国际学校之数学篇-学生分享
国际学校数学篇
冯柏锐，香港大学精算在读。数学强人一枚。
Two years ago, when I was fresh into the International Baccalaureate Diploma and said that I had chosen to study HL Mathematics, Physics and Economics, my peers often responded with “Wow, so Asian?”. Indeed, the IB Diploma are often referred to by students as “two years of suffering, torture and pain” due to its rigour and determined pursuit for one’s holistic development. The HL Mathematics (hereinafter ‘HL Maths’) programme has proved itself to be no different, although in earnest it is a challenging yet enriching and definitely worthwhile course.
两年前，当我告诉我的同学我选择的科目是高级数学、物理和经济学时，我的同伴都这样回应：“哇，极具亚洲学生特色的选择啊”。（西方学生数学很少选高级程度的，因为学不懂，其它的很多会从自身兴趣出发，如选择哲学、电影、设计等等）。的确， IBDP由于其严苛的学术要求被学生所描述成“两年的忍受、痛苦和折磨”。高级程度的数学（以下简称HL 数学）也是如此，充满挑战，但是富有意义，值得尝试。
As you may know, Mathematics plays an important role in the IB Diploma as it is acompulsory subject in the six subject groups (alongside TOK and the EE). The easiest option is the Mathematical Studies SL course, followed by the Mathematics SL course, then the Mathematical HL course. [These is also a Further Mathematics HL course taken outside of the six subjects, but that should only be considered by the most elite of young mathematicians as proficiency of the Mathematics HL core topics is an unwritten prerequisite before its commencing.]
数学在IB DP学科中是一门必选科目，一般来说，学生可以根据自身的程度选择其中三种不同的程度。最简单的是普通程度的数学研究，然后是普通程度数学最后是高级程度数学。当然，还有些极具数学天赋的学生可以选择高级程度的高等数学。
Below is the IB HLMaths syllabus in brief:
Core Topics (Compulsory) 1 Algebra and Complex numbers 2 Functions and Equations 3 Circular functions and Trigonometry 4 Vectors 5 Statistics and probability 6 Calculus | Options (Choose one) 7 Statistics and Probability 8 Sets, Relations and Groups 9 Calculus 10 Discrete mathematics
|
Topics are not equal in weighting andthe option is usually chosen by the teacher.
以上是高级程度数学的课程组成。
How does a Westerner teacher teach mathematics?
刺激的部分来了：西方人是如何教数学的？
This is not a question designed to promote racism or certain stereotypes, but is perhaps of interest to many parents especially Asian parents. Yes, of course they can teach maths, and to a very high standard too!
这个问题并非想刺激某种刻板印象，但是我想其实很多的亚洲家长都会很想知道。是的，当然，西方老师会教数学，而且会教的很好。（真的吗？下一篇等苏老师来吐个槽）
I do not wish to over-generalize, but one very notable difference is that such a teacher aims to teach not just the syllabus, but also mathematics and its application into the real world. It must be clarified and emphasized that ‘syllabus mathematics’ and ‘mathematics’ are two totally different concepts: the syllabusis merely mechanically memorizing different formulae then reproducing it for their end-of-study examination, whilst mathematics is learning about andunderstanding the intricacies of different mathematical techniques, logical problemsolving, and its utilization in life.
我不想以偏概全，但是一个非常明显的不同是西方老师目标不仅仅是只教大纲规定的内容，而且希望让数学可以真正运用到真实的生活中。在这里，我必须澄清和强调“大纲中的数学”和数学是完全不同的概念：大纲强调机械的记忆，复制，考试而真正的数学是理解不同数学工具的微妙之处，发展自己的解决问题的能力，确保数学在生活中的运用。
Both my teachers for iGCSE Maths and Additional Maths and IB Maths liked to start their lessons with a short fifteen minute problem solving activity, and these could be as simple as how to equally share a cake of irregular size (figure 1), or as complex asthe Monty Hall Problem and beyond. Our teachers took great care to find and create such activities of appropriate level; these were very stimulating in discovering maths in different real-life contexts and also were great warm-up preparations for the subsequent learning of maths in those lessons. Also, such teachers have a very different teaching style and the learning of the syllabus is not confined to the textbook; rather,our teachers taught the syllabus using a problem-based approach and emphasized its relevance in real life. Hence, this course is a very practical course and is also very useful for future life.
我的iGCSE和IB两位老师都喜欢用一个15分钟的数学活动作为上课前的热身练习。有时候是简单的如如何等分一块形状不规则的蛋糕，有时可以是复杂的“三门”问题。我们老师花了很大的力气发现设计这些活动以适合我们的数学程度。这些活动非常有益于我们在真实的生活语境中发现数学。老师的教学风格不同，并不拘泥于教学大纲。老师喜欢采用解决问题的方式来教学，并且特别强调它在实际生活中的关联。所以，我认为课程非常实用，对未来很有意义。
Also, my teachers tend to give minimal homework, not because they weren’t bothered to mark it, butbecause they believed that a set piece of homework did not necessarily caterwell for the learning pace of each individual student. Rather, it was their philosophy to leave revision and consolidational work up to the proactiveness of the individual student; I found this great as it meant that I could revise on previous exam questions at my own pace without the additional burden ofhomework.
我的老师们留很少的作业！（这个点，下篇文章我必吐槽），并不是因为他们懒得改，而是他们相信作业并不能完全兼顾学生的个体差异，保证学生的进步。他们的态度是学生按自己的情况来决定如何复习巩固。我觉得这点太棒了，这意味着我可以按自己的进度复习以前的考题而没有作业的负担。
Workload and Revision strategies工作量和复习策略
The workload of this course hugely depends on the individual student’s level of competency in mathematics. On average, students who took this course spent approximately 70% of study time on it and I found myself to spend a similar proportion of my study time.
这门课程的负担适不同学生的不同程度而定。平均来讲，大家一般花70%的时间复习，我自己也差不多。（其他课也在学，但数学绝对是耗时最长的）
Revision strategies include exposing yourself to different formats and contexts of mathematics. Joining competitions such as the United Kingdom Mathematical Trust Challenges, the Mathematical Olympiads or the American Scholastic Mathematics Association challengesare very good brain teasers and exercises for the application of mathematical techniques and skills learnt and can enhance mathematical understanding. In terms of revising for the actual syllabus, revising from past papers is an imperative to obtain a gist of examiners’ paper-setting patterns.
复习策略是熟悉不同的题目和各种数学语境。参加各种数学竞赛也是一种很好的训练大脑的方法，也可以运用课堂上所学过的技能，从而增进自己对数学的理解。对于具体的复习内容来说，复习历年真题毫无疑问是最好的学习方法，熟悉出卷官的出题模式。
[Additional] the Internal Assessment 再来谈谈校内评估
The Internal Assessment (IA) is a compulsory assignment for all IB courses, where maths is no different. For Mathematics HL, 20% of the final grade is accounted for by the IA which requires that students write an essay, of about 12 to 18 pages, on any aspect of mathematics (of course, to a certain extent of mathematicalcomplexity and rigour) they may be interested in; for example, my IA was about the connectivity of the Hong Kong Mass Transit Railway (MTR) system and required me to self-learn about Graph Theory in Discrete Mathematics. Someother chosen topics by my fellow students included investigating the projectile motion of kicking a football and mathematics in advanced computer programming.
IB课程的不同科目中都有校内评估项目，数学也不例外。对于高级程度数学来讲，最终成绩的20%是校内评估项目，要求学生写一篇数学论文。论文12-18页，可以就学生感兴趣的任何一个数学话题展开（当然要确保内容的深度）。举例来说，我的论文写的就是关于香港地铁的连通性。我的同学的选题则有研究足球的抛物线运动和计算机编程中的数学。
The IA was a very good opportunity to demonstrate creative thinking in mathematics as well as how to formulate mathematical ideas into formal written text.Teacher involvement was allowed, to improve the essay’s quality, provided that it did not detract from the originality of the student.
这种数学论文是培养数学创造力的很好的方式，也是把数学想法变成文字文本的一个好的方法。老师可以提供一些帮助，但要确保学生的原创性是其中最重要的部分。
(A more detailed guide to the syllabus can be found in thisweblink:
)
See
Junior, Intermediate and Senior Challenges: