東亞地區(qū)學(xué)生數(shù)學(xué)成績(jī)優(yōu)異的原因探究

1、東亞地區(qū)學(xué)生數(shù)學(xué)成東亞地區(qū)學(xué)生數(shù)學(xué)成績(jī)優(yōu)異的原因探究績(jī)優(yōu)異的原因探究 香港大學(xué)教育學(xué)院梁貫成香港大學(xué)教育學(xué)院梁貫成二二OO五年四月五年四月上海華東師范大學(xué)上海華東師范大學(xué) 東亞國(guó)家東亞國(guó)家*學(xué)生在學(xué)生在第三屆國(guó)際數(shù)學(xué)及科學(xué)研第三屆國(guó)際數(shù)學(xué)及科學(xué)研究究中中, 數(shù)學(xué)成績(jī)比其它國(guó)家優(yōu)勝數(shù)學(xué)成績(jī)比其它國(guó)家優(yōu)勝 本本報(bào)告報(bào)告嘗試就其原因作探究嘗試就其原因作探究 *國(guó)家一詞在本國(guó)家一詞在本報(bào)告報(bào)告指國(guó)家或地區(qū)指國(guó)家或地區(qū), 東亞地區(qū)東亞地區(qū)指日本南韓星加坡指日本南韓星加坡臺(tái)灣臺(tái)灣及香港及香港 與其它國(guó)家與其它國(guó)家比較比較, , 東亞?wèn)|亞五五國(guó)似乎國(guó)似乎沒(méi)有任何沒(méi)有任何特征特征 (例如例如: 國(guó)民生產(chǎn)總值國(guó)

2、民生產(chǎn)總值 教育開(kāi)支教育開(kāi)支 教育資源教育資源) 足以解釋其優(yōu)越的成績(jī)足以解釋其優(yōu)越的成績(jī) 教育制度亦然教育制度亦然,五五國(guó)沒(méi)有用較多國(guó)沒(méi)有用較多課時(shí)課時(shí)在數(shù)學(xué)在數(shù)學(xué)上上, 其學(xué)生也沒(méi)有用較多時(shí)間于其學(xué)生也沒(méi)有用較多時(shí)間于課外數(shù)學(xué)課外數(shù)學(xué)學(xué)習(xí)學(xué)習(xí),且且每班人數(shù)每班人數(shù)較多較多 GNP per Capita (1999)CountryGNP per CapitaJapanUS$38160 SingaporeUS$32810United StatesUS$29080Hong KongUS$25200AustraliaUS$20650CanadaUS$19640Chinese TaipeiUS$13

3、235KoreaUS$10550TIMSS-R countries average US$10584Education ExpenditureCountry% of GNPCanada6.9%Australia5.5%United States5.4%Chinese Taipei 4.9%Korea3.7%Japan3.6%Singapore3.0%Hong Kong2.9%TIMSS-R countries average 5.1%Home Educational Resources(International Avg.= 9%)051015202530AustraliaUnited Sta

4、tesKoreaCzech RepublicNetherlandsChinese TaipeiSingaporeHong Kong South AfricaThailand%Hours of Mathematics Instruction per Year (International Avg.= 129 Hours)050100150200ThailandHong KongUnited StatesCzech ReplublicAustraliaSouth Africa JapanChinese TaipeiSingaporeKoreaEnglandNetherlandsHoursOut-o

5、f-School Time Studying or Doing Mathematics Homework(International Avg.= 1.1 Hours per Day)00.511.52SouthAfricasingapore ThailandUnitedStatesAustralia ChineseTaipeiCzechRepublic HongKongJapanKoreaClass Size (International Avg.= 31) 五五國(guó)學(xué)生國(guó)學(xué)生并沒(méi)有特別并沒(méi)有特別重視重視數(shù)學(xué)數(shù)學(xué), , 甚至甚至對(duì)數(shù)對(duì)數(shù)學(xué)態(tài)度學(xué)態(tài)度相對(duì)負(fù)面相對(duì)負(fù)面, , 并并且對(duì)自己的且對(duì)自己的

6、數(shù)學(xué)能數(shù)學(xué)能力力缺乏信心缺乏信心學(xué)學(xué)生表達(dá)對(duì)數(shù)學(xué)缺乏自信心生表達(dá)對(duì)數(shù)學(xué)缺乏自信心, , 可能與其文可能與其文化背景有關(guān)化背景有關(guān) 東亞人東亞人( (特別是中國(guó)人特別是中國(guó)人) )向向來(lái)強(qiáng)調(diào)謙遜來(lái)強(qiáng)調(diào)謙遜, , 所謂所謂 滿招損滿招損, , 謙受益謙受益 這種謙遜雖然似乎叫學(xué)生失去自信這種謙遜雖然似乎叫學(xué)生失去自信, , 卻卻沒(méi)有影響其成績(jī)沒(méi)有影響其成績(jī) Students Report on Whether It is Important To Do Well in Mathematics(International Avg.= 96% of Students)Students Positive

7、 Attitudes towards Mathematics(International Avg.= 37% of Students )010203040506070South AfricaSingaporeEnglandThailandUnited StatesAustraliaHong KongTaipeiCzech NetherlandsJapanKorea% StudentsStudents Self-Concept in Mathematics(International Avg.= 18 % of Students )05101520253035United StatesAustr

8、aliaEnglandNetherlandsCzech SingaporeHong KongTaipeiKoreaSouth AfricaJapanThailand% Students 既然上述原因不能解釋東亞地區(qū)學(xué)生之優(yōu)既然上述原因不能解釋東亞地區(qū)學(xué)生之優(yōu)異成績(jī)異成績(jī), 我們下面將探討東亞學(xué)生成績(jī)優(yōu)我們下面將探討東亞學(xué)生成績(jī)優(yōu)異是否因?yàn)樗麄冇袛?shù)學(xué)能力較高的老師異是否因?yàn)樗麄冇袛?shù)學(xué)能力較高的老師 1.馮氏之研究馮氏之研究 (2000) 馮氏透過(guò)問(wèn)卷及訪談馮氏透過(guò)問(wèn)卷及訪談, 發(fā)現(xiàn)香港小學(xué)數(shù)學(xué)老發(fā)現(xiàn)香港小學(xué)數(shù)學(xué)老師之基礎(chǔ)數(shù)學(xué)知識(shí)非常薄弱師之基礎(chǔ)數(shù)學(xué)知識(shí)非常薄弱 Fungs questionnai

9、re (2000)1.Why is that multiplication takes precedence over addition in a numeric expression?2.Is it possible to find two positive integers whose quotient is an infinite non-recurring decimal? Why?3.What is direct proportion?4. The following calculation is done to find the L.C.M. of three positive i

10、ntegers.2) 12 20 90 2) 6 10 45 3) 3 5 45 3) 1 5 15 5) 1 1 5 1 1 1 Thus the L.C.M. of 12, 20 and 90 is 22325 = 180.Why is that it works? Can we apply this method to find the L.C.M. of more than three numbers?5.How can we determine which of the following two figures has a greater area?6.Why is that wh

11、en we divide two proper fractions, we simply invert and multiply?8.How do you arrange the following expressions in increasing order of magnitude? 200 , 1.08 x 0.59, 66% 3037.How do you explain the meaning of the following division algorithm with the help of counting cubes (of different colours)? _ 3

12、 3_ 6 ) 2 1 3 6 ) 2 1 3 6 ) 2 1 3 1 8 1 8 3 3 3 5 3 5 6 ) 2 1 3 6 ) 2 1 3 6 ) 2 1 3 1 8 1 8 1 8 3 3 3 3 3 3 3 0 3 0 3Question Number of studentsproviding good responsePercentage of students providingGood response 1 0 0 2 0 0 3 1 5.8 4 0 0 5 2 11.8 6 1 5.8 7 0 0 8 4 23.5Results (17 students) 馬氏比較美國(guó)及中

13、國(guó)馬氏比較美國(guó)及中國(guó) ( (上海上海) ) 小學(xué)數(shù)學(xué)老小學(xué)數(shù)學(xué)老師之?dāng)?shù)學(xué)能力師之?dāng)?shù)學(xué)能力, , 發(fā)現(xiàn)美國(guó)老師缺乏對(duì)基發(fā)現(xiàn)美國(guó)老師缺乏對(duì)基礎(chǔ)數(shù)學(xué)知識(shí)的深切了解礎(chǔ)數(shù)學(xué)知識(shí)的深切了解在香港之重復(fù)研究發(fā)現(xiàn)香港老師對(duì)數(shù)學(xué)知在香港之重復(fù)研究發(fā)現(xiàn)香港老師對(duì)數(shù)學(xué)知識(shí)的了解雖然不及上海老師那么深切識(shí)的了解雖然不及上海老師那么深切, , 但其數(shù)學(xué)能力仍遠(yuǎn)超美國(guó)老師但其數(shù)學(xué)能力仍遠(yuǎn)超美國(guó)老師 雖然如雖然如此此, , 他們的教學(xué)方法仍然偏重于教授計(jì)他們的教學(xué)方法仍然偏重于教授計(jì)算技巧多于數(shù)學(xué)概念的理解算技巧多于數(shù)學(xué)概念的理解 Task 2:Multi-digit Number Multiplication Some

14、sixth-grade teachers noticed that several of their students were making the same mistake in multiplying large numbers. In trying to calculate 123 645_ the students seemed to be forgetting to “move the numbers” (i.e., the partial products) over on each line. They were doing this: 123 645_ 615 492 738

15、_ 1845 instead of this: 123 645_ 615 492 738_ 79335 While these teachers agreed that this was a problem, they did not agree on what to do about it. What would you do if you were teaching sixth grade and you noticed that several of your students were doing this? 香港及韓國(guó)老師被問(wèn)及如何處理此學(xué)生錯(cuò)香港及韓國(guó)老師被問(wèn)及如何處理此學(xué)生錯(cuò)誤時(shí)

16、，其答復(fù)顯示他們的教學(xué)非常誤時(shí)，其答復(fù)顯示他們的教學(xué)非常程序程序性性但當(dāng)深入問(wèn)及為何要但當(dāng)深入問(wèn)及為何要 對(duì)位對(duì)位，補(bǔ)零補(bǔ)零 時(shí)時(shí)，我們卻發(fā)現(xiàn)他們對(duì)背后的數(shù)學(xué)概念是，我們卻發(fā)現(xiàn)他們對(duì)背后的數(shù)學(xué)概念是了解了解的的香港及韓國(guó)老師既有程序性的了解，也有香港及韓國(guó)老師既有程序性的了解，也有概念性的了解，但他們的教學(xué)卻偏于概念性的了解，但他們的教學(xué)卻偏于程程序性序性 I: What would you do if you find your students are making this kind of mistakes?T:Ill teach them that if you use the hun

17、dreds place times the unit place, your answer should be written under the “6”, align with this “6”. Now the textbook leaves two empty spaces there, and Ill tell them if you have empty spaces, it is easy for you to align wrongly. So Ill teach them after you get the product, say six times three equals

18、 eighteen, you should immediately put down two zeroes in the two empty spaces. If you have something there, you wont align wrongly. So the first thing is to align, (It is) the same with the second number. It is the tens place, so you should put your answer under the tens place, and you put a zero at

19、 the unit place I:Why are students making this kind of mistakes?T:Its a problem with place value. In fact the 6 here stands for 600. I can change 645 into 600 + 40 + 5, and then do it step by step, dividing (the multiplication) into 123 600 + 123 40 + 123 5. Then I will address the problem of aligni

20、ng the numbers (i.e. the partial products). I will tell them that you need to add in zeroes USShanghaiHong KongKoreaProcedural14 (61%)6 (8%)0 (0%)1 (11%)Conceptual+ procedural9 (39%)66 (92%)9 (100%)8 (89%)Table 2: Teachers Knowledge of Multi-digit Multiplication Algorithm USShanghaiHong KongKoreaPro

21、cedurallydirected16 (70%)9 (13%)7 (78%)6 (67%)Conceptuallydirected7 (30%)63 (88%)2 (22%)3 (33%)Table 3: Teaching StrategiesTask 3:Division by Fractions People seem to have different approaches to solving problems involving division with fractions. How do you solve a problem like this one?1 3/4 1/2 =

22、 Imagine that you are teaching division with fractions. To make this meaningful for kids, something that many teachers try to do is to relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of co

23、ntent. What would you say would be a good story or model for 1 3/4 1/2 ? 所有被訪老師皆能正確地完成此除數(shù)所有被訪老師皆能正確地完成此除數(shù)至少一半的上海、香港及韓國(guó)老師對(duì)此題至少一半的上海、香港及韓國(guó)老師對(duì)此題的理解足以作出一個(gè)與題目有關(guān)的故事的理解足以作出一個(gè)與題目有關(guān)的故事USShanghaiHong KongKoreaCorrect answer9 (43%)72 (100%)9 (100%)9 (100%)Correct story1 (4%)65 (90%)6 (67%)5 (50%)Table 4: Teach

24、ers Knowledge of Division by Fractions 有些韓國(guó)老師作以下的有些韓國(guó)老師作以下的處理方法處理方法（雖然（雖然教材建議另一種方法）教材建議另一種方法）這些老師明白程序背后的概念，卻選擇這些老師明白程序背后的概念，卻選擇程序性的教學(xué)程序性的教學(xué)Teacher E:This is the explanation provided in the (6th grade) textbook: But I dont use this formula because it is very hard for sixth graders to understand The s

25、tudents have already learned the rule of fraction multiplication which is basically the same. Im afraid if I introduce this principle, students might wrongly generalize this to addition of fractions like (a/b + c/d = (a + c)/(b + d). I prefer to use the previously promised rule. At the beginning of

26、teaching division by fractions, I introduce the example “1 1/7”. Assuming you are dividing a 1m rope into 1/7 m units, this results in 7. So, 1 divided by 1/7 is the same as 1 multiplied by 7. At this point, I set up a rule that “division by fraction is the same as multiplication by the reciprocal o

27、f the fraction”. Ill make my students recall this rule, and then use it here. 3411274127422742（）72（）41742144121Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as th

28、e perimeter of a closed figure increases, the area also increases. She shows you this picture to prove what she is doing: 4 cm 8cm 4 cm 4 cm Perimeter 16 cm Perimeter 24 cm Area 16 square cm Area 32 square cm How would you respond to this student? Task 4: Relationship between Perimeter and Area 沒(méi)有香港

29、老師及祇有一位韓國(guó)老師認(rèn)為命題沒(méi)有香港老師及祇有一位韓國(guó)老師認(rèn)為命題是正確的，祇有兩位老師一開(kāi)始就確定命是正確的，祇有兩位老師一開(kāi)始就確定命題是錯(cuò)的，并能舉出反例，大部分老師開(kāi)題是錯(cuò)的，并能舉出反例，大部分老師開(kāi)始時(shí)都始時(shí)都不肯定不肯定命題是否正確。命題是否正確。香港及韓國(guó)老師皆沒(méi)有利用此機(jī)會(huì)香港及韓國(guó)老師皆沒(méi)有利用此機(jī)會(huì)引導(dǎo)學(xué)生引導(dǎo)學(xué)生去探索有關(guān)的數(shù)學(xué)概念去探索有關(guān)的數(shù)學(xué)概念USShanghaiHong KongKoreaAccepted the claim2 (9%)6 (8%)0 (0%)1 (11%)Not sure18 (78%)0 (0%)2 (22%)0 (0%)Explore

30、usingproblematic strategy2 (9%)16 (22%)1 (11%)2 (22%)Explore usingcorrect strategy1 (4%)50 (70%)6 (67%)6 (67%)Table 5: Teachers Reactions to the Incorrect Statement I: How would you react to the students finding?Ms Ng:I would say: “yes, it works for your diagram, but is it always true? Have you trie

31、d more examples to see whether it works or not? What will happen if you change it to another diagram? Does it only hold true for squares and rectangles? Have you tried other diagrams? I havent thought about it, lets explore it together.”I: If you have some time to explore the problem with the studen

32、t, how would you explore it?Ms Ng:Frankly speaking, I have no idea how to explore it. Ill try (more examples) with the student, try different diagrams. May be first change the sides (of the rectangles), then change for another diagram, and then change the length and width, to see what will happen (t

33、o the area). We will explore together, but I have no idea what directions to take. (And after trying some diagrams, she by chance found a counter-example.) 根據(jù)馮氏之研究根據(jù)馮氏之研究, 香港老師數(shù)學(xué)知識(shí)薄弱香港老師數(shù)學(xué)知識(shí)薄弱, 而馬氏之重復(fù)研究又指出香港老師教學(xué)而馬氏之重復(fù)研究又指出香港老師教學(xué)時(shí)不著重理解概念時(shí)不著重理解概念, 那么香港及其它東亞那么香港及其它東亞地區(qū)學(xué)生地區(qū)學(xué)生, 為何仍能有優(yōu)異的成績(jī)?yōu)楹稳阅苡袃?yōu)異的成績(jī) ? 其實(shí)所謂偏重教授技巧之教學(xué)其實(shí)所謂偏重教授技巧之教學(xué), , 并不一定并不一定與理解概念相斥與理解概念相斥 中國(guó)或東亞的傳統(tǒng)中國(guó)或東亞的傳統(tǒng), , 似乎是強(qiáng)調(diào)透過(guò)演練而達(dá)至理解似乎是強(qiáng)調(diào)透過(guò)演練而達(dá)至理解 Teacher E:When I