ĵ > Holonomy??Costa c Garmston????Arthur Koestler??^c??????x???????????????e????????????

Holonomy??Costa c Garmston????Arthur Koestler??^c??????x???????????????e????????????

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??學??????與發????? Research and Development in Science Education qualterly 2011????????????75-96頁 2011, No.62, 75-96 75
國階???數學????設計???
蘇 ?? ??
臺匇??????????學數學? yiwen@tmue.edu.tw (????????2011.5.9???????????2011.6.30???????????2011.7.21)
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??論??闡??????設計國階???數學???????況?????????為現場?? 師???????級??學?參??????????詮釋學??????????課??標準??數學?識 內???課??編????????進??數學????尋訪??從??選擇??適??????數學 ?????進??詮釋??關數學?維????論發???過?????????????對????????? ??與???????????數學詵??兩??數學????設計?????鋪陳??內????紹? ??為??????????發??數學?????適????????????問???????請 24 ??國??職??師尌??適??適?????國????????問題????????????? ??對??數學????設計??? 1.????對學?????????????壓??????????譯 ???話???2.???設計與????領??連結?數學?????????? 關鍵???國??數學???數學詵

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??學??????與發?????????????? 76
??????
?????貫??數學學習領??????協??學?數學?????發???????數學? ?????????????師??學裡????進與??題??關??數學?題???對學???學習會?????????
義????????協??學?將????觀???體??????為??論????應??層???????????數學??? ???????進確????實?????????????????啟發???數學??????讀??實屬???????(????
????2008??頁 22)????過??????書??編???????們並沒??????關??題??歷? ??脈絡????????書裡??數學?????????????????數學??????數學?? 備?識較????適??????國課????還??現??????數學???????????? ???為???????適??國階??數學???????爭????實??雖?臺灣??學 ??數學??尌??現優異????數學??????學??討厭???????選(??????統計處?? 1998)???? TIMSS (Trends in International Mathematics and Science Study) 2007 為 ????臺灣????級學??數學??尌??????????級學????????????????臺灣學?? 對數學學習??????態?????????卻??顯???國際帄????績?? 經????數國數學?????數學??學實??????們發現?過數學???為學???? 開數學?????陎紗??讓學?????數學????????????陎??強??們??數學學習態 ??(???龍??蘇??????2009?????????蘇??????2009?????????蘇??????2009)?? 雖?數學?????數學??學??實???????經???紹數學?維??發?歷???讓學?? 體會數學????類???動????????數學??發?與??會???脈絡????關??進?? 開??學????視?(??萬????2000a????萬??????2009)??????國數學??????尋 ?????尌???師???????實??????個陎臨?????????處尋??問題?????? ????針對現??國?????關????題????尋???並設計???????級課???數學 ????????????國??師參?????數學??學時適當????????????詳?? 數學?????設計????並經????職國??師??檢驗???????發數學????應 該??????項??與對數學??????學??興????師?????

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國階???數學????設計??? 77
貳??數學?與數學??學??關??論
???運??數學??數學??學?????
??數學??學??輔??歷??????從 1970 ?????來??廣??國際數學??????? ??矚?????謂??數學?與數學??學??關聯?國際???(International Study Group on the Relations between the History and Pedagogy of Mathematics [HPM]) ?????隸屬?國際數學?????員會 (International Commission on Mathematical Instruction [ICMI])???個????????????動數學?與數學??學? 關聯??換?????????數學?學對數學???????種應????????????數學????? ????????數學?與數學???????動??來????數學??師???學?質與學????學 習???? ???為?????歷?維??????數學課?????????學???????????????? ??發學????數學???興?(Furinghetti & Paola, 2003)??讓??們對???????論?? ?????瞭??對??????????視??幫??數學學習??(Barbin, 2000; Gulikers & Blom, 2001; Tzanakis & Arcavi, 2000)??綜??來說????數學?????數學??學????達??? ?標???????蓋??個範疇????尌??????認????????動????麼實際???數學 ????國現??國??課??綱領????????麼樣???????Fasanelli (2000)???? ?????個國???現???????????????認????????層陎??內??????????? (?)??????????關????(??????國???臘??義????????蘭????蘭??國??) ??發學習動機??????學習興???????數學??傳記啟發學??????長?? (??)????認???關????(????) 幫??學??經??歷?脈絡?????數學????發??????????讓學????較????? 題????????????????對數學??單????????? (??) ?????????動??關????(????????????麥????國??紐??蘭?????????國??國 ??) 經??認識?????????????數學發?????視數學??????達??????義??學 習??

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??學??????與發?????????????? 78
??????????數學??????????????????學?????k??蓋????認????? ????動??陎??????來??陎臨??問題??????? HPM ??學???師應該???準備 ??適??數學????????關?課?????????歷?????準備徑??Furinghetti ?? Paola (2003)??????????認為應該??瀏覽數學??????????????興???時間 點???????????尋訪???????來????準備計畫?????????????????? ???參????????準備徑???為?線進??尌頗??????????為??對??師實 ??????????????????單?????????????????紹???書內?????數學??? ??開?瀏覽?????尋????????????????為這?連????準備過???應??雙 ??遞迴?????動態???當尋訪???時????們??????會?????們????興?????? 數學???????????時???????????????????????們?????????瀏 覽數學????????處???????????數學???關??參??資???當???這???? ????????點???????尋?????時????師??頇??茲??茲??????頇???????? ???????檢視???與 HPM ?????關係????????編製??來??????????????? 實 HPM ?????達????學????????實????? HPM ?????來????尌??? Tzanakis ?? Arcavi (2000)????????????????獻???學??????????學???????為? ??????????學?????佔??????????????來???????????將針對 HPM ?? 學????開發??????詳細??說????
????HPM ??學??????發?????
??關 HPM ??學??????製?????????????點???????學????學習興??? ??????????尌??說?????從純數學????????來??????數學????帄傳 記????????????????無??據??傳聞軼?????????????傳記??內??????? 應該隨時?????????傳記資??(Pycior, 1987)???????數學?????課???學??? ??應????????尌?????師????????與???????當?????????????來???? ??????發???Hans Freudenthal ??張????導?????發????(guided re-invention)?? ????數學??學時??????們??應該???發?????歷???跡??????經????過?時 ??????導??歷?過?????(Freudenthal, 1973, pp. 101, 103)??????學??應該?? 機會??類???數學???????發???????經驗??數學?????發???????時學 ??並???????????發???????????師????導??獲????????識??當??讀數學?? 題時????們??頇設?????學????????????歷???獲???????師?????歷???

Page 5
國階???數學????設計??? 79
?????學????隨??歷???徑??瞭????????數學????發????????們??學 習???????????發??這??題(van Maanen, 2001)??這個??????????????設 計數學????時??參???? 從 HPM ??????來??????師??????數學發????參??與???與???書單? ??關?適當??歷?????讓學?????觀?????為??張?????????(??萬????2002)?? ??張????????????譯??啟????????數學??????啟??????學??尌?????學 時??當學習????數學????時?????體????????動學習??啟??????學???????? ???????????尌??????確??達?????數學??????發??????尌?????續?? ??關?????學習時?????還????學?????????讓學????????????????樂學 ????????????學????學習動機????個屬??(鄭?????2000)???????貫課????強 調??????????????????項???進???學習與國際瞭????認識並?????? ???????????與??賞??國?????????歷??????????數學學習領????連結?? ??題來說?????覺?????則?標???C-R-04 ???覺數學與??類?????動??關???? 這???? HPM ??????場?處????數學課??????數學????????讓學?????數學 ????類???動??瞭????????樣?????養????????關懷?????????讓學??體 會數學??????陎????適當????????????????????當??數學??學??? 範????幫??學??對?數學????????體會??啟發?? ????瞭??歷???????尋??準備????????麼??實???????師應該??? ???數學??數學??學???當??師??課????運??數學?時?????????兩種??? (Menghini, 2000)???種??舉??????????題???沒??強調??處(implicit use of history)??這??較為簡單??????尌??僅採??????這????歷?並?????標?? ????個??學??計畫???????????種尌???????????(explicit use of history)????????訴學????????處????紹數學????處時??????????會??經濟 脈絡??並?????????歷??????進??當??師採??????進??學時????們??頇 ??慮??課時間?????????????還??實時????師們??????對歷?????? ?過??隨??????歷?徑????????現??數學?識???質??當???無論???種?? ?進????師????頇謹記????????????過強調歷?陎??????歷?與數學 ??開????為??數學課??????進歷?維???????????為??幫??數學??學(???? ???2002????萬????1999)?? ??闡????數學?與數學??學???關??論???????將對??????設計?? 進???????說????

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??學??????與發?????????????? 80
參????????
?????設計
??數學?????設計???????從詮釋學???????發???????? Jahnke ???雙 ???(twofold circle)?????(Jahnke??1994)?? Horng (2004)??詮釋學(?環)??陎體 ??發???來?? HPM 實??????(圖 1)(蘇??????2004??2007??Su, 2006)?????釋?? ????????????? T ?????師??學????圓?? E ??????書編???S ????課??標 準??K ????數學?識??C ??????書內?????圓?? M ????數學???O ??數學 ?????T ??數學??論?????? I 則??體會???環????????詮釋???? E??S/K ?? C 這??????環組???? C1 ??圓??M??O ?? T ??????環則組???? C2 ??圓????? ??????構???????將??????????????對???選??兩個????級數學??? ???設計????????闡釋??
T ??師??學 E S/K C M O T I ????詮釋 C1 ???書編???課 ??標準與???書 內????構?? C2 ???數學???數 學??論??數學?? ????構??
圖 1 HPM 實??????

Page 7
國階???數學????設計??? 81
(?)??????與??????數學????設計???? 對??????與???????????題????????????尋?????貫課??綱?數學 學習領?????關????細???對?????????強調???????????????????? 貫數學學習領??????細? 5-n-14 ?????認識????????????????應??(?????? ????????????)???????????說?????????????????數課題????????學 習????強調???????佔?體????與????????????課綱????學?????處 ?????????與?????????????問題??????觀??現???書???單?數學 ?????????????????????數學?????????????國問??????????數學 ??簡????—????數書?????題?為???????????????????賞??讓閱讀??領?? 數學風?????讓學????讀題??練習處???????與????????????????問 題??????課綱????細? 5-n-12 ??說???????????????????????????? 學?應???????義??記??與應????????????????????????????記號 ??發?過???????學???????
?? C2 ?環????????選??????數書????為????題?????數學????????
書????? 1983 ??發現???國??匇???張???漢?????據????????曆譜記 載??????????漢呂?????(???? 186 ??)寫????????簡????????????數書?? ????僅???????????????國??學?????這??書???????對???國????時 ????學風?????????????????????關??問題???現??這??書??????? ??????運????數學???????學??領??數學????風?????數學????類?? ??動??獲????靈????與?????????數書??????數學問題???????編??????? ?運???條???????題????尌???現???個??題????這類??題??????? ?????幫??學習?????體????題舉??????養????題???????????????決實際 ??????????????問題??????????數書????題??????題????????幾個??為???? ????編號為 12 ??題??尌??????關????題????淺顯??動??????題??????????? ??數學??問題??歷歷?????????認為????發學??興????????實際??題過 ??領??數學與???????關?????將該題佈???????? ??關?????數學????????????????????????號??學??????????? ??會???????????對??????變?????慣????號??%????這???發???陏 ???書?????????闕???實?????????這個詞來?????朮為 100 ???數?? ??開???????????紀????業??術???????????尋資?????學????紹這? ????進????????讓學????????學習時?????覺??進????運???處?????

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??學??????與發?????????????? 82
????問題??????經過 C1 ?? C2 兩個圓???環????經過????詮釋????????? 編寫??????與??????數學????? (??)???數學詵??數學????設計???? 關??數學詵??題??設計???????? C1 ?環????????????????貫?? 課??綱???連結??????題???????題???????覺??轉??????題??溝?與評?? ????項???????覺????強調????????覺??????與數學??關?????????覺數 學與????領???間????連結???????數學????應????????學????會??學?? ?????數學???進??類???發??????體???????轉????強調????????? ??與問題??關??數?????????????????????數?????????關係??數學語??? 達???????????與數學??關??資??資訊???????????問題轉????數學??問 題??????????嘗詴????數學詵??????讓學???曉數學?識與??類??動? 關聯??????詵????轉換??數學題???????複雜??問題為??????題 ???????(??題)????數學語???現??題??過??(溝?)????????題??結??闡釋? 來????問題(評??)??????????????貫課??綱??強調??學習領???統???? ???????????尋訪???數學詵??佈??????????並????????????尋 ?????????與紀曉嵐????????應???對聯??與??國語??結???讓學????語 ??課?????賞???????戰並???數學???? ?? C2 ?環???????????????關數學?資???????決??從????數學??? ??(1592)?????統????書??擷?????????????統???????國???數學 ??發?過????佔?????????傳?極為廣????長???對??國????間????? ???????響?????????這??書還傳?????鮮???????東??亞?????對? 這???傳????????????????????這??書???????數學詵題屬?簡 單???????題???????認為???????學????????術?????????體會??術 ???數???異??開??學????視??????????????時??讀書極為廣????對書 ????數學?????興??????沒?????????從 20 歲?????長?????????帶經??? ??為??業計????????隨時????數學????訪??師????????數學書?????鑽?? ???????????約 40 歲時??????????鄉??專??????????參??????學 說????????????見??? 60 歲時???????傑??????統?????????認為這 ????????????陰????????????????經歷????????學???????讓學??緬 懷????????興????為???????????壯????????數學詵?????????? ????帄??紹???????樣經過 C1 ?? C2 兩個?環??????????詮釋???????編

Page 9
國階???數學????設計??? 83
寫???數學詵??數學?????
????資?????與????
為??????????發??數學?????適???????數學??????????? ????請國??職??師 24 ??閱覽??並尌??適??適?????國????????????適?? ????國????適???幾??級??學???????個單??????問題??寫???????評論 ??見??????結???節?????????????設計?數學?????內????並 ??????職??師????寫???見??對?????兩??國數學???????檢驗?? ???????????????問??調??進??關???數學??????見??????? ?????? 24 國??師???????師?? 7 ??????師?? 17 ??帄????學??資約 12 ??????與???數??國??師?樣????並????經驗過數學?????數學??學 ?歷?????資????????????????國??師對??????與??????????? 數學詵??兩?????內?????見??進??????與????????????現??????實?? 達??師???????編碼???????????碼 T ?????師?????碼?????碼?? ???師???號?????? T17 ???編號為 17 號???師??
??????結??
???節???????????現兩??國數學?????設計內???????????國 ??職??師???見對??數學???????檢驗??????尌兩??數學?????別???? 詳????
???設計內??
(?)??????與??????數學???? ??? 1983 ??????國??匇???張???漢???????學??發現????僅??? ?????????????國??學?????????數書????????數書?????????簡書????據 ????????曆譜記載????????????漢呂?????(???? 186 ??)寫????????簡 ??????????數書?????????對???國????時????學風???????????????

Page 10
??學??????與發?????????????? 84
?????數學????討並??這??數學???????過??????????對?????時 ???數學?識??動??總????????????術????內???????????????現????? 據????數書????內?????們對????數書????????????術??與????數學????? 間??關係??????獲???較??體??觀????(??萬????2000b)??????數書??????簡 ???? 190 ???長 29.6??30.4 ?????寬 0.6??0.7 ????????????數學問題?????? ?編????????運???條???????題????尌???現???個??題????這類 ??題???????????幫??學習?????體??舉??????養????題??????????????? 決實際??????????????問題??????數書????題??????題????????幾個??為???? ????編號為 12 ??題??尌??????關??這???題與??????關??問題??
?????????關????????錢???謂????謂????爾??????????當?????問????幾?? ???????????錢?????????????錢?????????????錢???????術??????????? ???為????????????為實??實??????
??時??攜帶貨???關??頇繳??關稅??關?題?????????們???????????? ???????攜帶??????????????關???????繳?? 111 錢??關稅??帶???? ????對帶??????????帶?????????對帶?????????說??????帶??????價?? ?????兩???????????繳??兩????關稅??????請問?????稅???????? 這???題簡單?????問題??????瞭題???????????尌???????帶?????? ??應繳???稅???
7 1 ??帶?????????應繳???稅??? 7 2 ??帶?????????應繳?
??稅???
7 4 ?????尌???????????????國數學???????現??風?????問
題???????????????術???????說????題??????? ?????????這個詞來?????朮為 100 ???數????開???????????紀 ????業??術??當時?????????????????????這樣??習慣?????業?? ??????來??並進?????????????????(???????)為??礎貨幣?統??國?? ??強????????確?????????????????為?進??術?個??別?????? ???????號經歷???幾個??紀????進????約????? 1425 ??時??????????寫?? ??per 100???????????? 1650 ????漸變??per
0 0 ?????????簡???? 0 0 ????
????尌變???????們??慣????號??%??(Berlinghoff & Gouvea, 2004????萬???? ??????? HPM 團隊譯??2008)??

Page 11
國階???數學????設計??? 85
(??)???數學詵??數學???? 數學??????詵嗎????? 14 ??紀?? 17 ??紀??約??????間????國數學??顯 ????點??尌??從????????論??幾???頓?????應?????術????????????? ??詵?݌??訣????這種??況??實與當時??業??廣????會???????關??這? ??時???稱為??業數學???時??????????????????????時??讀書極為廣 ????對書????數學?????興??????沒?????????從 20 歲?????長???????? ?帶經?????為??業計??????????隨時????數學????訪??師????????數學 書?????鑽??時??會????????約 40 歲時??????????鄉??專?????? ????參??????學說????????????見??? 60 歲時???????傑??????統??? (1592)??這??書???? 17 ???記載????詳備???????識??????????稱??參會諸 ???說?????????????????編??????????國???數學??發?過???????? ??統????????????????????????傳極為廣????長???對??國????間????? ???????響?????????還傳?????鮮???????東??亞?????對?這?? ??傳????????????????????萬曆???????(1598)?????對????統???進 ??刪減???????????????編為????篹??????? ????統????書??????數學詵題屬?簡單???????題?????????? 學????????術?????????體會??術???數???異??開??學????視???????? ??饅頭?|???饅頭???????????個??無爭??????????個????????幾??????題? ????????????個???????個饅頭?????????????個饅頭??????????? ?個??請問???????????個饅頭?????????????????????饅頭?? ???個?????????????????饅頭?????個????這個題?????術???????? ?? 100??(3??1)??25?????????? 25 ???饅頭數 75 個??????????尌顯?? ??見???當???題???數????設??????? x ???????(100-x)?????
100 3 ) 100( 3
= - +
x x
????????????適??????腦????盪??還???題??書 ?????}??講??????????書?裡????學??搖頭??腦????讀書????們讀 ?麼書??????詵????????秓????????????種書??????為書????夠?????? ??? 3 個???讀??????詵????4 個????????????秓????5 個???讀??????? ?????現????經???這??類書??????? 94 ??????????學??總??????????? 動腦??????????讓??們????賞????詵??

Page 12
??學??????與發?????????????? 86 ?詩????????書???????冊??無餘?? ?詩?冊???????????????????? ????????讀?????????樣幾??書?? ??見學??????數??請???????躊????
這??題?????? x ????學??????數????麼??為 3 個?????????詵??????? 詵尌??
3 x
?????樣????????秓??
4 x
??????????
5 x
????題???經??訴??們?詵?? ??秓????????種書???? 94 ??????????們??????
3 x
??
4 x
??
5 x
??94 這個???? ?????o?????現過饒??興??對聯??????????????????歲??????雙慶???????
???????這????????紀曉嵐??對????????參????????賓???????長
?????紀(???????1997)??????聯??題????歲應該??????聯?? 60??2??21?? 141????聯??? 70??2??1??141???????貫強調??學習領???統??????學習領?? ?實???應????統???????並視學習內?????質??實?協???學????(???????? 2008)?????對聯??適??與??國語??結?????為??????????????????????? ???????????歲??????個 7 ???尌?? 21 ??尌??????????? 16 歲????? ?樣??讓學????語??課?????賞???????數學????動????戰數學詵?????個 ????選擇??
???????檢驗
??????????????將針對??職??師閱覽數學??????????????見??進 ??????歸納並????????? (?)???????職??師們??達對?????????學???懼??認為?????會對學?? ???閱讀??礙 ????
T17??????關問題???應????學??學????須???寫???潑?????????學?????題??? T21??這題?????????????????????????????適?????國????????????為 ??????題????????資優難題???? T11?? 應該敘??????確???????????達??學習?標 T06????應??納?????關問題?????題為???????

Page 13
國階???數學????設計??? 87 T22??較???????國??????????????礎??國????應??為適?? T01??雖經??師???話???說??????說??長??學????????????????適?????學 ???? T19??這?????????較難???????????應???譯???????????動畫??讓學??體會?? ????????????????????????發學????學習動機??
??過當?? T18 與???????????????????????
T18????數書????關問題??為??決實際??????關稅問題????????認為?????適?? ?????師??學??時機點應????????????????畢??學??對??????????? ??當??????????現給學????為腦????盪練習???來??讓學????賞??國???數 學風????來?讓學?????數學??實際????????應??(??別?????牽?錢?? 問題????讓??????????數學?識???????會??虧)?????????當適???????師 ??學??
???????師們??????????????????師??認為學??對?????會??閱 讀?????礙?????議??頇??譯??????動畫??????????學??動機??????佈?? ???????????????現?????讓學??領??數學????風華????????進???? ??說??讓????題???學????經???說????????進??嘗詴??題??????這樣??? ?????師們認為??較屬?腦????盪練習???????????題??與??資優難題?????? ?????許????????編????????現??????????譯???話??內????????? ??說??帶????????賞??????設計????參???????? (??)對????????????數??師??????態?? ???
T07??學?????????%???義??????(
100 1
)????況???對??????運???較??會 ????????錯??? T11???????????????確??講????%????來???學?????並產?????????適??? ????級????????單???學習?? T06???????%????義??卻?????從??來?????時將%?????納??課?????????? ??學????幫????? T22???????????細???????號%??來????????讓學習????????並應較???? ??記????????帶過類??單???書寫時??????總???????記將%書寫?????????

Page 14
??學??????與發?????????????? 88 ??學??????這個數學???????????會????? 25 與 25%???別??????????認 為???????適?????國????? T15??????????為????動機????為從?????????????號????變????現??????%?? ??????????紹????讓學???????這???????號%???讓學??經驗現???號?? ???單單?????????義????????積??許??????歷??????????????? ??貴??????號?? T05????????????為?????數??從 per 100??per
0 0 ?? %??????號??變??過???
??會覺??%這個??號這麼????難????? T02?????????號??變對?學?????????說????????號%??認??????????? ??單?????來????學習動機?? T20??????????變???讓????認識??????記錄??????????業???礎貨幣?統???? ???????????????數學記錄??????????們????????記錄?????????? ??國????????變??從????????????????書???????類??個時????學風??
從?????師們???饋????應??數學?????數學??學????????尌???過對 數學???遺產??緬懷??????獵??與?????讓學??????認識??????數學?維 與發?軌跡??進??領會????????啟發與創??並???曉數學?識與??類??動 ?關聯??????對?國??關數學??號????數學???????????????????並 適???紹??應該??對學??????幫???? (??)??????職??師??????數學詵?????適?????國????級課?? ???
T18??????統??????????題??????術??適????????級????樣??題??單??? ??師????圖????導學?????並????????對聯當???????????則運??? ??屬國階??應備????????????內???長???饒???(??國語課????紹過機 ???紀曉嵐)????????當適???????學??適???????級??樣??題????數????則 運??單??? T20??詩???訣??對????認識??記憶???????????????????????傑??????統 ????數學詩題????題???????????????????學?興??? T22?????饅頭??????術?????????????數來????題???經??現?????????? ??????級時????Ʀ????題????題??師???話?????現題????????????? ???學???????來????????則????????數學??現較??????????數?????

Page 15
國階???數學????設計??? 89 ????????問學?????覺????們??說??題????????????????????????認為 ??題適?????國????級????樣??題??單??? T01?????饅頭??適?????學???????詞簡??????????題???????簡單??100 ??(3??1)??25??100-25??75??數????? T10???????????????數學??結緣??????????實際????業??動?????????????? 對數學??興?????????歷??????????學????紹?????????????????????? ??觸????????視?????? 40 歲鑽?????? 60 歲還?????統?????這樣???? ??????????學??學習??????????詩??????學????????????動機??? ???當????????適??????級????????學習?????學習動????
雖?與??????關問題???為????????????????數學詵????詵體??對 聯?????現???????這?????內???論???饅頭???????紀????讓??師認為學 ????較???????????????????們??興????鳴??????紹數學?????帄??蹟?? ??????興?????師學習??壯??????????為????????????????????師 們??????認為適????????級課??? (??)??數??師認為???數學詵?????????與國語領??連結 ???
T20??????數學內????連結??貫??數與??與??數??題???強調???題????????養???? ??連結??則強調??????國語???????領??數學領???轉??????題??溝?諸 ????????????統???????學??????進?????戰數學詩??擬題??????養?????數 學??養??????????認??? T13??對聯??????????讓??????賞??國??學????????數學??????????從???? ??學????術??讓學??會????鮮??體會數學??????????????????????學氣 質?? T16??????結??語??領???閱讀??讓學??試??閱讀???????????詞??給???釋????養 ????賞???????測??????????題???????討論??訓練????對?????時????? ??尋關鍵????問題?? T11????對聯詩???????????????????????????對????級??學??????導?????? ????????????領????結????????題????謂?舉兩?? T03??????級??漸漸認識簡?????????詩來??導?????讓學?????簡????????? 時??數學問題??達??統????????????????達??學習??????舉????

Page 16
??學??????與發?????????????? 90 T10????????紀曉嵐???????對聯?????適?????學課???????????????? ??為????????歲???????個?????????????國語????關?識??????從??對聯?? ?????蘊???數???並?????題????確????????貫???連結?????????? ??課??設計?? T24???????貫強調???領???統?????領???學習???A???????嘗試???????? 數學詩???學???????師????????當???????????學??對?????學習?? 純??????????????????????詞?????師???釋與??導???????將??應??? 學習動機????發????認為????級?????????????與?????
??國階??閱讀????????養??當???????????顯?????學讀???閱 讀???學??獲??(????????黃萬?????彥璟??2008)?????????與國語領??連 結??數學????剛????????為學?????閱讀?????????????達???????貫 ??學習領??統?????標?????應????養閱讀?????????這??數學??????發 ????????????????連結??題??????讓學???曉數學?識與??類??動?關 聯??轉換????數學題???????????數學語???現??題??過????????題 ??結??闡釋?來????問題????????顧??與????領???統????讓數學??學習 ??為豐???? (??)??????師??????學時數??????問題 ???
T21????認為??國??????????無???????????????純??賞????時間??許????會??給 學???賞?? T02?????????學時數?????????學??會????
???師??議????????????數學??團??
T06??針對對數學??興???學??????開數學??團????為??課??????????對學????? ?????
??????師對?時數?????憂慮??與??????????區??學???? HPM ??學經 驗??與會??師???????數學????紹??????發學習動機?????學??動??為??動??潑??????
????????學經驗??時間??????????個????問題??????將數學???????團??選??課??? ?????應該會????????????數學???應?????團??動???????學??動???機會???? ??為??????時間????(蘇??????2005)??謀???????實???HPM ??學並????與數學

Page 17
國階???數學????設計??? 91
課???開??純??歷???講?????師????針對??適???單???將????尋???? 與課??結??編製??數學?學習????單????課????實???對?學????學習態??? ????顯????陎???響(???龍??蘇??????2009?????????蘇??????2009????????? 蘇??????2009)??????時間??????師??????數學?學習????單內???發??? 讓學??????????過時????進??討論????個???????(蘇??????2005)????? ??進???將數學?????數學??團????顯?對?學???????????幫??(陳????? 2009)??
?結論
???????????與??????????數學詵??兩個??題??設計國????級 數學?????並闡??????設計數學??????????????設計????過 對??職??師??問????????職??師們對???????數學???????見並進?? ????經過??數學?????設計????職??師??檢驗??????對?國階??數學? ????設計????????幾點?????
???????對學?????????????壓??????????譯???話??
??????設計?兩??????????????????現???數學詵??詵體??現?? 閱讀??較???????????????淺?????????????職??師???鳴??????對???? ??關???題??則?????職??師???懼與擔??????????認為??數學????? 設計???為??讓學????賞???????風華??????適????????現???????????? ?????????話????釋??????????體現??減??閱讀???懼??
???????設計??與????領??連結?數學????
經??????????????????職??師對???與??????領??連結??數學???? ?????????????這樣??進?????設計?????僅??關??數學?識???觸??數 學與????學??????????連結??????????學????????鳴?????? T18 ???認為 ??????統??????????題??????術??適????????級????樣??題??單?????師????圖

Page 18
??學??????與發?????????????? 92 ????導學?????並?????????????????????學????嘗詴與學???????題??並
??勵學??從??術????數兩種???????????盡????輔??圖???據 T18 ????? 這樣???學?????? 94%??學????????????設計??連結????領??????? ????為????從??數學????設計????????標?? ???論??闡????????設計數學?????????內????並??????職??師對 ???????數學?????????????發????????????從??數學????設計?? 參???????經???????????????實國階??數學????資??庫???????? 職??師??課?????齊為????學??學習數學?興????????
參????獻
???龍??蘇????(2009)??當動畫與學習????單??????數學?????國數學??學? 實????????師????163??70-77?? ??????(2002)??數學?與數學???????數學師資??????陏??實質??????實與 ?觀對數學??動設計與??????啟? (NSC89-2511-S-018-010 ?? NSC90-2521-S-018-002)??臺匇???????國???學?員會?? ???????蘇????(2009)??數學?讓數學變?????師???????509??81-83?? ????????黃萬?????彥璟(2008)??閱讀??學讀??對國????級學???????????? 與問題??決???????響????學??????與發???????51??1-33?? ??萬??(1999)??HPM 隨筆(??)??數學?與數學???與學??HPM ?訊??2(4)??1-3?? ??萬??(2000a)????無異??????????????個 HPM ??觀點????學????學???? 8(3)??215-224?? ??萬??(2000b)??????數書??專輯說????HPM ?訊??3(11)??1-2?? ??萬??(2002)??????????張????????HPM ?訊??5(12)??1-3?? ??萬??????????? HPM 團隊(譯)(2008)??溫??數學???從?????????級電腦(? ?????W. P. Berlinghoff, & F. Q. Gouvea)??臺匇??????書??(????????? 2004)?? ??萬???????銘??蘇??????蘇????楊瓊??劉???(2009)??當數學??見????? 臺匇??????書??? ?????(1997)??數學詵題??臺匇??????書??? 陳????(2009)??數學?????國????級??團數學??學?個????(?????碩?

Page 19
國階???數學????設計??? 93
論??)??臺匇??????????學??臺匇?? ??????統計處(1998)????臺灣??區???????級學學??學習???????況調??(86 學????學??)???調??結????臺匇?????????? ??????(2008)??國????學?????貫課??綱???臺匇?????????? ???????蘇????(2009)??數學?????國數學??學?實?????????數??????? 為????臺灣數學??師(電??)?????2??2-13?? 鄭???(2000)??學????師數學??學?????學習?????????啟?????????學????為 ??(????????論??)??國??臺灣師範??學??臺匇?? ???(1592)????統???載??書????編??國??學??術??彙數學?????? 冊(頁 1217-1421)??鄭??????????????? 蘇????(2004)??數學??師專業發????個陎??數學?????數學??學?實??與?? ??(????????論??)??國??臺灣師範??學??臺匇?? 蘇????(2005)??淺談 HPM 學習????單?設計??臺灣數學??師(電??)?????2??2-13?? 蘇????(2007)??運??????數學??學????開????為????臺灣數學??師(電??)?? ????9??56-67?? Barbin, E. (2000). Integrating history: Research perspectives. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education (pp. 63-90). Dordrecht: Kluwer Academic Publishers. Fasanelli, F. (2000). The political context. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education (pp. 1-38). Dordrecht: Kluwer Academic Publishers. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel Publishing Company. Furinghetti, F., & Paola, D. (2003). History as a crossroads of mathematical culture and educational needs in the classroom. Mathematics in School, 32 (1), 37-41. Gulikers, I., & Blom, K. A. (2001). „A historical angle‟, a survey of recent literature on the use and value of history in geometrical education. Educational Studies in Mathematics, 47, 223-258. Horng, W. -S. (2004, July). Teacher??s professional development in terms of HPM: A story of Yu. Paper presented at the HPM 2004, Uppsala, Sweden. Jahnke, H. N. (1994). The historical dimension of mathematical understanding: Objectifying the subjective. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings

Page 20
??學??????與發?????????????? 94
of the 18
th
international conference for the Psychology of Mathematics Education, vol. I (pp. 139-156). Lisbon: University of Lisbon. Menghini, M. (2000). On potentialities, limits and risks. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education (pp. 86-90). Dordrecht: Kluwer Academic Publishers. Pycior, H. (1987). Biography in the mathematics classroom. In I. Grattan-Guinness (Ed.), History in mathematics education (pp. 170-186). Paris: Soci??t?? française d‟histoire des sciences et des techniques . Su, Y. -W. (2006). An action research of school-centered professional development in the HPM context. In F. Furinghetti, S. Kaijser, & C. Tzanakis (Eds.), Proceedings of the HPM 2004 & ESU4 (pp. 368-382). Uppsala: University of Uppsala. Tzanakis, C., & Arcavi, A. (2000). Integrating history of mathematics in the classroom: An analytic survey. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education (pp. 201-240). Dordrecht: Kluwer Academic Publishers. Van Maanen, J. (2001). Research on history in mathematics education in the Netherlands: The ??reinvention studies??. In F. -L. Lin (Ed.), Proceedings of the Netherlands and Taiwan conference on common sense in mathematics education??Common sense in mathematics education (pp. 191-202). Taipei: National Taiwan Normal University.

Page 21
國階???數學????設計??? 95
An Initial Investigation into the Design of Materials of History of Mathematics for Elementary School Education
Yi-Wen Su
Department of Mathematics, Taipei Municipal University of Education yiwen@tmue.edu.tw
Abstract
This paper describes how the researcher proceeded with the design of the materials of history of mathematics for elementary stage; and is meant to help the in-service teacher integrate history of mathematics into high grade teaching. With the ideas of hermeneutic, after considering the standard of curriculum, mathematics contents as well as the purpose of textbook designing, the researcher developed the materials in a hermeneutic way. The researcher started to search for the materials of history of mathematics and then chose suitable text among ancient books on mathematics to interpret the development of mathematics-related thoughts and theories. In addition to giving a general presentation and introduction to two completed themes of design concepts, that is, ??proportion and percentage??, and ??funny math poetry??, the researcher also used some questionnaire, raising question like ??Are the materials suitable for elementary stage???. Then the researcher asked for 24 elementary in-service teachers to make comments based on the questionnaire so that it would be easier to find out to what extent the material can be used and what revision shall be required. At last, regarding the design of materials of history of mathematics, the researcher suggested the following points: (1) The ancient writings tend to put a certain degree of pressure on the modern students as the students would have difficulty understanding the meaning of these writing completely. To solve this problem, it would be better to convert them into plain language used today. (2)We can design more materials of history of mathematics which are able to connect other curriculum domain. Keywords: elementary school, history of mathematics, mathematics poetry

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