臺北市立教育大學九十九學年度第一學期 理學院週末研習活動 提升數學教學創造力之國際研討會 International workshop of Enhancing mathematical teaching for creativity 23 October, 2010 The Learner’s Perspective Study: A Hong Kong Story 學習者角度研究： 香港的故事 Ida Ah Chee Mok Associate Professor Faculty of Education The University of Hong Kong Email: iacmok@hku. hk 莫雅慈副教授 香港大學教育學院 1 致謝: （1) 主辦機構; (2) CRCG, HKU 香港大學
Outline 大綱 • An introduction of Hong Kong School Curriculum 香港課程簡介 • An exemplary lesson 一堂示範課 • Conclusion: A challenge for teachers 結論：對於教師的一大挑戰 2
Hong Kong School Curriculum 香港學校課程 (primary and secondary schools) 中學及小學 Seven learning goals for 10 year strategic plan (20012011) (CDC, 2001) 七個宗旨學習 (課程發展會議, 2001) 3
The Hong Kong School Curriculum – Life-wide learning 香港學校課程 – 全方位學習 Five essential learning experience 五個基要的學習經歷 Moral and Civic Education 德育及 公民教育 Physical and Aesthetic Development 體藝發展 Career-related Experience 與 作 有關的經驗 Physical education Art Education Technology Education Curriculum development Institute Education department HKSAR Science Education Personal, Social and Humanities Education Mathematics Education 價值觀和態度 English Language Education VALUES AND ATTITUDES 中 國 語 文 教 育 Community Services 社會服務 KEY LEARNING AREAS 學者領域 個 科 科 藝 英 數 體 人 社 學 技 術 國 學 育 會 教 教 教 語 教 教 及 人 育 育 育 文 教 教 育 育 Chinese Language Education GENERAL SKILLS 共同能力 Intellectual Development 智能發展 Communication skills 溝通能力 Critical thinking skills 批判性思考能力 Creativity 創造力 Collaboration skills 協作能力 Information technology skills 運用資訊科技能力 Numeracy skills 運算能力 Problem-solving skills 解決問題能力 Self-management skills 自我管理能力 Study skills 研習能力 Perseverance 堅毅 Respect for others 尊重他人 Responsibility 責任感 National Identity 國民身份認同 Commitment 承擔精神 …… 4
Diagrammatic representation of the framework of the mathematic curriculum 數學課程架構圖 Mathematics Curriculum 數學課程 provides content knowledge which can serve as a means to develop students’ thinking abilities and foster students’ generic skills and positive attitudes towards mathematics learning 提供學習內容，藉以發展學生的思維能力及培養學生的共通能力和學習數學的正面態度 Strands 學習範疇 provides a structured framework of learning objectives in different areas of the Mathematics Curriculum 提供課程內一個包含不同課題的學習重點的組織架構 9 Generic Skills 九項共通能力 Number 數 S 1 – S 3 (中一至中三) S 4 – S 6 (中四 至 中六) Algebra 代數 Number and Algebra 數與代數 (Extended Part) (伸延部分) Module I (Calculus and Statistics) 單元一 (微積分與統計) Measures 度量 Shapes and Spaces 圖形與空間 Measures, Shapes and Spaces 度量、圖形與空間 Number and Algebra 數與代數 S 1 – S 3 (中一至中三) Data Handling 數據處理 (Extended Part) (伸延部分) (Compulsory Part) (必修部分) Measures, Shapes and Spaces 度量、圖形與空間 P 1 – P 6 (小一至小六) Data Handling 數據處理 Module II (Algebra and Calculus) 單元二 (代數與微積分) S 4 – S 6 (中四 至 中六) Values and Attitudes 價值觀和態度 P 1 – P 6 (小一至小六) Future Learning Unit 進階學習單位 Effective Linkage of learning, teaching and assessment 學與教及評估的連繫 5 Overall Aims and Learning Target of Mathematics 數學科的教學宗旨和學習目標
Primary Mathematics Curriculum Organization 小學數學課程組織 Dimension 範疇 Number 數 Shape and Space 圖形與空間 Measure 度量 Data Handling 數據處理 Algebra 代數 Content 學習內容 • Whole number 整數 • Nature of Number 數的性質 • Fractions, decimals and percentages 分數、小數和百分數 • Calculating devices 計算 具 • Three dimensional shapes 立體圖形 • Lines 線 • Two dimensional shapes 平面圖形 • Angles 角 • Directions 方向 • Money 貨幣 • Length 長度 • Time 時間 • Weight 重量 • Capacity 容量 • Perimeter 周界 • Area 面積 • Volume 體積 • Speed 速度 Statistics 統計 • Algebraic symbols 代數符號 • Equations 方程 6
A comparison for the old and the current mathematics curricula (1983 and 2000 curricula) 小學數學科新舊課程比較 (1983年的課程和2000年的課程) Topics that are deleted after 2000年已刪去的課程 (1)Temperature 溫度 (P 2 小二) (2)Direct proportion 正比例 (P 4 小四) (3)Square brackets 方括 (P 4 小四) (4)Division： 3 -digit divisor, 4 -digit dividend除法：除數三個位，被除數四個位 (P 4 小四) (5)Operations in time scale時間單位的四則運算 (P 5 小五) (6)Inverse proportion反比例 (P 5 小五) (7)Line chart 直線圖樣 (P 5 小五) (8)Factorization, scientific notation 因數分解、指數記數法 (P 5 小五) (9)Making pie chart 圓形圖製作 (P 6 小六) (10)Positive and negative numbers 正負數 (P 6 小六) (11)L. C. M. (factorization, short division) 最少公倍數 (因數分解、短除法) (P 6 小六) (12)Area of circles 圓面積 (P 6 小六) (13)Simple measurement 簡易測量 (P 6 小六) (14)Use of protractor, degree 量角器的使用、度 (P 6 小六) (15)Simple interest 單利息 (P 6 小六) (16)Story of calculating tools計算 具的故事 (P 6 小六) (17)Simple equation: operation of same items 簡易方程式：同類項運算 (P 6 小六) (18)Magic square 幻方 (P 6 小六) (19)H. C. F. (factorization, short division) 最大公因數 (因數分解、短除法) (P 6 小六) (20)Curves graphs 曲線圖 (P 6 小六) 7
A comparison for the old and the current mathematics curricula (1983 and 2000 curricula) 小學數學科新舊課程比較 (1983年的課程和2000年的課程) Topics that are added after 2000年已增加的課程 (1)Operation (1): solving word question 四則運算(一)：解答應用題 (P 3 小三) (2)Perpendicular lines 垂直線 (P 3 小三) (3)Angle (2): Acute angle, obtuse angle 角(二) ：銳角、鈍角 (P 3 小三) (4)Blocks chart: making frequency chart 方塊圖：製作頻數表 (P 3 小三) (5)Operation (2): solving word question 四則運算(二)：解答應用題 (P 4 小四) (6)Understanding of modern tools 現代 具的認識 (P 4 小四) (7)Understanding of modern tools 現代 具的認識 (P 6 小六) Topics as an enrichment in 2000年後，課程改為增潤的課題或內容 (1)Tessellation 密鋪 (P 3 中三) (2)Prime number and composite number 質數及合成數 (P 5 中五) (3)Divisible number (Divisor are 3, 4, 6, 8, 9, 11) 整除數 (除數為 3, 4, 6, 8, 9, 11) (P 6 中六) (4)Number type (square number and triangle number) 數型 (正方形數和三角形數) (P 6 中六) (5)Use of protractor, degree 量角器的使用、度 (P 6 中六) (6)Rotational symmetry 旋轉對稱 (P 6 中六) (7)Embroidery curves 繡曲線 (P 6 中六) (8)Repeating decimals 循環小數 (P 6 中六) (9)Square and square root (exclude factorization) 平方及平方根 (不包括用因數分解法) (P 6 中六) (10)Ancient number 古代數字 (P 6 中六) 8
A comparison for the old and the current mathematics curricula (1983 and 2000 curricula) 小學數學科新舊課程比較 (1983年的課程和2000年的課程) Topics in 1983 that are not be as an individual in 2000, but are arranged in related topic. 2000年後，課程不作獨立課題 而編排在有關課程。 項目 Curriculum in 1983年課程 Curriculum in 2000 200年課程 Year 年級 課程 課程 (1) P 1 小一 初步活動 數及圖形與空間的範疇 (2) P 1 小一 統計活動 象形圖 (一) (3) P 2 小二 貨幣的加減運算 加與減 (4) P 3 小三 貨幣的四則運算 四則計算 (一) (5) P 3 小三 括號 四則計算 (一) (6) P 4 小四 十進制度量單位化聚 度量的課題 (7) P 5 小五 乘法分配性質 四則計算 (一) (8) P 5 小五 時間單位的四則運算 在學習時間，進行單位化聚及計算的 練習。 (9) P 6 小六 整除性 (除數為 2、5和 10) 除法 (二) 9
A Comparison of the Old and the Current Secondary Academic Structures 新舊中學架構的比較 Old Structure 舊學制架構 (3+2+2+3) 3 -year Undergraduate Degree 三年大學課程 HKALE 香港高級程度會考 HKCEE 香港中學會考 New Structure 新學制架構 (3+3+4) 4 -year Undergraduate Degree 四年大學課程 Secondary 7 中七 Secondary 6 中六 Secondary 5 中五 Secondary 4 中四 Secondary 3 中三 Secondary 2 中二 Secondary 1 中一 2010 complete secondary education 2010年完成 中學課程 Secondary 3 中三 Secondary 2 中二 2016 1 st cohort of graduates 2016年完成 四年大學課程 HKDSE 香港中學文憑 2006/07 school year 10 2006/07 入學年
A Comparison of the Old and the Current Mathematics Curriculum 新舊數學課程比較 Secondary Mathematics Curriculum (S 4 –S 5) 中學數學課程 (中四至中五) Additional Mathematics Curriculum (S 4 – S 5) 附加數學課程 (中四至中五) ASL/AL Mathematics Curricula 高級程度/高級補充程度 數學課程 (中六至中七) Old curriculum (S 4 – S 7) 舊有課程 (中四至中七) Compulsory Part 必修部份 Extended Part (Module I or Module II) 伸延部份 (單元 一 或 單元二) Current curriculum (S 4 – S 6) 現行課程 (中四至中六) Students may take the Compulsory Part only, the Compulsory Part with Module 1 (Calculus only and Statistics) or the Compulsory Part with Module 2 (Algebra and Calculus). Students are only allowed to take at most one module from the Extended Part. 學生可只修必須部分，亦可修讀必修部分及單元一 (微積分與統計) 或必修部分及單元二( 11 代數與微積分)。學生最多只能從延部分中修讀其中一個單元。
Compulsory Part 必修部分 Foundation Topics 基礎課程 Cover topics with different areas to enable students to develop a Include basic concepts and knowledge necessary for simple applications in real-life situations 提供學習基礎和知識， 讓學生在日常生活中應用。 cohere nt body of knowledge 覆蓋不同範疇，使學生建立有 Non-foundation Topics 非基礎課程 Cover a wider range of contents and provide students with a foundation for their future studies and career development 覆蓋廣泛的內容﹐ 為日後進修 及 作中需要更多數學知識和技 能的學生作基礎 12
Extended Part 伸延部分 Module 1 (Calculus and Statistics) 單元一 (微積分與統計) Module 2 (Algebra and Calculus) 單元二 (代數與微積分) Emphasize on statistics and applications, the understanding of mathematics for a wider knowledge and deeper further progress in mathematical-related understanding of the application of disciplines mathematics 旨在微積分與統計的直觀概念，擴闊學 旨在學生在高中階段學習更深的數學知 識，幫助學生將來深造和就業作準備 生在數學方面有更深入的知識和應用 13
Mathematics Curriculum about Creativity 有關創造力的數學課程重點 Emphases on Mathematics Curriculum 數學課程重點： To focus on the fundamental knowledge and skills, capability to learn how to learn, think logically and creatively, develop and use knowledge, analyze and solve problem, asses and process information, make sound judgment and communicate with others effectively. 重視知識和技巧，以及學會如何學者、具邏輯和創意思考、 建構和運用知識，分析和解決問題、獲取和處理資訊、作出 正確判斷、以及善於與人溝通的能力。 (Mathematics Education, Key Learning Area Curriculum Guide (Primary 1 to Secondary 6), Curriculum Development Council, The Education Department HKSAR 2000, Key Message, p. iii 數學教育 學習領域課程指引 (小一至中三) 香港特別行政區教育署課程發展議會, 14 提要第iii頁 )
Mathematics activity about creativity 有關創造力的數學活動 Measurement (P 3) 量度 (小三) Activity: Measure the following items in the playground by means of measuring tools (such as wheels and meter rule) and record the result: 運用不同的量度 具(如滾輪、米尺等)，量度和記錄在操場上的物件： Width 濶度 (1) The perimeter of the basketball court is _____ m. 籃球場的周界是 _____ 米。 (2) The height and the width of soda machine are _____ m and ____ m respectively. 汽水機的高度和濶度分 別是 _____ 米和 ____ 米。 (3) The height of the door of store room is _____ m. 雜物房門的高度是_____ 米。 Height 高度 Acclaim image. com (4) The length of the tabletennis table is _____ m. 乒乓球桌的長度是 _____ 米。 15
Mathematics activity about creativity 有關創造力的數學活動 There are 8 sinks in the school in which 4 of them are out of order. We have 45 students to wash their hands. The time of washing is different in each sink. Find the place of the sink and check which sink could be used. Then answer the following questions: a. The time of washing in sink A is ______ second. b. The time of washing in sink B is ______ second. c. The time of washing in sink C is ______ second. d. The time of washing in sink D is ______ second. e. The arrangement of 45 students to wash their hands i. Sink A has ____ students, the time for washing is ____ second. ii. Sink B has ____ students, the time for washing is ____ second. iii. Sink C has ____ students, the time for washing is ____ second. iv. Sink D has ____ students, the time for washing is ____ second. v. The minimal time for 45 student in washing their hands is ___Second The worksheet is from the web sites of Po Chui Catholic Secondary School 16
LPS Research Series Publications Clarke, D. , Keitel, C. and Shimizu, Y. (Eds. ) (2006). Mathematics Classrooms in 12 Countries: The Insiders’ Perspective. Rotterdam: Sense Publishers. Clarke, D. , Emanuelsson, J. , Jablonka, E. , and Mok, I. A. C. (Eds. ) (2006). Making Connections: Comparing Mathematics Classrooms Around the World. Rotterdam: Sense Publishers. Shimizu, Y. , Kaur, B. , Huang, R. , & Clarke, D. J. (Eds. ) (2010). Mathematical Tasks in Classrooms Around the World. Rotterdam: Sense Publishers.
An Exemplary Lesson in HK 香港的一堂示範課 Factorization of Polynomial 多項式因式分解 Mok, I. A. C. (2009). In search of an exemplary mathematics lesson in Hong Kong: An algebra lesson on factorization of polynomials. Zentralblatt fuer Didaktik der Mathematik (ZDM Mathematics Education). 41, 319 -332. 18
Objectives of this presentation 報告目標 What may a lesson with quality instruction look like in the Hong Kong education system? 在香港教育體制下的高質量的課堂教學狀況 • A researcher’s perspective (Variation Theory): How the teacher’s teaching may possibly create learning opportunities for the learners to see the mathematical object in a certain way. 研究者的視角（變異理論）：該教師的教學將怎樣給學生 創造學習機會使其以某特定方式感知數學目標。 • The researcher’s analysis was placed in contrast with the teacher’s and students’ commentary of the same lesson. 研究者的分析置於該教師和學生對該堂課的評價差異中。 19
School HK 1 香港某中學 • The school uses Chinese as the medium of teaching, which was the choice of the majority of the schools in Hong Kong. 授課語言：中文 （多說香港中學的教學語言為中文） • Among the schools in the city, the school belonged to average standard. 學校層次：中等 • According to the teacher, the students could be counted as a combination of average and slightly above average in mathematics based on the internal screening of the school. 學生層次：中等及中等偏上 • The average International Benchmark Test IBT score was 40. 26 out of 50. 國際基準考試均分： 40. 26至 50 20
The teacher 教師基本情況 • Mr. X is a teacher with more than twenty years of experience in both primary and secondary mathematics teaching. X先生在港小學、中學有多達 20多年的數學教學經歷 • He is active in teaching, curriculum development and research activities. 積極參與教學、課程發展以及其它研究活動 • He is recommended as a very good teacher by local mathematics educators, his school principal, colleagues and students. 被當地數學教育研究者、校長、同事和學生一致推薦 為一個非常好的教師 21
The students liked their lessons. 學生對課堂的喜歡程度 • Out of the 34 student interviews, 31 students expressed that they liked their mathematics lesson, two did not express any opinion and only one said that he did not like the lesson. 34位被訪學生中，有31位學生表示他們喜歡數學課、 2位未發表任何意見、1位表示不喜歡數學課。 22
The Theory of Variation 變異理論 • Marton, et al. (2004): Learning is a kind of experiencing in which the learners develop a way of seeing or experiencing. 馬登等：學習是學生發展一種經驗和理解方式的體驗過 程 • Learning in lessons is centred on objects of learning and it is very important for the learners to discern critical features of the object of learning. A key feature enhancing learning and awareness is variation. 課堂教學中的學習是以相關學習目標為中心的，由此教 師應能讓學生能區分該目標的關鍵特徵。而變異是一個 可加強學生學習和提升學生意識就的關鍵 23
Looking for variations in the analysis 分析過程中尋找“變異” • Contrast - a comparison between what the object is and what it is not, e. g. , “three” and not three, such as “two” or “four”. 對比 – 比較目標是什麽、不是什麽，例如，3 和 非 3， 比 如 2 或者 4 • Separation - separating a certain aspect of the object from the other aspects. To experience this, one aspect must vary while other aspects remain invariant. 區分 – 將目標的某一特徵與其它特徵區分開來。為此，就 應當對該特徵進行變異而將其它特徵保持不變 • Fusion – an experience of taking several critical aspects into account. 24 糅合 - 同時考慮該目標的多個方面
The lesson: Factorization of Polynomials 一堂課：多項式因式分解 • Introduction in the form of teacher-led whole class discussion (4 minutes). • 課堂引入 （4分鐘）：教師引導下全班討論為主 Main part (16 minutes) of teaching in the form of teacher-led whole class discussion, which included a very brief small group discussion (less than one minute). • 主體（新課講授）（16分鐘）：教師引導下全班討論為主，短 暫的學生小組活動（為時不到一分鐘） Supporting activities included individual seatwork, teacher’s between-desk-instruction students working on the board and the teacher’s comments on the students’ board work (7 minutes). • 課堂輔助活動（7分鐘）：學生獨自練習、師生間小組討論，學 生黑板演示隨後教師給予評論 25
Introduction of the name of the topic “Factorization of Polynomials” “多項式因式分解”含義介紹 T: What does it remind you of? ” 各位同學你睇個題目，會諗起d乜呢？會諗起乜？ Associated meaning of the name and old knowledge Patrick : Addition and subtraction. 加減。 將新知識的含義和舊知識聯繫在一塊 T: It reminds you of adding and subtracting numbers. Thank you. Why do you think about addition and subtraction? 諗起數字加減。唔該。點解你會諗起呢啲 數字加減呢？ Patrick: Simplify complicated things. 因為由複雜轉為簡單。 Invitation of elaboration 尋求進一步解釋 T: Who’s more imaginative? What comes to your mind when you’re reading this question, factorizing polynomials? What do you know and what do you want to know? 邊個同學有d想像力，你見到呢條題目多項式因式分解，你會諗起乜？或者你已 經知道乜，想學d乜？ Mark: Learning Factors. 學到d因數。. Invitation of participation & probing 邀請參與和進一步的促進學生思考 26
Starting from a very simple example 簡單例子“m(a+b)=ma+mb” 的導入 T: This is a very simple expansion of polynomial. From here to here [the teacher writing two arrows of opposite directions on the blackboard] multiplication starts from left to right. But from right to left…you know it’s correct once you see it. This has a name. What’s the name? [The teacher wrote the word “factorization” below the identity on the blackboard. ] 呢個係一個簡單到不得了…多項式展開式。由呢度過呢度[T在黑板畫上箭咀]， 由左邊變形到右邊就係乘式展開，但係由右變番左邊…你一望就知實啦，我 有個名，呢個名叫乜呢？[T寫因式分解] T: The name is factorization. Expanding two or more polynomials. If you do it in the reversed way and make the answer…make the answer into the question, this is called factorization. 呢個名叫因式分解。換言之，將兩個或者以上多項式乘開佢 ，將個結果… 將個結果當係題目，要你調轉頭寫，呢個過程就叫因式分解，或者有d書叫 分解因式。 A contrast between multiplication and factorization: 乘法和因式分解的比較: 符號iconic (箭頭arrows), 讀reading (L-R and R-L) 27 27
Correction/Feedback for a student’s example: 對學生例子“ 2(m+s)=2 m+2 s” 的糾正/回饋 T: It’s absolutely correct. Can you explain which direction you should go in factorization? You’ve given me the example, but it’s just an example of expansion. I won’t count it as an example of factorization. 一定，咁你解釋一下，點樣方向先係因式分解。你家俾你條題 目我，我當你係乘法，我唔當你做因式分解。 Mark: 個2係大家公因數。 促進學生的解釋 Two is their common factor. Probing for the student’s explanation T: Is it called factorization if I go from left to right or from right to left? 我依家問你由左邊做到右邊，叫因式分解，定由右邊做到左邊 係因式分解？ Mark: From right to left. 促使學生比較 Probing for the contrast 右邊到左邊。 28
Correction/Feedback for a student’s example: 對學生例子“ 2(m+s)=2 m+2 s” 的糾正/回饋 T: Thank you. I want to emphasize this. In order to…explain this more clearly…we use to read from left to right. This is our habit. When we look at the graphs, whether it ascends or descends, we read from the left to the right. In order to show you what is factorization, we write in this way. [The teacher wrote 2 m+2 s=2(m+s) replacing the student’s on the blackboard. ] 唔該。我要強調呢一點。為…為方便解釋…因為我習慣由左睇到右架麻，呢 個係我習慣，包括我地睇圖，升呀、降呀，左邊睇到右邊，為強調咩叫因 式分解，就咁寫。 T: From left to right. We simply call this factorization. You may think that this is very easy. You only have to reverse it and change the answer into a question. If the question asks you to multiply…eight, that’s it. Is it that simple? Theoretically, it is, but there’re many techniques in the questions. 由左邊做到右邊，呢個過程呢，我簡單講做因式分解。你話好簡單，將佢 調 番轉，好似將答案當係題目，原來乘… 8題目當答案，咁就搞掂，係咪咁 簡單，概念就可以咁講，但係題目就有好多唔同技巧，我家分別睇先。[T 在黑板寫na+nb]. 29 Starting from the simple but more to follow 由易入難
Variation in a sequence of 7 examples 練習中的變異 Q. 1 Q. 2 Q. 3 Q. 4 Q. 5 Q. 6 Q. 7 na+nb 2 a+2 b 2 na+2 nb -2 na-2 nb -2 na+2 nb 2 na+2 n 2 b 2 Variation is built in the design of the tasks: the contrast between the questions 該練習中有設置一定的 變異：題目間的對比 30
Fusion: Observing different aspects of the process of factorization together 糅合 ：因式分解過程中觀察不同的方面 T: Pay attention now! Questions one and two are easy! Questions one and two are similar to Question three. It involves "two", and it involves "n". Is it embarrassing? Which one should we deal with first? 留心喎，第一、第二題太淺，第一第二題同第三題大家幾似樣，又 有2，又有n，會唔會好尷 尬呢，都唔知做邊…做邊個。 • [After getting the three answers, “n(2 a+2 b)”, “ 2 n(a+b)” and “ 2(na+nb)”, from the students, ] [在從學生處得到如下三個答案后：“n(2 a+2 b)”, “ 2 n(a+b)” 和 “ 2(na+nb)”] Q. 1 na+nb, Q. 2 2 a+2 b, Q. 3 2 na+2 nb 31
Fusion: Observing different aspects of the process of factorization together 糅合 ：因式分解過程中觀察不同的方面 T: know…these three answers…are they the same as the original formula? ） n(2 a+2 b)，好，我首先問先，呢三個答案，同埋原來多項式一唔一 樣先？ S: The same! 一樣 T: Then are all three answers correct? 既然一樣的話，係咪三個答案都啱 呀？ S: No! 唔係。 T: No? You said they are the same and now they are not all correct? How come? Discuss with your classmates and then tell me your conclusions! 又唔係。又話一樣，又話唔係，究竟點樣？你地可以討論。 32 Q. 1 na+nb, Q. 2 2 a+2 b, Q. 3 2 na+2 nb
![Expecting more…than intuition 預期多於直覺 T: [The discussion lasted for less than one minute. ]](https://present5.com/presentation/f3d9294d2e322d46eb34f7ecf0f93753/image-33.jpg "Expecting more…than intuition 預期多於直覺 T: [The discussion lasted for less than one minute. ]")
Expecting more…than intuition 預期多於直覺 T: [The discussion lasted for less than one minute. ] Time’s up. Some of you have no conclusion! Well…in fact…these three answers…leave out those things, they are actually the same as the original. Which one is the factorization then? The first one, second, or the third? [討論不到一分鐘] 俾d時間你，討論到呢度。有d同學話冇 結論。 好，三個答案其 實呢，如果我 唔計佢 屬於乜，數清楚兩個係一樣。但係我 做因式分解，寫第一個、第二個抑或第三個？ S: The first one! 寫第一個。 T: The first one! Why is it the first one? 寫第一個，點解寫第一個？ S: 順眼d。 It seems to be! 33 [全班大笑All laughing]
Q. 1 na + nb Q. 2 2 a + 2 b Q 3: 2 na + 2 nb Contrast with 12=4 3 與12=4 3對比 “n(2 a+2 b)”, “ 2 n(a+b)”, “ 2(na+nb)” T: It seems to be. Okay, Thank you. Let's think back when we were in primary school. How did we do the factorization? …Can we write in that way? [The teacher wrote 12=4 3 on the blackboard] 順眼d。唔該。我回憶小學時候，我做因數分解我點做呢？[T在黑 板上寫 12=4 x 3] T: Can we? That’s factorization, right? What then? How to rewrite it then? Two times three. Okay! We can factorize further, and further. The question is that whether this question can be further factorized. [The teacher pointing at 2(na+nb). ] 可唔可以寫成咁就算？因數分解囉，搞掂囉。得唔得？會點呀？ 可以點呀？要點寫？再寫成點？ 2乘 2乘 3。我可以分解落去，就一 路分落去，唔可以停，問題係，睇吓喎。呢個有冇 得再分呢？[指 著 2(na+nb)] E: Yes! 有！ T: And this one, can it be further factorized? 即係試到啦，第一題嘛，係嗎？呢個可唔可以再分呀？ 34 E: Yes﹗ 可以。
So, which one should we choose? 那麼，我們應該選哪個呢？ “n(2 a+2 b)”, “ 2 n(a+b)”, “ 2(na+nb)” LEO: The middle one﹗ 中間個。 T: The middle one. Give me a reason. 中間個，你再講一次個道理係乜。 LEO: Because it is the simplest one! 因為中間個係最簡。 T: The biggest? 最大？ LEO: The simplest! 最簡！ T: The simplest? You should say it is…what factor? It shouldn’t be that complicated. (…) 最簡。最簡潔，不如你話呢一係…即係乜因式重好啦，駛乜複雜 (…). 35 LEO: True factor. 質因式，質因式。
So, which one should we choose? 那麼，我們應該選哪個呢？ “n(2 a+2 b)”, “ 2 n(a+b)”, “ 2(na+nb)” T: It should be …it should be what factor? Didn't you learn this in the previous lesson? 有應該係乜因式，上一堂學過。 T: What factor? How to spell it in English? 乜因式呀？或者乜點串呀英文？ LEO: LCM. 最小公倍數。 LEO: HCF. 最大公因數。 T: To say in simple words…in fact…it is a HCF. But this is not a HCF, just a CF, that is the common factor. This is the common factor, but not the highest common factor! If you want to have a complete factorization, you must have the HCF. 簡單講淨係呢一個…呢一個[指著 2 n(a+b)]係…呢一個係HCF，你話呢 個(指著 2(na+nb))唔係咩？呢個只能算CF，公因數，呢個係(n(2 a+2 b) 公因式，但係唔係最大，如果要做到分解完整呢，要係最大公因式， 36 即係HCF。
The teacher’s perspective 教師的角度 The topic as basic and important for paving the way of learning in future. 該知識點是以後學習的重要基石之一 I have to build up the meaning of factorization of polynomial in their mind and pave the way of future. This is a very important beginning. That is if the beginning is not good, they do not know what is the use of factorization of. 我必須為學生後繼學習奠定基礎，這個是非常重要的開始。如果開頭基 礎不打好，他們不知道因式分解有什麽用 Well, actually, mm, for a new topic, this is the basic work. Well… if we can do it well, this would be… helpful to the smooth flow of …the teaching of the lesson. 還有，事實上，恩，對於一個新的知識點，這是最基本的 作。好了， 如果我們做好了它，那麼它就會…有助於後面順利的完成今天的教學任 37 務
Summarizing his pedagogical style 該教師教學風格總結 • Developing new knowledge from the old knowledge 在已有知識基礎上發展新知識 • Developing habits in mathematics 培養學生的數學習慣 • Developing a method of comparison 培養學生比較的方法 • An awareness of the difficult points 清楚地意識到教學難點 • Catering for the need of the students 關注學生需求 • Being reflective on his lesson 對教學的反思 38
Student: Naomi 學生: 拿俄米 He saw the sequence of the six questions from easy to difficult as important. 他感知到了這六個問題的難易程度 Naomi: From now on…from question 1 to 6…these examples help us to understand how to solve these problems step by step. 從現在開始…從問題1到問題6…這個例子有助于我們理解怎樣一步一步地解 決這些問題 Int. : How do you understand? 你是怎麼知道的 Naomi: He goes through the examples from the easier ones to the difficult ones very slowly. It’s easier to understand. He won’t give you difficult examples at the beginning. He explains well too. The students…I think his explanation is very clear and thorough. 他演示這些例子，從點單的問題到複雜的問題非常緩慢。這個理解起來比較 容易。他不會一開始就給你難的問題。他的解釋得也很好。學生… 我認為他的解釋很清楚和完整。 39
Student: Leo 學生: 里奧 Leo had some kinds of general rules for seeing some forms of the lesson activity important. 里奧有自己的一套方式來判斷學習活動的重要性 – The revision and the beginning was important. 複習和開頭是很重要的 – Difficult questions were important. 比較難的問題是重要的 – The checking of answers was important. 核對答案是重要的 40
Class Atmosphere: T-Ss 課堂氣氛：師生關係 T: We want to teach the students to understand all the knowledge and make them happy and relaxed. 我想讓學生理解所有的知識，而且讓他們能快樂和輕鬆地掌 握這些知識。 Naomi: It’s not related to mathematics. We’re just laughing, so that we are not that bored, just to relax a bit. 這個跟數學無關。我們只是在笑，我們不會覺得枯燥，僅僅 是放鬆了一下。 Leo: We’re doing a lot of other things, but the teacher doesn’t see it. …Such as…chatting. 我們在做很多其它的事，但是老師沒有發現… 比如…聊天。 41
A directive approach 一個直接引導的方式 • The teacher talk was a major input component of teaching. 教師是一個主要的輸入載體 • The teacher was directive in most part of the discourse showing a clear guidance for students to observe certain features. 在大多時候，教師都是直接講授，直接引導學生去觀察某些特 徵 • The guidance was planned by a careful design of the mathematical problems and the key oral questions. 教師通過設計數學問題和關鍵的口頭問題以到達對課堂的直接 引導 • This directive approach: keeps the focus on the mathematical content; following the teacher’s plan; good time control; and relatively efficient. 直接引導方式：聚焦于數學知識、嚴格執行教學計劃、教學節 奏控制到位、相對高效 42
The technique of using variation 運用變異的方法 • The technique of variation was used in the design of the mathematical problems. 問題設計是運用變異 • The technique of variation was used in the design of the mathematical problems. 在課堂交流中進一步創造變異維度 • These variations were obviously well picked up by the teacher in the actual lesson as he pointed to the contrast between questions and answers in the class discourse. 在實際課堂教學和交流中，該教師通過比較問題和答案實現 了相關變異 • They were also mentioned in his reflection of the lesson and they were referred to as advanced planning. 相關變異在其課後反思中也有提及並且進一步被運用到後繼 43 備課中
Similarity and diversity in the students’ perspectives 學生視角的相似性和差異性 • Naomi may have focused on the variation of the levels of difficulty in a simplistic way, whereas Leo may have focused on the same episodes based on his principles: revision, difficult point, checking (reflective). 拿俄米可能以一種簡單的方式在認識難度層次上的變異認識， 然而Leo可能根據自己的方式來判斷：複習、難點、核對練習 （反思） • In both cases, the students were very attentive and ready to receive clear explanation from the teacher. 這兩個學生都很關注且隨時準備接受教師的解釋 44
Is this what we want? 這是我們所追求的嗎？ To promote critical reflection and a positive attitude that implies modesty to prepare for ratification and correction, open-mindedness to receive alternative views and possibility, and a caring heart to share their learning process with the others. 為促進學生批判性的反思和積極的態度: 讓他們能有隨時接受批 評和指正的謙虛的心、有接受不同觀點和可能性的開放的頭腦、 有與其他學生分享其學習歷程的關懷的心。 A challenge for many teachers in their daily teaching!!!! 這是個大多數老師每天面對的挑戰！！！！ 45
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The Learner’s Perspective Study LPS
Where is the Paradox? A Mismatch between Images and Outcomes • Classrooms in Asian regions were sometimes described as teacher-dominating with passive learners. This was an essentially negative image of learning environment. • The students in some of these cultures give very good performances in comparative studies such as the Third International Mathematics and Science Study (TIMSS) (Hiebert et al. , 2003) and the Programme for International Student Assessment (PISA) (OECD, 2004).
Some findings for the nature of Chinese teaching and learning • Leung (1995) compared mathematics classrooms between Hong Kong, Beijing and London; • Ma (1999) found that the Chinese teachers demonstrated profound understanding of the subject matter (PCK); • Gu, Huang and Marton (2004) described teaching with variation as a Chinese way of promoting effective teaching; • Lopez-Real, et al. (2004) studied a Shanghai teacher's teaching style and found varying patterns. (LPS) • Huang et al. (2006), based on the analysis of the teaching of the particular procedural method of elimination in Hong Kong, Macau and Shanghai – Practice can go beyond drilling by rote but help building up an interrelated knowledge structure. (LPS) • Mok (2006 a) analysis of a SH 3 teacher. (LPS) • Mok (2006 b) analysis of a SH 2 teacher. (LPS) • Mok (2009) analysis of a HK teacher. (LPS) • Mok (2010) analysis of mathematical tasks
Learner’s Perspective Study (LPS) http: //www. edfac. unimelb. edu. au/DSME/lps/ • • • Australia China – Beijing, Hong Kong, Macau, Shanghai Czech Republic England Germany Israel Japan Korea New Zealand • • • Singapore South Africa Sweden The Philippines USA
Key features in the LPS design • Document the teaching of sequences of lessons, rather than single lessons like the TIMSS video study. • “Record” lessons from multiple perspectives: video, teacher- and student -interviews. • Eighth grade lessons were recorded in three classrooms (one for each schools) for each region.
Methods • A minimum of 10 consecutive lessons were recorded for each class/teacher. • Videos from three cameras with a technique of on-site mixing of the images to provide a splitscreen record of both teacher and students. • Two students were interviewed after each lesson. The video-stimulated recall technique was used. • The teacher was interviewed three times during the whole period of recording. The teacher was asked to comment on one of his/her lessons chosen by himself/herself.
Selection of Sample • School Selection: Schools in urban/metropolitan communities (Shanghai, Hong Kong, Macau and Beijing) • Teacher Selection: Three competent teachers in each city (at least five years of experience as a qualified teacher) • Class Selection: One secondary-2 class per teacher, in order to match the database of TIMSS Video Study and the Learner's Perspective Study. • Lesson Selection: A continuous sequence of at least 10 lessons for each class.
Camera Configuration 1. 3 cameras are employed – a “Teacher Camera”, a “Student Camera” and a “Whole Class Camera”. 2. On-site mixing of images from two video cameras to provide a split-screen record of both teacher and student actions. 3. A third camera recorded “corporate” student practices which is common to the whole class group. Teacher Image Student image a split-screen record
Video-stimulated recall in interviews n Conduct student interviews immediately after the lesson n To obtain participants' reconstructions of the lesson and the meanings which particular events held for them personally.
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