琀栀攀洀愀琀椀挀猀⸀ 吀栀攀猀攀 樀漀甀爀渀愀氀猀 挀漀渀琀愀椀渀 栀椀猀琀漀爀椀挀愀氀 愀爀琀椀挀氀攀猀Ⰰ 愀爀琀椀挀氀攀猀 攀砀瀀漀猀椀渀最 椀猀猀甀攀猀 眀椀琀栀 挀甀爀爀攀渀琀 爀攀猀攀愀爀挀栀Ⰰ 猀甀挀栀 愀猀 琀栀攀 昀漀甀爀 挀漀氀漀甀爀 琀栀攀漀爀攀洀 愀渀搀 䘀攀爀洀愀琀✀猀 䰀愀猀琀 吀栀攀漀爀攀洀Ⰰ 愀渀搀 倀爀漀戀氀攀洀 挀漀爀渀攀爀猀Ⰰ 眀栀攀爀攀 渀攀眀 瀀爀漀戀氀攀洀猀 愀爀攀 瀀漀猀攀搀Ⰰ 漀琀栀攀爀 挀甀爀爀攀渀琀 瀀爀漀戀氀攀洀猀 昀爀漀洀 伀氀礀洀瀀椀愀搀猀 愀爀攀 搀椀猀挀甀猀猀攀搀 愀渀搀 猀琀甀搀攀渀琀猀 洀愀礀 猀甀戀洀椀琀 琀栀攀椀爀 漀眀渀 猀漀氀甀琀椀漀渀猀⸀ 䔀砀愀洀瀀氀攀猀 漀昀 猀甀挀栀 樀漀甀爀渀愀氀猀 椀渀 琀栀攀 䔀愀猀琀攀爀渀 䈀氀漀挀Ⰰ 眀栀攀爀攀 琀栀攀 琀爀愀搀椀琀椀漀渀猀 愀爀攀 漀氀搀攀爀Ⰰ 愀爀攀 㰀䤀㸀䬀洀愀氀㰀⼀䤀㸀 ⠀䠀甀渀最愀爀礀⤀ 愀渀搀 㰀䤀㸀䬀瘀愀渀琀㰀⼀䤀㸀 ⠀刀甀猀猀椀愀⤀⸀ 䤀渀 琀栀攀 圀攀猀琀 漀甀琀猀琀愀渀搀椀渀最 攀砀愀洀瀀氀攀猀 愀爀攀 㰀䤀㸀䌀爀甀砀 䴀愀琀栀攀洀愀琀椀挀漀爀甀洀㰀⼀䤀㸀 ⠀䌀愀渀愀搀愀⤀Ⰰ ഀഀ Mathematics Magazine and Mathematical Spectrum (UK).

    ഍਀㰀倀㸀㰀䈀㸀䈀漀漀欀猀㰀⼀䈀㸀㰀⼀倀㸀ഀ਀਀㰀倀㸀吀栀攀爀攀 愀爀攀 洀愀渀礀 瀀甀戀氀椀挀愀琀椀漀渀猀 眀栀椀挀栀 攀渀爀椀挀栀 愀渀搀 挀栀愀氀氀攀渀最攀 琀栀攀 猀琀甀搀攀渀琀✀猀 椀渀琀攀爀攀猀琀 椀渀 洀愀琀栀攀洀愀琀椀挀猀⸀ 䤀渀 琀栀攀 䔀渀最氀椀猀栀 氀愀渀最甀愀最攀 琀栀攀 䴀愀琀栀攀洀愀琀椀挀愀氀 䄀猀猀漀挀椀愀琀椀漀渀 漀昀 䄀洀攀爀椀挀愀 栀愀猀 愀 洀愀猀猀椀瘀攀 挀愀琀愀氀漀最甀攀 愀渀搀 琀栀攀 䄀甀猀琀爀愀氀椀愀渀 䴀愀琀栀攀洀愀琀椀挀猀 吀爀甀猀琀 栀愀猀 愀 猀椀最渀椀昀椀挀愀渀琀 渀甀洀戀攀爀 漀昀 瀀甀戀氀椀挀愀琀椀漀渀猀⸀ 䤀渀 刀甀猀猀椀愀渀 琀栀攀爀攀 椀猀 愀氀猀漀 愀 瘀攀爀礀 爀椀挀栀 爀攀猀漀甀爀挀攀Ⰰ 琀爀愀搀椀琀椀漀渀愀氀氀礀 瀀甀戀氀椀猀栀攀搀 琀栀爀漀甀最栀 䴀椀爀⸀ 䤀渀 琀栀攀 䘀爀攀渀挀栀 氀愀渀最甀愀最攀 琀栀攀 㰀䤀㸀䬀愀渀最漀甀爀漀甀㰀⼀䤀㸀 愀渀搀 漀琀栀攀爀 瀀甀戀氀椀猀栀攀爀猀 栀愀瘀攀 愀 瀀爀漀搀椀最椀漀甀猀 挀愀琀愀氀漀最甀攀Ⰰ 愀猀 搀漀攀猀 琀栀攀 䌀栀椀甀 䌀栀愀渀最 䴀愀琀栀攀洀愀琀椀挀猀 䔀搀甀挀愀琀椀漀渀 䘀漀甀渀搀愀琀椀漀渀 椀渀 琀栀攀 䌀栀椀渀攀猀攀 氀愀渀最甀愀最攀⸀ 吀栀椀猀 樀甀猀琀 爀攀昀攀爀猀 琀漀 洀愀樀漀爀 氀愀渀最甀愀最攀猀⸀ 䤀琀 椀猀 攀砀瀀攀挀琀攀搀 琀漀 戀攀 椀洀瀀漀猀猀椀戀氀攀 琀漀 琀爀礀 琀漀 氀椀猀琀 椀渀搀椀瘀椀搀甀愀氀 爀攀昀攀爀攀渀挀攀猀 椀渀 琀栀椀猀 匀琀甀搀礀⸀ 圀攀 攀砀瀀攀挀琀 椀琀 眀椀氀氀 戀攀 搀椀昀昀椀挀甀氀琀 攀渀漀甀最栀 琀漀 椀搀攀渀琀椀昀礀 琀栀攀 洀愀樀漀爀 瀀甀戀氀椀猀栀攀爀猀⸀㰀⼀倀㸀ഀഀ

    Internet

    ഍਀㰀倀㸀吀栀攀爀攀 愀爀攀 愀 渀甀洀戀攀爀 漀昀 攀砀愀洀瀀氀攀猀 椀渀 眀栀椀挀栀 瀀攀漀瀀氀攀 挀愀渀 樀漀椀渀 愀 挀氀愀猀猀爀漀漀洀 戀礀 椀渀琀攀爀渀攀琀⸀ 吀栀攀 ✀攀ⴀ挀氀愀猀猀爀漀漀洀✀ 挀漀渀搀甀挀琀攀搀 戀礀 一漀爀椀欀漀 䄀爀愀椀 椀猀 愀 瘀椀爀琀甀愀氀 挀氀愀猀猀爀漀漀洀 椀渀 眀栀椀挀栀 攀瘀攀爀礀戀漀搀礀 椀渀琀攀爀攀猀琀攀搀 椀渀 洀愀琀栀攀洀愀琀椀挀猀 挀愀渀 樀漀椀渀 戀礀 爀攀最椀猀琀爀愀琀椀漀渀⸀ 吀栀攀 挀氀愀猀猀爀漀漀洀 椀猀 爀甀渀 愀渀搀 猀甀瀀攀爀瘀椀猀攀搀 戀礀 愀 昀攀眀 洀愀琀栀攀洀愀琀椀挀椀愀渀猀 挀愀氀氀攀搀 洀漀搀攀爀愀琀漀爀猀⸀ 唀猀甀愀氀氀礀 漀渀攀 漀昀 琀栀攀洀 最椀瘀攀猀 愀 瀀爀漀戀氀攀洀 猀甀挀栀 愀猀 怀✀栀愀爀愀挀琀攀爀椀猀攀 愀 昀爀愀挀琀椀漀渀 眀栀椀挀栀 椀猀 愀 昀椀渀椀琀攀 搀攀挀椀洀愀氀✀⸀ 吀栀攀渀Ⰰ 搀椀猀挀甀猀猀椀漀渀猀 猀琀愀爀琀⸀ 䄀 猀琀甀搀攀渀琀 最椀瘀攀猀 愀 瘀愀最甀攀 椀搀攀愀 琀漀 猀漀氀瘀攀 琀栀攀 瀀爀漀戀氀攀洀Ⰰ 愀 瀀愀爀琀椀愀氀 愀渀猀眀攀爀 漀爀 愀 焀甀攀猀琀椀漀渀Ⰰ 愀渀搀 漀琀栀攀爀 猀琀甀搀攀渀琀猀 最椀瘀攀 挀漀洀洀攀渀琀猀 漀渀 椀琀 漀爀 椀洀瀀爀漀瘀攀 昀漀爀洀攀爀 椀搀攀愀猀⸀ 䴀漀搀攀爀愀琀漀爀猀 攀渀挀漀甀爀愀最攀 琀栀攀 搀椀猀挀甀猀猀椀漀渀猀Ⰰ 最椀瘀椀渀最 栀椀渀琀猀 椀昀 渀攀挀攀猀猀愀爀礀⸀  唀猀甀愀氀氀礀 琀栀攀 搀椀猀挀甀猀猀椀漀渀猀 攀渀搀 眀椀琀栀 挀漀洀瀀氀攀琀攀 愀渀猀眀攀爀⸀ 匀漀洀攀琀椀洀攀猀 愀 渀攀眀 瀀爀漀戀氀攀洀 愀爀椀猀攀猀 昀爀漀洀 搀椀猀挀甀猀猀椀漀渀猀⸀ 伀琀栀攀爀眀椀猀攀Ⰰ 愀渀漀琀栀攀爀 瀀爀漀戀氀攀洀 眀椀氀氀 戀攀 最椀瘀攀渀 戀礀 愀 洀漀搀攀爀愀琀漀爀⸀㰀⼀倀㸀 ഀ਀਀㰀倀㸀一⸀ 䄀爀愀椀 搀攀瘀攀氀漀瀀攀搀 猀漀昀琀眀愀爀攀 猀漀 琀栀愀琀 漀渀氀礀 猀琀甀搀攀渀琀猀 漀昀 琀栀攀 挀氀愀猀猀爀漀漀洀 挀愀渀 栀愀瘀攀 愀挀挀攀猀猀 琀漀 琀栀攀 搀椀猀挀甀猀猀椀漀渀猀⸀ 䤀渀 琀栀椀猀 攀渀瘀椀爀漀渀洀攀渀琀 愀 猀栀礀 挀栀椀氀搀 漀爀 愀渀 攀氀搀攀爀 瀀攀爀猀漀渀 眀栀漀 椀猀 渀漀琀 猀漀 猀琀爀漀渀最 椀渀 洀愀琀栀攀洀愀琀椀挀猀 洀愀礀 昀攀攀氀 洀漀爀攀 挀漀洀昀漀爀琀愀戀氀攀 椀渀 樀漀椀渀椀渀最 搀椀猀挀甀猀猀椀漀渀猀⸀㰀⼀倀㸀ഀഀ

    "MATHEMATICɓ ASSEMBLIES"

    ഍਀㰀倀㸀吀栀攀猀攀 愀挀琀椀瘀椀琀椀攀猀 愀爀攀 愀椀洀攀搀 愀琀 最爀漀甀瀀猀 漀昀 瀀攀漀瀀氀攀 眀栀漀 最攀渀攀爀愀氀氀礀 愀猀猀攀洀戀氀攀 琀漀最攀琀栀攀爀 椀渀 漀渀攀 瀀氀愀挀攀 琀漀 戀攀 攀搀甀挀愀琀攀搀 戀礀 愀渀 攀砀瀀攀爀琀 漀爀 最爀漀甀瀀 漀昀 攀砀瀀攀爀琀猀⸀ 圀攀 栀愀瘀攀 椀渀 洀椀渀搀 栀攀爀攀 猀甀挀栀 琀栀椀渀最猀 愀猀 洀愀琀栀攀洀愀琀椀挀猀 挀氀甀戀猀Ⰰ 洀愀琀栀攀洀愀琀椀挀猀 搀愀礀猀Ⰰ 猀甀洀洀攀爀 猀挀栀漀漀氀猀Ⰰ 洀愀猀琀攀爀 挀氀愀猀猀攀猀Ⰰ 洀愀琀栀攀洀愀琀椀挀猀 挀愀洀瀀猀Ⰰ 洀愀琀栀攀洀愀琀椀挀猀 昀攀猀琀椀瘀愀氀猀 愀渀搀 猀漀 漀渀⸀ 䘀椀瘀攀 猀瀀攀挀椀昀椀挀 攀砀愀洀瀀氀攀猀 愀爀攀 最椀瘀攀渀 戀攀氀漀眀 琀栀愀琀 爀攀昀攀爀 琀漀 洀愀琀栀攀洀愀琀椀挀猀 搀愀礀猀Ⰰ 爀攀猀攀愀爀挀栀 挀氀愀猀猀攀猀Ⰰ 愀渀搀 椀渀搀甀猀琀爀礀 挀氀愀猀猀攀猀⸀㰀⼀倀㸀ഀ਀਀ഀഀ

    School mathematics days

    Therɥ arɥ manɹ exampleɳ arounɤ thɥ worlɤ oɦ mathematicɳ dayɳ iɮ whicɨ teamɳ oɦ studentɳ froɭ variouɳ schoolɳ iɮ ɡ districɴ comɥ togetherȮ Durinɧ thɥ daɹ theɹ wilɬ participatɥ iɮ variouɳ individuaɬ anɤ teaɭ eventɳ iɮ aɮ enjoyablɥ atmosphereȬ anɤ therɥ maɹ bɥ expositorɹ lectures.

    ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀䴀愀琀栀攀洀愀琀椀挀猀 䌀氀甀戀猀㰀⼀䈀㸀㰀⼀倀㸀ഀഀ

    The world has many examples of mathematics clubs (or circles as they are sometimes known) of students who meet at regular intervals in their town to solve new problems. Often these clubs use a correspondence competition such as the <ɡ hreɦ Ƚ "tt/imtot.html">Internationaɬ Mathematicɳ Tournamenɴ oɦ Towns ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀䴀愀琀栀攀洀愀琀椀挀猀 䠀漀甀猀攀猀㰀⼀䈀㸀㰀⼀倀㸀ഀഀ

    In Iran, a team of teachers and university staff have established what are called <ɡ hreɦ Ƚ "http://www.mathhouse.org">Mathematicɳ Houses ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀刀攀猀攀愀爀挀栀 䌀氀愀猀猀攀猀㰀⼀䈀㸀㰀⼀倀㸀ഀഀ

    In Germany for many years the prize for the winners of a mathematical competition is an invitation to a <ɡ hreɦ Ƚ "http://zfm.mathematik.tu-darmstadt.de">ModellierungswocheȮ Iɮ thisȬ groupɳ oɦ ȸ studentɳ togetheɲ witɨ twɯ teacherɳ worɫ oɮ ɡ reaɬ applicatioɮ probleɭ poseɤ bɹ locaɬ industryȮ Manɹ oɦ thɥ problemɳ arɥ optimisatioɮ problemsȮ Thɥ solutioɮ normallɹ requireɳ modellingȬ mathematicaɬ analysiɳ anɤ makinɧ ɡ computeɲ program.

    ഍਀ഀ਀਀㰀倀㸀䄀猀 愀渀漀琀栀攀爀 攀砀愀洀瀀氀攀Ⰰ 椀渀 㰀椀㸀䴀愀琀栀 攀渀 樀攀愀渀猀㰀⼀椀㸀 攀愀挀栀 琀攀愀洀 眀漀爀欀猀 椀渀 挀漀氀氀愀戀漀爀愀琀椀漀渀 眀椀琀栀 愀 甀渀椀瘀攀爀猀椀琀礀 爀攀猀攀愀爀挀栀攀爀 眀栀漀 栀愀猀 瀀爀漀瀀漀攀搀 愀 瀀爀漀戀氀攀洀Ⰰ 椀搀攀愀氀氀礀 挀漀渀渀攀挀琀攀搀 琀漀 栀椀猀⼀栀攀爀 爀攀猀攀愀爀挀栀Ⰰ 漀渀 眀栀椀挀栀 琀栀攀 猀琀甀搀攀渀琀猀 眀漀爀欀 昀漀爀 愀 氀漀渀最 瀀攀爀椀漀搀 ⠀漀昀琀攀渀 甀瀀 琀漀 愀 昀甀氀氀 猀挀栀漀漀氀 礀攀愀爀⤀⸀㰀⼀倀㸀ഀഀ

    (bȩ Trends

    ഍਀㰀倀㸀䤀琀 猀攀攀洀猀 琀栀愀琀Ⰰ 眀椀琀栀 昀攀眀 攀砀挀攀瀀琀椀漀渀猀Ⰰ 琀栀攀 漀瘀攀爀愀氀氀 琀爀攀渀搀猀 愀爀攀 瀀漀猀椀琀椀瘀攀⸀ 䘀漀爀 攀砀愀洀瀀氀攀Ⰰ 琀栀攀爀攀 愀爀攀 洀愀渀礀 渀攀眀 挀漀洀瀀攀琀椀琀椀漀渀猀 琀栀愀琀 挀愀琀攀爀 昀漀爀 愀 眀椀搀攀爀 爀愀渀最攀 漀昀 猀琀甀搀攀渀琀猀 琀栀愀渀 琀栀攀 洀漀爀攀 琀爀愀搀椀琀椀漀渀愀氀ഀ਀਀伀氀礀洀瀀椀愀搀ⴀ猀琀礀氀攀 挀漀渀瀀攀琀椀琀椀漀渀 愀渀搀 椀渀挀氀甀搀攀 礀漀甀渀最攀爀 挀栀椀氀搀爀攀渀 琀栀愀渀 戀攀昀漀爀攀⸀ 䴀愀渀礀 挀漀洀瀀攀琀椀琀椀漀渀猀 渀漀眀 椀渀瘀漀氀瘀攀 最爀漀甀瀀猀 漀昀 猀琀甀搀攀渀琀猀 爀愀琀栀攀爀 琀栀愀渀 樀甀猀琀 椀渀搀椀瘀椀搀甀愀氀猀⸀㰀⼀倀㸀ഀഀ

    Iɮ recenɴ yearɳ tooȬ probleɭ solvinɧ haɳ beeɮ addeɤ tɯ thɥ curriculɡ oɦ ɡ numbeɲ oɦ countriesȮ HoweverȬ withouɴ somɥ professionaɬ developmenɴ foɲ teachersȬ iɴ maɹ noɴ appeaɲ iɮ thɥ actuaɬ curriculuɭ delivereɤ iɮ class.

    ഍਀ഀ਀਀㰀倀㸀䤀渀 琀栀攀 猀愀洀攀 瘀攀椀渀Ⰰ 琀栀攀爀攀 愀瀀瀀攀愀爀猀 琀漀 戀攀 愀渀 椀渀挀爀攀愀猀椀渀最 渀甀洀戀攀爀 漀昀 洀愀琀栀攀洀愀琀椀挀愀氀 攀砀栀椀戀椀琀椀漀渀猀⸀ 䘀漀爀 愀 眀栀椀氀攀Ⰰ 洀愀琀栀攀洀愀琀椀挀愀氀 攀砀栀椀戀椀琀猀 最攀渀攀爀愀氀氀礀 愀瀀瀀攀愀爀攀搀 椀渀 猀挀椀攀渀挀攀 挀攀渀琀爀攀猀 戀甀琀 渀漀眀 琀栀攀爀攀 愀爀攀 洀漀爀攀 攀砀栀椀戀椀琀椀漀渀猀 搀攀瘀漀琀攀搀 猀漀氀攀氀礀 琀漀 洀愀琀栀攀洀愀琀椀挀猀⸀ 䄀渀搀Ⰰ 椀渀猀琀攀愀搀 漀昀 戀攀椀渀最 栀攀氀搀 椀渀 洀甀猀攀甀洀ⴀ氀椀欀攀 猀攀琀琀椀渀最猀Ⰰ 洀愀琀栀攀洀愀琀椀挀猀 攀砀栀椀戀椀琀椀漀渀猀 攀砀椀猀琀 琀栀愀琀 愀爀攀 瀀漀爀琀愀戀氀攀 漀爀 愀瀀瀀攀愀爀 椀渀 猀甀挀栀 甀渀甀猀甀愀氀 猀攀琀琀椀渀最猀 愀猀 猀栀漀瀀瀀椀渀最 挀攀渀琀爀攀猀Ⰰ 猀甀戀眀愀礀猀Ⰰ 愀渀搀 琀栀攀 漀瀀攀渀 愀椀爀⸀㰀⼀倀㸀ഀഀ

    Aɳ foɲ publicationsȬ therɥ recentlɹ seemɳ tɯ havɥ beeɮ aɮ increasinɧ numbeɲ oɦ bookɳ anɤ filmɳ oɦ ɡ mathematicaɬ naturɥ foɲ thɥ generaɬ publicȮ Somɥ oɦ theseȬ sucɨ aɳ Fermat'ɳ Lasɴ TheoremɁ Beautifuɬ MindȬ havɥ beeɮ extremelɹ successfulȮ ഍਀伀渀 琀栀攀 戀漀漀欀 猀椀搀攀 琀栀漀甀最栀Ⰰ 琀栀攀爀攀 洀愀礀 戀攀 愀 琀爀攀渀搀 愀眀愀礀 昀爀漀洀 挀氀愀猀猀椀挀愀氀 瀀爀漀戀氀攀洀 戀漀漀欀猀 琀漀 戀漀漀欀猀 琀栀愀琀 搀椀猀挀甀猀猀 洀愀琀栀攀洀愀琀椀挀愀氀 琀漀瀀椀挀猀 愀渀搀 愀爀攀 洀攀愀渀琀 琀漀 戀攀 爀攀愀搀 爀愀琀栀攀爀 琀栀愀渀 眀漀爀欀攀搀 漀渀⸀ 吀栀攀猀攀 戀漀漀欀猀 洀愀礀 愀琀琀攀洀瀀琀 琀漀 挀漀渀瘀攀礀 搀攀攀瀀 愀渀搀 挀漀洀瀀氀椀挀愀琀攀搀 洀愀琀栀攀洀愀琀椀挀猀 戀甀琀 琀栀攀礀 搀漀 猀漀 戀礀 挀爀攀愀琀椀渀最 愀渀 椀洀瀀爀攀猀猀椀漀渀 爀愀琀栀攀爀 琀栀愀渀 最漀椀渀最 椀渀琀漀 最爀攀愀琀 搀攀琀愀椀氀猀⸀㰀⼀倀㸀ഀ਀਀ഀഀ

    In recent years the Moscow Centre for Continuous Mathematical Education has published a series of books The library of mathematical education. These are small books (20-30 pages) written by professional mathematicians and addressed to interested high school students. They include popular explanations of various areas of mathematics, challenging problems for students and history. The small size of the volumes, good illustrations, and popular style of writing attract a lot of readers.

    ഍਀㰀倀㸀䤀琀 愀瀀瀀攀愀爀猀 琀栀愀琀 洀愀最愀稀椀渀攀猀 愀渀搀 渀攀眀猀瀀愀瀀攀爀猀 愀爀攀 挀甀爀爀攀渀琀氀礀 挀愀爀爀礀椀渀最 洀漀爀攀 洀愀琀栀攀洀愀琀椀挀猀Ⰰ 戀漀琀栀 眀椀琀栀 猀琀漀爀椀攀猀 愀戀漀甀琀 挀漀渀琀攀洀瀀漀爀愀爀礀 洀愀琀栀攀洀愀琀椀挀猀 愀渀搀 眀椀琀栀 瀀爀漀戀氀攀洀猀 漀爀 瀀甀稀稀氀攀猀⸀㰀⼀倀㸀ഀ਀਀ഀഀ

    Mathematics can be found in many sites on the internet. These sites range from discussions of specific topics to problem sites, to the history of mathematics, to teacher professional development, to games (including sites that claim to read your mind), to emergency rooms where you can ask for mathematical help. There are even more and varied sites that all help to make mathematics more accessible, if not popular.

    ഍਀㰀倀㸀㰀䈀㸀⠀挀⤀ 倀爀漀戀氀攀洀猀 䤀搀攀渀琀椀昀椀攀搀㰀⼀戀㸀㰀⼀倀㸀ഀ਀਀ഀഀ

    The difficulties that these contexts produce fall into two categories: development and applications. In the former category most new initiatives depend on a small number of people for their success. This makes them fragile. It seems often easier to find money to begin new projects than to find continuing support for them.

    ഍਀㰀倀㸀䈀礀 愀瀀瀀氀椀挀愀琀椀漀渀猀 眀攀 洀攀愀渀 愀瀀瀀氀椀挀愀琀椀漀渀猀 椀渀 猀挀栀漀漀氀猀⸀ 䤀琀 椀猀 渀漀琀 挀氀攀愀爀 琀栀愀琀 洀甀挀栀 漀昀 琀栀攀 渀攀眀 洀愀琀攀爀椀愀氀 愀瘀愀椀氀愀戀氀攀 椀猀 戀攀椀渀最 甀猀攀搀 猀甀挀挀攀猀猀昀甀氀氀礀 戀礀 最爀攀愀琀 渀甀洀戀攀爀猀 漀昀 琀攀愀挀栀攀爀猀 椀渀 琀栀攀 爀攀最甀氀愀爀 挀氀愀猀猀爀漀漀洀⸀ 吀栀椀猀 洀愀礀 戀攀 昀漀爀 愀 瘀愀爀椀攀琀礀 漀昀 爀攀愀猀漀渀猀⸀ 䘀椀爀猀琀Ⰰ 琀攀愀挀栀攀爀猀 愀爀攀 昀爀攀焀甀攀渀琀氀礀 瀀氀愀最甀攀搀 眀椀琀栀 琀椀洀攀 挀漀渀猀琀爀愀椀渀琀猀 愀猀 洀漀爀攀 洀愀琀攀爀椀愀氀Ⰰ 攀猀瀀攀挀椀愀氀氀礀 椀渀瘀漀氀瘀椀渀最 渀攀眀 猀甀戀樀攀挀琀猀 漀甀琀猀椀搀攀 漀昀 洀愀琀栀攀洀愀琀椀挀猀Ⰰ 攀渀琀攀爀猀 琀栀攀 猀挀栀漀漀氀 挀甀爀爀椀挀甀氀甀洀⸀ 吀栀攀猀攀 猀甀戀樀攀挀琀猀 爀攀搀甀挀攀 琀栀攀 琀椀洀攀 愀瘀愀椀氀愀戀氀攀 昀漀爀 洀愀琀栀攀洀愀琀椀挀猀⸀ 匀攀挀漀渀搀Ⰰ 攀猀瀀攀挀椀愀氀氀礀 椀渀 猀攀渀椀漀爀 猀攀挀漀渀搀愀爀礀 猀挀栀漀漀氀Ⰰ 栀椀最栀 猀琀愀欀攀猀 攀砀愀洀椀渀愀琀椀漀渀猀 昀漀爀挀攀 琀攀愀挀栀攀爀猀 椀渀琀漀 琀攀愀挀栀椀渀最 昀漀爀 琀栀攀 攀砀愀洀椀渀愀琀椀漀渀 爀愀琀栀攀爀 琀栀愀渀 搀攀瘀攀氀漀瀀椀渀最 洀愀琀栀攀洀愀琀椀挀愀氀 椀搀攀愀猀⸀ 䄀渀搀 琀栀椀爀搀Ⰰ 琀攀愀挀栀攀爀猀 洀愀礀 氀愀挀欀 琀栀攀 挀漀渀昀椀搀攀渀挀攀 琀漀 搀攀愀氀 眀椀琀栀 琀栀攀 渀攀眀 洀愀琀攀爀椀愀氀 琀栀愀琀 眀愀猀 渀漀琀 瀀愀爀琀 漀昀 琀栀攀椀爀 甀渀搀攀爀最爀愀搀甀愀琀攀 琀爀愀椀渀椀渀最⸀ 吀栀攀礀 洀愀礀 愀氀猀漀 戀攀 甀渀挀漀洀昀漀爀琀愀戀氀攀 眀椀琀栀 琀栀攀 洀漀爀攀 漀瀀攀渀 瀀攀搀愀最漀最礀 爀攀焀甀椀爀攀搀 昀漀爀 挀栀愀氀氀攀渀最椀渀最 猀椀琀甀愀琀椀漀渀猀 眀栀椀挀栀 愀爀攀Ⰰ 戀礀 琀栀攀椀爀 渀愀琀甀爀攀Ⰰ 氀攀猀猀 猀琀爀甀挀琀甀爀攀搀 琀栀愀渀 琀栀攀 琀爀愀搀椀琀椀漀渀愀氀 瀀攀搀愀最漀最礀⸀㰀⼀倀㸀ഀ਀਀㰀⼀氀椀㸀ഀഀ
  • ഍਀ഀ਀਀㰀倀㸀伀渀攀 最漀愀氀 昀漀爀 琀栀攀 匀琀甀搀礀 䌀漀渀昀攀爀攀渀挀攀 眀椀氀氀 戀攀 琀漀 最攀琀 愀 最漀漀搀 瀀椀挀琀甀爀攀 漀昀 眀栀愀琀 ഀഀ is the state of the art. Here are some examples of issues that may be considered in the context of this Study.

    ഍਀㰀倀㸀䤀洀瀀愀挀琀 漀昀 琀攀愀挀栀椀渀最 愀渀搀 氀攀愀爀渀椀渀最 椀渀 琀栀攀 挀氀愀猀猀爀漀漀洀㨀㰀⼀瀀㸀ഀ਀਀㰀唀䰀㸀ഀഀ
  • How do challenges contribute to the learning process?
  • Hoɷ caɮ challengeɳ bɥ useɤ iɮ thɥ classroom?
  • ഍਀㰀䰀䤀㸀䠀漀眀 洀甀挀栀 挀栀愀氀氀攀渀最攀 椀猀 瀀爀漀瘀椀搀攀搀 椀渀 挀甀爀爀攀渀琀 挀甀爀爀椀挀甀氀愀㼀㰀⼀氀椀㸀ഀ਀਀㰀䰀䤀㸀圀栀愀琀 昀甀爀琀栀攀爀 漀瀀瀀漀爀琀甀渀椀琀椀攀猀 琀漀 挀栀愀氀氀攀渀最攀 眀漀甀氀搀 攀渀栀愀渀挀攀 琀攀愀挀栀椀渀最 愀渀搀 氀攀愀爀渀椀渀最 椀渀 琀栀攀 爀攀最甀氀愀爀 挀氀愀猀猀爀漀漀洀㼀㰀⼀氀椀㸀 ഀഀ
  • How can teachers be made aware of the existence of the different types of challenges?
  • Hoɷ caɮ wɥ ensurɥ thaɴ thesɥ challengeɳ arɥ compatiblɥ witɨ thɥ mandateɤ syllabus?
  • ഍਀㰀䰀䤀㸀䠀漀眀 挀愀渀 琀椀洀攀 挀漀渀猀琀爀愀椀渀琀猀 椀渀 琀栀攀 挀氀愀猀猀爀漀漀洀 戀攀 栀愀渀搀氀攀搀㼀㰀⼀氀椀㸀ഀ਀਀㰀䰀䤀㸀䠀漀眀 挀愀渀 挀栀愀氀氀攀渀最攀猀 戀攀 攀瘀愀氀甀愀琀攀搀㼀㰀⼀氀椀㸀ഀഀ
  • How can students be evaluated in challenges?
  • Hoɷ caɮ thɥ effectivenesɳ oɦ usinɧ challenginɧ materialɳ bɥ supporteɤ bɹ thɥ gradinɧ system?
  • ഍਀㰀氀椀㸀圀栀愀琀 猀漀爀琀猀 漀昀 挀栀愀氀氀攀渀最攀猀 愀爀攀 愀瀀瀀爀漀瀀爀椀愀琀攀 昀漀爀 爀攀洀攀搀椀愀氀 愀渀搀 猀琀爀甀最最氀椀渀最 猀琀甀搀攀渀琀猀㼀㰀⼀氀椀㸀ഀ਀਀㰀䰀䤀㸀圀栀愀琀 愀爀攀 琀栀攀 椀洀瀀氀椀挀愀琀椀漀渀猀 昀漀爀 琀攀愀挀栀攀爀 琀爀愀椀渀椀渀最 漀昀 挀栀愀氀氀攀渀最攀猀 眀栀椀挀栀 愀爀攀 椀渀 琀栀攀 挀氀愀猀猀爀漀漀洀㼀㰀⼀氀椀㸀ഀഀ
  • What are the implications for teacher training for challenges which exist outside of the classroom?
  • Whaɴ backgrounɤ dɯ studentɳ neeɤ tɯ handlɥ challengɥ materiaɬ anɤ hoɷ caɮ thiɳ bɥ introduceɤ intɯ thɥ classroomȿ Thiɳ includeɳ familiaritɹ witɨ mathematicaɬ notatioɮ anɤ conventionsȬ abilitɹ tɯ reasoɮ anɤ draɷ conclusionsȬ abilitɹ tɯ observɥ anɤ classifɹ anɤ skilɬ aɴ communication.
  • ഍਀㰀䰀䤀㸀䠀漀眀 挀愀渀 ✀戀攀礀漀渀搀 挀氀愀猀猀爀漀漀洀 愀挀琀椀瘀椀琀椀攀猀✀ 氀椀欀攀 挀漀洀瀀攀琀椀琀椀漀渀猀Ⰰ 攀砀栀椀戀椀琀椀漀渀猀Ⰰ 挀氀甀戀猀Ⰰ 洀愀琀栀猀 昀愀椀爀猀 攀琀挀 椀渀昀氀甀攀渀挀攀 琀栀攀 挀氀愀猀猀爀漀漀洀 愀挀琀椀瘀椀琀椀攀猀 愀渀搀 氀攀愀爀渀椀渀最 椀渀 猀甀挀栀 愀 眀愀礀 琀栀愀琀 愀氀氀 猀琀甀搀攀渀琀猀 椀渀 琀栀攀 挀氀愀猀猀 愀爀攀 挀栀愀氀氀攀渀最攀搀 愀渀搀 洀漀琀椀瘀愀琀攀搀㼀㰀⼀氀椀㸀ഀ਀਀㰀䰀䤀㸀䠀漀眀 挀愀渀 琀攀愀挀栀攀爀猀Ⰰ 瀀愀爀攀渀琀猀 愀渀搀 猀琀甀搀攀渀琀猀 戀攀 洀愀搀攀 愀眀愀爀攀 琀栀愀琀 琀栀攀猀攀 欀椀渀搀猀 漀昀 愀挀琀椀瘀椀琀椀攀猀 愀渀搀 挀栀愀氀氀攀渀最攀猀 愀氀猀漀 眀椀氀氀 猀琀爀攀渀最琀栀攀渀 琀栀攀 氀攀愀爀渀椀渀最 愀渀搀 甀渀搀攀爀猀琀愀渀搀椀渀最 漀昀 戀愀猀椀挀 挀漀渀挀攀瀀琀猀 愀渀搀 猀欀椀氀氀猀 椀渀 洀愀琀栀攀洀愀琀椀挀猀㼀㰀⼀氀椀㸀ഀഀ
  • Can experience with competitions, maths fairs etc be part of teacher training and in service teacher education? And will this help to engage teachers in `beyond classroom activities' or implement these kinds of activities in classroom practice?
  • Hoɷ caɮ textbookɳ bɥ writteɮ sɯ thaɴ challenginɧ activitieɳ iɳ thɥ philosophɹ anɤ leadinɧ ideɡ behinɤ thɥ textbookȬ anɤ noɴ onlɹ fragmentaɬ partɳ oɦ thɥ contenɴ oɦ thɥ book?
  • ഍਀㰀䰀䤀㸀䠀漀眀 挀愀渀 琀攀挀栀渀漀氀漀最礀 戀攀 甀猀攀搀 戀礀 琀攀愀挀栀攀爀猀 愀渀搀 猀琀甀搀攀渀琀猀 琀漀 挀爀攀愀琀攀 挀栀愀氀氀攀渀最椀渀最 攀渀瘀椀爀漀渀洀攀渀琀猀㼀㰀⼀氀椀㸀ഀ਀਀㰀⼀甀氀㸀ഀഀ

    Beyonɤ classrooɭ activitiesHow can one make visible the mathematics behind everyday technological devices, and how can this be put into a context that is accessible and mathematically challenging for different groups of people? ഍਀ഀ਀਀㰀倀㸀㰀䈀㸀刀攀猀攀愀爀挀栀㰀⼀戀㸀㰀⼀倀㸀ഀഀ

    ഍਀㰀倀㸀㰀戀㸀䴀漀爀攀 最攀渀攀爀愀氀 焀甀攀猀琀椀漀渀猀㰀⼀戀㸀㰀⼀瀀㸀ഀ਀਀㰀倀㸀㰀甀氀㸀ഀഀ
  • How can the mathematics and mathematics education community be involved in this kind of challenging activity that goes beyond their own research interests?
  • Arɥ therɥ somɥ brancheɳ oɦ mathematicɳ thaɴ arɥ morɥ suitablɥ foɲ producinɧ challenginɧ problemɳ anɤ situations?
  • ഍਀㰀䰀䤀㸀䠀漀眀 挀愀渀 搀椀昀昀攀爀攀渀琀 搀攀猀椀最渀猀 漀昀 挀栀愀氀氀攀渀最椀渀最 愀挀琀椀瘀椀琀椀攀猀Ⰰ 椀渀 瀀愀爀琀椀挀甀氀愀爀 挀漀洀瀀攀琀椀琀椀漀渀猀Ⰰ 愀琀琀爀愀挀琀 搀椀昀昀攀爀攀渀琀 最爀漀甀瀀猀 漀昀 瀀攀漀瀀氀攀 ⠀琀栀攀 瘀攀爀礀 愀戀氀攀 猀琀甀搀攀渀琀猀Ⰰ 最攀渀搀攀爀Ⰰ 挀甀氀琀甀爀愀氀 搀椀昀昀攀爀攀渀挀攀猀Ⰰ 搀椀昀昀攀爀攀渀琀 愀挀栀椀攀瘀攀洀攀渀琀 攀琀挀⤀㰀⼀氀椀㸀ഀ਀਀㰀䰀䤀㸀圀栀愀琀 挀愀渀 戀攀 搀漀渀攀 琀漀 椀搀攀渀琀椀昀礀Ⰰ 猀琀椀洀甀氀愀琀攀 愀渀搀 攀渀挀漀甀爀愀最攀 琀栀攀 洀愀琀栀攀洀愀琀椀挀愀氀氀礀 琀愀氀攀渀琀攀搀 猀琀甀搀攀渀琀猀㼀㰀⼀氀椀㸀ഀഀ

    ഍਀ഀ਀਀㰀䠀刀㸀ഀഀ ഍਀㰀䰀䤀㸀㰀栀㐀㸀䌀愀氀氀 昀漀爀 䌀漀渀琀爀椀戀甀琀椀漀渀猀㰀⼀栀㐀㸀ഀ਀਀ഀഀ

    The work of this Study will take place in two parts. The first consists of a Conference to take place in Trondheim, Norway, from 27 June to 03 July 2006. The Conference will be a working one. Every participant will be expected to be active. Participation is by invitation only, based on a submitted contribution. Among the attendees, it is planned to represent a diversity of expertise, experience, nationality and philosophy. Such attendance should be drawn broadly from the mathematics and mathematics education community. It is hoped that the Conference will attract not only long term workers in the field but also newcomers with interesting and refreshing ideas or promising work in progress. In the past, ICMI Study Conferences have included about 80 participants.

    ഍਀㰀倀㸀吀栀攀 䤀倀䌀 栀攀爀攀戀礀 椀渀瘀椀琀攀猀 椀渀搀椀瘀椀搀甀愀氀猀 漀爀 最爀漀甀瀀猀 琀漀 猀甀戀洀椀琀 挀漀渀琀爀椀戀甀琀椀漀渀猀 漀渀 猀瀀攀挀椀昀椀挀 焀甀攀猀琀椀漀渀猀Ⰰ 瀀爀漀戀氀攀洀猀 漀爀 椀猀猀甀攀猀 爀攀氀愀琀攀搀 琀漀 琀栀攀 琀栀攀洀攀 漀昀 琀栀攀 匀琀甀搀礀 昀漀爀 挀漀渀猀椀搀攀爀愀琀椀漀渀 戀礀 琀栀攀 䌀漀洀洀椀琀琀攀攀⸀  吀栀漀猀攀 眀栀漀 眀漀甀氀搀 氀椀欀攀 琀漀 瀀愀爀琀椀挀椀瀀愀琀攀 猀栀漀甀氀搀 瀀爀攀瀀愀爀攀 ഀ਀਀ഀഀ

    (a) a one-page listing of their current position and contact information, as well as of their past and present publications and activities pertinent to the theme of the Study;

    (bȩ ɡ papeɲ oɦ 6-1Ȱ pageɳ addressinɧ matterɳ raiseɤ iɮ thiɳ documenɴ oɲ otheɲ issueɳ relateɤ tɯ thɥ themɥ oɦ thɥ Study.

    ഍਀ഀ਀਀㰀倀㸀 倀爀漀瀀漀猀愀氀猀 昀漀爀 爀攀猀攀愀爀挀栀 琀栀愀琀 椀猀 漀渀 椀琀猀 眀愀礀Ⰰ 漀爀 猀琀椀氀氀 琀漀 戀攀 挀愀爀爀椀攀搀 漀甀琀Ⰰ 愀爀攀 愀氀猀漀 眀攀氀挀漀洀攀⸀  刀攀猀攀愀爀挀栀 焀甀攀猀琀椀漀渀猀 猀栀漀甀氀搀 戀攀 挀愀爀攀昀甀氀氀礀 猀琀愀琀攀搀 愀渀搀 愀 猀欀攀琀挀栀 漀昀 琀栀攀 漀甀琀挀漀洀攀 ⴀ 愀挀琀甀愀氀 漀爀 栀漀瀀攀搀 昀漀爀 ⴀ 猀栀漀甀氀搀 戀攀 瀀爀攀猀攀渀琀攀搀Ⰰ 椀昀 瀀漀猀猀椀戀氀攀 眀椀琀栀 爀攀昀攀爀攀渀挀攀 琀漀 攀愀爀氀椀攀爀 愀渀搀 爀攀氀愀琀攀搀 猀琀甀搀椀攀猀⸀㰀倀㸀ഀഀ

    Thesɥ documentɳ shoulɤ bɥ submitteɤ nɯ lateɲ thaɮ Augusɴ 31Ȭ 2005Ȭ tɯ botɨ co-chairɳ oɦ thɥ Studɹ eitheɲ bɹ postȬ bɹ facsimilɥ oɲ (preferablyȩ bɹ e-mailȮȠ Alɬ sucɨ documentɳ wilɬ bɥ regardeɤ aɳ inpuɴ tɯ thɥ planninɧ oɦ thɥ Studɹ Conferencɥ anɤ wilɬ assisɴ thɥ IPɃ iɮ issuinɧ invitationɳ nɯ lateɲ thaɮ Januarɹ 31Ȭ 2006ȮȠ Alɬ submissionɳ musɴ bɥ iɮ EnglishȬ thɥ languagɥ oɦ thɥ Conference.

    ഍਀ഀ਀਀㰀倀㸀吀栀攀 挀漀渀琀爀椀戀甀琀椀漀渀猀 漀昀 琀栀漀猀攀 椀渀瘀椀琀攀搀 琀漀 琀栀攀 䌀漀渀昀攀爀攀渀挀攀 眀椀氀氀 戀攀 洀愀搀攀 愀瘀愀椀氀愀戀氀攀 琀漀 漀琀栀攀爀 瀀愀爀琀椀挀椀瀀愀渀琀猀 戀攀昀漀爀攀栀愀渀搀 愀猀 瀀爀攀瀀愀爀愀琀椀漀渀 洀愀琀攀爀椀愀氀⸀  倀愀爀琀椀挀椀瀀愀渀琀猀 猀栀漀甀氀搀 渀漀琀 攀砀瀀攀挀琀 琀漀 瀀爀攀猀攀渀琀 琀栀攀椀爀 瀀愀瀀攀爀猀 漀爀愀氀氀礀 愀琀 琀栀攀 䌀漀渀昀攀爀攀渀挀攀Ⰰ 愀猀 琀栀攀 䤀倀䌀 洀愀礀 搀攀挀椀搀攀 琀漀 漀爀最愀渀椀稀攀 椀琀 椀渀 漀琀栀攀爀 眀愀礀猀 琀栀愀琀 昀愀挀椀氀椀琀愀琀攀 琀栀攀 匀琀甀搀礀✀猀 攀昀昀攀挀琀椀瘀攀渀攀猀猀 愀渀搀 瀀爀漀搀甀挀琀椀瘀椀琀礀⸀㰀⼀倀㸀ഀഀ

    Unfortunatelɹ aɮ invitatioɮ tɯ participatɥ iɮ thɥ Conferencɥ doeɳ noɴ implɹ financiaɬ supporɴ froɭ thɥ organizersȬ anɤ participantɳ shoulɤ financɥ theiɲ owɮ attendancɥ aɴ thɥ ConferenceȮȠ Fundɳ arɥ beinɧ soughɴ tɯ providɥ partiaɬ supporɴ tɯ enablɥ participantɳ froɭ non-affluenɴ countrieɳ tɯ attenɤ thɥ ConferenceȬ buɴ thɥ numbeɲ oɦ sucɨ grantɳ wilɬ bɥ limited.

    ഍਀ഀ਀਀㰀倀㸀吀栀攀 猀攀挀漀渀搀 瀀愀爀琀 漀昀 琀栀攀 匀琀甀搀礀 椀猀 愀 瀀甀戀氀椀挀愀琀椀漀渀 眀栀椀挀栀 眀椀氀氀 愀瀀瀀攀愀爀 椀渀 琀栀攀 䤀䌀䴀䤀 匀琀甀搀礀 匀攀爀椀攀猀⸀  吀栀椀猀 匀琀甀搀礀 嘀漀氀甀洀攀 眀椀氀氀 戀攀 戀愀猀攀搀 漀渀 猀攀氀攀挀琀攀搀 挀漀渀琀爀椀戀甀琀椀漀渀猀 猀甀戀洀椀琀琀攀搀 愀猀 眀攀氀氀 愀猀 漀渀 琀栀攀 漀甀琀挀漀洀攀猀 漀昀 琀栀攀 䌀漀渀昀攀爀攀渀挀攀⸀  吀栀攀 攀砀愀挀琀 昀漀爀洀愀琀 漀昀 琀栀攀 匀琀甀搀礀 嘀漀氀甀洀攀 栀愀猀 渀漀琀 礀攀琀 戀攀攀渀 搀攀挀椀搀攀搀 戀甀琀 椀琀 椀猀 攀砀瀀攀挀琀攀搀 琀漀 戀攀 愀渀 攀搀椀琀攀搀 挀漀栀攀爀攀渀琀 戀漀漀欀 眀栀椀挀栀 椀琀 椀猀 栀漀瀀攀搀 眀椀氀氀 戀攀 愀 猀琀愀渀搀愀爀搀 爀攀昀攀爀攀渀挀攀 椀渀 琀栀攀 昀椀攀氀搀 昀漀爀 猀漀洀攀 琀椀洀攀⸀㰀⼀倀㸀ഀഀ

    Internationaɬ Programmɥ Committee

    ഍਀㰀倀㸀䌀漀ⴀ挀栀愀椀爀㨀 䔀搀眀愀爀搀 䨀⸀ 䈀愀爀戀攀愀甀ഀ਀਀㰀⼀戀爀㸀唀渀椀瘀攀爀猀椀琀礀 漀昀 吀漀爀漀渀琀漀Ⰰ 䌀䄀一䄀䐀䄀㰀⼀倀㸀ഀഀ

    Co-chair: Peter J. Taylor
    Universitɹ oɦ CanberraȬ AUSTRALIA

    ഍਀㰀瀀㸀䌀栀愀椀爀Ⰰ 䰀漀挀愀氀 伀爀最愀渀椀猀椀渀最 䌀漀洀洀椀琀琀攀攀㨀 䤀渀最瘀椀氀氀 䴀⸀ 匀琀攀搀礀ഀ਀਀㰀戀爀㸀一漀爀眀攀最椀愀渀 唀渀椀瘀攀爀猀椀琀礀 漀昀 匀挀椀攀渀挀攀 愀渀搀 吀攀挀栀渀漀氀漀最礀Ⰰ 吀爀漀渀搀栀攀椀洀Ⰰ 一伀刀圀䄀夀㰀⼀瀀㸀ഀഀ

    Mariolina Bartolini Bussi
    Universitͷ dɩ Modenɡ eɴ Reggiɯ EmiliaȬ ITALY

    ഍਀㰀倀㸀䄀氀戀爀攀挀栀琀 䈀攀甀琀攀氀猀瀀愀挀栀攀爀ഀ਀਀㰀䈀刀㸀䴀愀琀栀攀洀愀琀椀猀挀栀攀猀 䤀渀猀琀椀琀甀琀Ⰰ 䜀椀攀尀猀猀 攀渀Ⰰ 䜀䔀刀䴀䄀一夀㰀⼀倀㸀ഀഀ

    Patricia Fauring
    Buenoɳ AiresȬ ARGENTINA

    ഍਀㰀倀㸀䐀攀爀攀欀 䠀漀氀琀漀渀ഀ਀਀㰀䈀刀㸀唀渀椀瘀攀爀猀椀琀礀 漀昀 伀琀愀最漀Ⰰ 䐀甀渀攀搀椀渀Ⰰ 一䔀圀 娀䔀䄀䰀䄀一䐀㰀⼀倀㸀ഀഀ

    Martine Janvier
    IREMȬ Lɥ MansȬ FRANCE

    ഍਀㰀倀㸀嘀氀愀搀椀洀椀爀 倀爀漀琀愀猀漀瘀ഀ਀਀㰀䈀刀㸀䴀漀猀挀漀眀 匀琀愀琀攀 唀渀椀瘀攀爀猀椀琀礀Ⰰ 刀唀匀匀䤀䄀㰀⼀倀㸀ഀഀ

    Ali Rejali
    Isfahaɮ Universitɹ oɦ TechnologyȬ IRAN

    ഍਀㰀倀㸀䴀愀爀欀 䔀⸀ 匀愀甀氀ഀ਀਀㰀䈀刀㸀䜀愀琀攀眀愀礀 䤀渀猀琀椀琀甀琀攀Ⰰ 䌀椀琀礀 唀渀椀瘀攀爀猀椀琀礀 漀昀 一攀眀 夀漀爀欀Ⰰ 唀匀䄀㰀⼀倀㸀ഀഀ

    Kenji Ueno
    Kyotɯ UniversityȬ JAPAN

    ഍਀㰀倀㸀䈀攀爀渀愀爀搀 刀⸀ 䠀漀搀最猀漀渀Ⰰ 匀攀挀爀攀琀愀爀礀ⴀ䜀攀渀攀爀愀氀 漀昀 䤀䌀䴀䤀ഀ਀਀㰀䈀刀㸀唀渀椀瘀攀爀猀椀琀 䰀愀瘀愀氀Ⰰ 儀甀戀攀挀Ⰰ 䌀䄀一䄀䐀䄀㰀⼀倀㸀ഀഀ

    Advisorɳ froɭ ICMɉ Executivɥ Committee

    ഍਀㰀倀㸀䴀愀爀椀愀 䘀愀氀欀 搀攀 䰀漀猀愀搀愀ഀ਀਀㰀䈀刀㸀唀渀椀瘀攀爀猀椀琀礀 䄀渀琀漀渀椀漀 一愀爀椀䠀漁Ⰰ 䈀漀最漀琀愀Ⰰ 䌀伀䰀伀䴀䈀䤀䄀㰀⼀瀀㸀ഀഀ

    Petar Kenderov
    Academɹ oɦ SciencesȬ SofiaȬ BULGARIA

    ഍਀ഀ਀਀㰀栀㐀㸀䤀渀焀甀椀爀椀攀猀㰀⼀栀㐀㸀ഀഀ

    Inquiries on all aspects of the Study, suggestions concerning the content of the Study Conference and submission of contributions should be sent to both co-chairs:

    ഍਀㰀倀㸀倀爀漀昀⸀ 䔀搀眀愀爀搀 䨀⸀ 䈀愀爀戀攀愀甀ഀ਀਀㰀䈀刀㸀䐀攀瀀愀爀琀洀攀渀琀 漀昀 䴀愀琀栀攀洀愀琀椀挀猀ഀഀ
    University of Toronto
    TorontɯȠ M5ɓ 3G3 ഍਀㰀䈀刀㸀䌀䄀一䄀䐀䄀㰀⼀瀀㸀ഀ਀਀ഀഀ

    Tel: +1 416 653 1961
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