在斯坦福旁觀中美學生數學對抗賽
The notorious Gaokao, (高考, or “High Test”) is China’s SAT on steroids, with a score on the nine-hour test being the sole criterion for admission to Chinese universities. Preparing for the test is a years-long obsession for both students and parents. (In case you missed it: Brook Larmer’s Inside a Chinese Test-Prep Factory.) And for many, the unfortunate consequence is that the lengthy preparation destroys, rather than enhances, academic ability. Student enthusiasm and curiosity are crushed.
聲名狼藉的中國高考相當於美國的學術能力評估測試(SAT)，但強度高得多。長達九個小時的高考的成績是中國大學錄取的唯一標準，備考是一件讓學生和家長操心多年的事情。（如果你還沒讀過Brook Larmer的報道“毛坦廠中學，中國應試教育工廠”，請點擊此處。）對很多人來說，高考的不幸後果是，學術能力被冗長的備考過程毀掉而不是增強了。學生們的熱情和好奇心受到摧殘。
Although many in China are aware of the Gaokao’s impact, the test has a 1,300-year history and will not be easily killed. Instead, perhaps the best way to break the Gaokao’s life-draining grasp is indirectly, through clubs and activities that rejuvenate kids’ sense of curiosity and fun. And two weeks ago I discovered one such extra-curricular activity that’s becoming popular among Chinese math-lovers: The Math League, an organization based in New Jersey committed to having kids worldwide enjoy math and discussions about problem-solving.
雖然在中國有很多人知道高考的巨大影響，但這種考試有1300年的歷史，並不會被輕易取締。因此，打破高考令人窒息的控制的最好方法或許是間接的，靠通過各種興趣小組和活動，重新激發孩子們的好奇心和趣味感。大約兩週前，我發現了一項這種課外活動，它正逐漸受到中國數學愛好者的歡迎：數學大聯盟杯賽(The Math League)。這家總部設在新澤西的機構致力於讓世界各地的孩子喜歡數學，喜歡有關解決問題的討論。
The Math League organizes math contests around the world, including an annual China vs. United States event held annually at Stanford University. After this competition I had the opportunity to spend a day with the Chinese team introducing the some of the topics (cognitive bias, critical thinking, mathematical creativity, intuitive problem-solving) the students may find useful if they pursue math or computer science as a career. The team was wildly enthusiastic (see photo at the top of this post), perhaps a bit too rowdy at times, but with an abundance of curiosity and self-expression. Any negative impact from the Gaokao seemed, for the moment at least, to be gone.
數學大聯盟杯賽在世界各地舉辦競賽，包括在斯坦福大學舉行年度中美對抗賽。今年的比賽結束後，我有機會和中國隊花了一天的時間，向隊員介紹一些話題，比如認知偏見、批判性思維、數學創造力，以及直覺性的解決問題方法等。如果學生們想從事數學或計算機科學方面的工作的話，可能會覺得這些話題有用。中國隊熱情洋溢（見本文頂部的照片），有時可能有點兒過於吵鬧，但是他們充滿了好奇心，會充分表達自己。任何高考造成的消極影響至少在那個時刻看不到。
Certainly the Math League experience was at the core of this enthusiasm, and this week it’s my pleasure to share the questions used for the competition. (The Math League has copyrighted the questions, and they are used here with permission.) There are quite a few questions, so for our main challenge this week we’ll consider the following five:
參加數學大聯盟杯賽的經歷當然是這種熱情的核心，本週我有幸與大家分享競賽中使用的試題。（數學大聯盟杯賽擁有這些試題的版權，本文經過許可之後使用。）試題很多，因此，作爲本週的主要挑戰，讓我們來考慮下面的五個：
Fourth Annual Stanford Math League Competition (Sample Questions):
第四屆斯坦福數學大聯盟杯賽年度競賽（樣題）：
1. If my age 24 years ago was 24, what will my age be 24 years from now?
1、如果24年前我是24歲，那麼24年後我多大？
2. What is the minimum number of people needed in a room to be certain that at least 3 of them were born in the same month?
2、一間屋子裏至少要有多少人才能確保至少有3個人是在同一個月份出生的？
3. If the sum of three prime numbers is 20, what is the largest possible value of one of the three primes?
3、如果三個質數之和爲20，那麼這三個質數中可能的最大值是多少？
4. If my brain weighs 0.90 kilogram plus half its weight in kilograms, how many kilograms does my brain weigh?
4、如果我的大腦的重量等於0.90千克加上其一半的重量，那麼我的大腦究竟有多重？
5. Working alone, a pump fills an empty pool in 3 days. If it works with a second pump, the two pumps together can fill an empty pool in 2 days. In how many days can the second pump, working alone, fill an empty pool?
5、一個水泵獨自工作需要3天充滿一座空水池。如果加上第二個水泵，兩個水泵一起兩天就可以充滿這座空水池。那麼，倘若第二個水泵獨自工作的話，需要幾天才能充滿？
These questions will serve as our official challenge for the week. They aren’t too tough, but how about 60 of them — in 45 minutes? That is the idea behind the Math League’s Speed Test 60. Is the test actually fun? Give it a try with this year’s test.
這些問題是我們本週的正式挑戰。它們並不難，但要是有60道題，而且只給45分鐘。這就是數學聯盟60題快速測試背後的理念。這個測試的確很好玩吧？來試一試今年的試題！
If your pencil’s still sharp and you’re up for more, you might consider the following challenges — also from this year’s Stanford contest. These have tougher questions, but there are only 10 of them. Time limit: 70 minutes for all 10 questions. Here are the Grade 6 and 7 test (10 questions) and the Grade 8 test (10 questions).
如果你還意猶未盡想做更多的題，可以考慮接下來的挑戰——也是出自今年的斯坦福競賽。它包含更難的問題，但是隻有10道。此處是六七年級的測試（共10題）和八年級的測試（共10題）。
The average score for Chinese Math League team members, by the way, was 30 out of 60 for the speed test and 5 out of 10 for the individual tests. (Speed Test answers posted at the end of this post. Individual test solutions to be posted on Friday, Sept. 4.)
順便說一下，中國數學聯盟隊員的平均成績是：60道速答題答對30題，個人賽10道試題答對5題。（速答題答案見本文文末。個人賽試題的答案將於9月4日的週五公佈。）
I was impressed by the Math League program and asked its co-founder Dan Flegler about its impact. Here’s his response:
數學大聯盟杯賽的試題給我留下了深刻的印象。我向這一活動的聯合創始人丹·弗雷格勒(Dan Flegler)詢問了賽事的影響。下面是他的回答：
I believe that contests such as ours instill a love of mathematics in students and spur them on in their study of mathematics. The best math student I ever taught was getting Cs in the honors ninth-grade math course I was teaching when his father bribed him with a $5 bill to join the math team. On the first contest he ever took, he missed 5 of the 6 questions, but got the very hardest question correct. I was amazed since the problem he got correct was a geometry question that I thought he lacked the requisite knowledge to solve. It turned out that he found a novel way to solve the problem that was completely different from the way I thought it should be done. He went on to complete algebra 1, geometry, algebra 2 and trigonometry in about four months, studying independently under my direction. He eventually won virtually every high school math competition in New Jersey and later got his Ph.D. from Stanford University in applied mathematics. But this never would have happened if he hadn’t been inspired by a math contest.
我覺得，我們舉辦的這類競賽，向學生們灌輸了對數學的熱愛，激勵了他們的數學學習。我見過的一名最好的數學學生，他在我教的九年級（相當於中國高一）數學優等班裏的成績只是C，後來他父親用5美元賄賂他加入了數學競賽。他首次參賽時，6道題做出了5道，但答對了最難的一道。我很驚訝，因爲他做對的那道是幾何題，我以爲他缺乏必要的解題知識。原來，他發現了一種全新的解題方法，與我認爲應該用的方法截然不同。他在我的指導之下，在四個月的時間裏，獨立完成了代數1、幾何、代數2和三角的學習。他後來還在新澤西州的幾乎所有高中數學競賽中獲勝，並最終獲得斯坦福大學的應用數學博士學位。然而，如果不是因爲他的靈感在一次數學競賽中被激發，這一切也許永遠不會發生。
The only students who ever got perfect scores on the speed round were two fraternal Chinese twins (a boy and a girl). Neither I nor any of my colleagues can get a perfect score on the speed round when we take the speed round for fun (even when I write many of the questions myself!). Trying to do 60 questions in 45 minutes isn’t an easy task for anyone. The speed round is really very different from anything else we do in our competitions — we save it for last since it’s like dessert after a heavy meal.
有史以來，只有兩名學生做速答題得了滿分，他們是來自中國的一對異卵雙胞胎（一個男孩和一個女孩）。無論是我、還是我的任何同事，我們爲了好玩做速答題時，都得不了滿分（儘管其中有不少道題是我自己出的！）。試圖在45分鐘內做60道題對任何人來說都不是一件容易的事。速答題與我們競賽中的其他試題完全不一樣，我們將速答題保留到最後，因爲它們像是一頓大餐後的甜點。
And the Chinese student experience with Math League contests? I asked Rui Hu, coach of the China team (Dr. Hu is at the far right in the photo at the top of this post). Here’s his response:
中國學生參加數學大聯盟杯賽的經歷如何？我問中國隊主教練胡銳博士（本文頂部照片後排最右邊的那位）。下面是他的回答：
The Math League and the Stanford International Youth Program enable Chinese students to view and learn math and science from a completely different perspective from what is available in China.
數學大聯盟杯賽和斯坦福國際青年計劃(Stanford International Youth Program)讓中國學生能從另一種角度觀察和學習數學及科學，與在中國能得到的角度完全不同。
I find two major problems with current Chinese education. The first is math education. All students are forced to study Math Olympiad problems, very hard problems, even though a lot of these students have no interest in studying these hard problems designed only for math geniuses. This makes Chinese students the least happy students in the world. But Math League contests are well-known math competitions in North America and in the world, with more than 30 years’ history. They present fun and creative problems that promote critical thinking and problem-solving skills, the two most important skills that Chinese students are missing in the current exam-oriented, cram education system in China. It is much more fun working on Math League questions than on extremely hard math problems. Math League contests present a complement to the current Chinese math education. To me, I think the best way is American-style math plus Chinese-style rigorous practice and sound, systematic curriculum.
我覺得目前中國的教育存在兩個主要問題。首先是數學教育。所有的學生都被迫學習奧林匹克數學競賽試題，這些題非常難做，儘管學生中很多人對這些專門爲數學天才設計的難題毫無興趣。這讓中國學生成了世界上最不幸福的學生。而數學大聯盟杯賽是北美和全球有名的數學競賽，有30多年的歷史。他們出的題既有樂趣，又有創意，有利於提高批判性思維和解決問題的能力，這是在目前中國以考試、應試爲主的教育體系中，中國學生所缺少的兩個最重要的技能。做數學大聯盟杯賽的試題，比做極難的數學題有趣多了。數學大聯盟杯賽對目前中國的數學教育是一種補充。在我看來，我認爲最好的結合是美國式的數學，加上中國式的嚴謹練習和完善的系統性課程。
The second problem is English education. Chinese students spend a lot of time on studying English. But Chinese teachers teach students English with the goal to help them achieve high scores in various English tests, such as the TOEFL (Test of English as a Foreign Language) and SAT, not the goal to use English in daily life. So Chinese students achieve very high scores on English tests, but can’t communicate with English speakers. This is true in math education, too. The goal is scoring high in contests like Math Olympiad, not developing students’ true interests in math. So ironically, Chinese students hate math and English, although they spend a lot of time studying math and English. And this is where Math League can help as well. Students can learn native English and have fun solving Math League problems.
第二個是英語教育。中國學生花很多時間學習英語。但中國老師教授學生英語，是以幫助他們在諸如託福（Test of English as a Foreign Language的中文簡稱）和SAT等各種英語考試上得高分爲目標，不是以在日常生活中使用英語爲目標。因此，中國學生的英語考試得分很高，但不能與說英語的人交流。在數學教育中也有這個問題。我們的目標是在如奧林匹克數學競賽上得高分，而不是激發學生對數學學習的真正興趣。因此有一種諷刺性的後果，那就是中國學生討厭數學和英語，雖然他們花了很多時間學習數學和英語。而這也是數學大聯盟杯賽能提供幫助之處：學生能在學習純正英語的同時，享受解決數學競賽試題的樂趣。
And do students focused on the Gaokao really want to take time out for activities like Math League? Here’s Dr. Hu:
集中精力高考的學生真能抽出時間來參加像數學大聯盟杯賽這樣的活動嗎？下面是胡銳的回答：
The current Gaokao system, which was restored in the winter of 1977 by Chinese leader Deng Xiaoping, played the most important and critical role in lifting China from poverty and making China the second-largest economy in the world. Thanks to the Gaokao, millions of quality students are able to enter universities in China; most are trained to be the quality engineers who helped create modern China. But the Gaokao and the education system in both K-12 and university levels have their shortcomings. It is basically a cram system. It doesn’t really encourage critical thinking, creative thinking or intuitive problem-solving. This partially explains why China is very good at copycatting, but not necessarily good at innovation or creativity.
目前的高考制度是中國（已故）領導人鄧小平在1977年冬季恢復的，這個制度在讓中國脫離貧困、成爲世界第二大經濟體中起了重要和關鍵的作用。由於高考，中國數以百萬計的優秀學生得以進入大學讀書；他們中的大多數都成爲中國現代化過程中訓練有素的工程師。但高考、以及中小學和大學教育體制也有自己的缺點。基本上是一種應試教育。沒有真正鼓勵批判性思維、創造性思維，以及直覺地解決問題的能力。這也部分解釋了爲什麼中國能非常好地山寨別人的東西，但創新能力或創造力不一定行。
China has produced many, many quality engineers, but not enough innovators or independent thinkers. The Chinese government realizes this problem, so do Chinese parents and teachers. But it is extremely hard to change the current Gaokao system. No one expects the current Gaokao system will change in foreseeable future. So they (Chinese parents, students and teachers) are looking at alternatives or complementary, and the United States is their first choice, as it is the most powerful and innovative country in the world.
中國已經產生了很多很多訓練有素的工程師，但沒有足夠多的創新者和獨立思考者。中國政府意識到了這個問題，中國的家長和老師也意識到了。但是，很難改變目前的高考制度。沒有人認爲目前的高考制度會在可預見的未來能有改變。因此，他們（中國家長、學生和老師）正在尋找提供替代或補充的東西，美國是他們的第一選擇，因爲美國是世界上最強大、最有創新力的國家。
Not all the students who participate in Math League contests in China are planning to go to non-Chinese universities. I would say more than half of them are still planning to go to Chinese universities. But these Chinese students want to try something new, something that is innovative and creative, something that is complementary to the current Chinese education system.
並非所有參加數學大聯盟杯賽的中國學生打算去國外上大學。我覺得他們中一半以上的人仍打算在中國上大學。但是，這些中國學生想試試新的東西，試一些具有創新性和創造性的東西，一些爲目前中國的教育體系提供補充的東西。
Thank you, Dan Flegler and Rui Hu. For more about The Math League, check out its website, which offers information about contests as well as books with past that final picture of enthusiasm we wrap up this week’s challenge. As always, once you’re able to read comments for this post, use Gary Hewitt’s to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles to .
作者感謝丹·弗雷格勒和胡銳。有關數學大聯盟杯賽的更多內容見其網站，網站提供競賽信息以及歷屆競賽的試卷。本週的挑戰以最後這張熱情洋溢的照片結束。與往常一樣，如果你能讀這篇文章的評論，還可用加里·休威特(Gary Hewitt)提供的工具來正確地顯示公式和圖形。（點擊此處查看工具的有關介紹）。請把你喜歡的智力遊戲用電子郵件發給。
Solutions
試題答案
The Speed Test brief solutions can be found here. Check back on Friday, Sept. 4, for solutions to the Individual Problems from The Math League.
速答題的簡單答案可以在這裏找到。請於9月4日的週五到本欄查看數學大聯盟杯賽個人賽試題的解答。
