Faculty Feature | Paulo Tirao教授：像享受足球和艺术一样享受数学
三尺讲台春风化雨，门下自是桃李成辉。广东以色列理工学院拥有世界一流的教师团队，老师们全情投入，潜心钻研，成为学生们科研思维的开启者、科研道路的引路人，也因此收获学生们敬重和爱戴。本期开始，我们推出教师系列专访，分享广以的师长大家们在治学修身、教书育人上的真知灼见和感悟箴言。让更多人走进广以课堂，走近广以教师，了解广以的办学特色与育人理念。
在采访Paulo Tirao教授之前，我们已经从众多学生口中听闻他们对这位阿根廷教授课堂的热爱。他对于数学、对于教育，就像阳光从不吝惜他的家乡阿根廷的土地一样，永远充满了热情。他向人们讲解数学，又仿佛在讲解哲学。
谈到数学，你会首先想到什么？是试卷上难以计算的数学题，还是人类至今仍然无法完全验证的哥德巴赫猜想，抑或是应用广泛的金融产品定价模型？而Paulo Tirao教授告诉我们，数学更像发明，在某种程度上与艺术、体育相仿，需要充分的热情去探索。
以下，是Paulo Tirao教授的讲述。
1
在广以做数学
2020年，我来到美丽宁静的海滨城市汕头，开启了在广东以色列理工学院的教学。
我的最大感受是，这里非常适合做数学。因为思考数学问题就像培育树木，需要好几代人的共同努力，需要时间，更需要耐心。而在这里，或许是因为潮汕的品茶文化，这里的人们生活节奏没有那么快，也更愿意等待打磨的结果。我常常在散步、交流、品茶的过程中思考，能够静下心来做有价值的研究，这样的生活对数学家来说是非常完美的。
Paulo Tirao教授（左二）与同事、学生合影
而对于广以的学子们，这里除了有浓郁的学术氛围，也真正让学生能够各学所需。与以色列理工学院本部的课程设置相同，广以的数学课也分为不同层级的训练。各个系都提供了数学课，但相似的课程名称却包含着很大的区别。化工、材料等其他系的学生所学的数学课是“给工程师的数学课”，公式和术语对于工程师而言更像解决所在学科问题的工具。但数学系的学生上的是“给数学家的数学课”，广以提供非常严谨的系统性训练，学生们不仅需要学会使用数学公式和术语，更需要理解它们是如何被发明和创造的。扎实跨出去的每一步，都为未来的数学家之路铺垫基石。
2
数学到底是门怎样的学科？
任何时候学生问我这个问题，我都会回答他们，数学是一门非常特别的学科。
数学在许多方面与任何其他学科完全不同。数学是唯一一门由人类发明的学科。生物学和化学的研究是基于自然的规律，而经济是关于人类社会经济行为的研究，它们都有一个非常具体的目标。而数学研究更像开发新游戏。设想一下，国际象棋在被发明出来之前是不存在的。而当第一个人定下规则，推广玩法，国际象棋才逐渐成为人类文明生活的一部分。所以外行很难想象数学世界将要发生什么，甚至数学家都不知道未来将有何新发现。
所以我告诉学生，学数学要有耐心。当然，在广以数学系毕业后的选择可以是学术深造，也可以是进入行业工作。对于选择工作的毕业生，数学系所训练的技能将使他们能够快速适应不同的环境。而对于想要深造的毕业生，本科学习只是帮助他们完成了数学世界中基本的数学语言训练。
3
很多人都想知道，学数学到底能干什么
Paulo Tirao教授幽默的课堂深受学生喜爱
不管你是想要继续学术道路或是投身实际工作，数学系的训练都将使你拥有三项受益一生的能力：学习能力、严密思考能力和创新能力。
如果你去图书馆的书架上找到数学书，会发现它们大多数都只有100页或200页。和生物、化学、法律等其他学科的大部头相比，数学书显得如此单薄。这是因为数学是非常、非常凝练的一门学科。我的课堂上，每一学年的课程教材可能只有100页。然而，每一页内容可能需要花一个星期的时间才能充分理解。因此，在广以的数学学习中，除了特定的代数和计算机科学知识之外，还有另一部分：学会如何学习、学会从浓缩的内容中学习。
在广以数学系的培养目标中，每一个学生最相关的技能不是知识本身，而是学习新事物的能力。我常常告诉学生，当你开始学习数学时，你不仅是在学习代数。你是在学习“如何学习数学”。这是你在其他学科中可能不需要的技能：知道如何学习。这种可迁移的学习能力让学生在其他知识框架中也能够如鱼得水。这项技能似乎比较抽象，但实际上，每一个数学家已经被训练成以一种非常深刻的方式学习新事物。假设一个优秀的数学家进入汽车行业，尽管最初一无所知，但他将能够利用他的学习能力，在半年内从一个全新的环境中快速学习和掌握工具。
Paulo Tirao教授介绍数学与计算机科学专业
严密思考，也是数学研究所需要的重要能力。在数学中常常有这样的描述：区分某些事情是“在所有情况下发生”还是“在某些情况下发生”；“有一个数字不满足条件，就不是‘真’的”。这是数学的论述中必不可少的。严密思考，意味着提出准确的问题，做精确的陈述，佐以精确的论据和理由，这就是数学证明的核心，也是数学家独特的能力。科技在高速发展，超级计算机能够处理大量的数据，但数学中还有许多开放性问题至今仍未解决，这说明数学体系的思考框架并不能简单被机器替代。同时，数学严密思考的体系能够帮助解决许多其他实际问题，比如固定优先级、战略设计、组织关系等等，在医学、股票、工业、化学、物理等领域都能被广泛运用。这也是我们为什么一直注重训练学生严密思考能力的原因。
创新能力是数学系所教会和要求的另一种能力。学生通过学习证明所有已被证明的定理，并基于所学对某一定理做出新的证明，这是每一个数学家由学生到研究员、数学家的必经之路。但数学是人类发明的学科，从“学会”到“拓展”，设计新的证明方法、突破新的发现是非常困难的。而每个研究数学的人也正是这过程中不断训练自己的创新能力，偶然地发现一个前人未至的结果，便成了数学家。所以我们在数学教育中也不断强调创新能力，这是以色列理工学院的传统，也是我们在广以的追求。
4
对抗挫折感，享受数学这门艺术
许多学生来到广以之后或许会感觉不太适应，不仅是对全英的教学环境，更是对全新的教学方式。在这里，上课不是单纯的聆听，学数学不是单纯的计算。同学们需要对以往应试教育中所默认背诵使用的公式进行严格的证明。
我带过不同国家的学生，发现中国的孩子们总是很焦虑。他们永远想在两分钟之内得到答案，并根据自己的计算结果来对自己的学习情况做出评价，但这并不是学习数学的方法。因为数学世界需要一个更开放的思维，并习惯于忍受挫折带来的焦虑。即使是非常简单的公式，也可能需要你花1天、3天、甚至1个月的时间证明，这是正常的。但我觉得大部分中国学生无法接受这一点。
我每天都会告诉我的学生，数学世界的很多开放性问题，人类等待了2000年，等到了计算机的发明，却仍然没有得到答案。这很正常，而且正是数学之美的一部分。数学不像经济和法律一样，在电脑上搜索就能得到匹配的结果。真正的数学在某种程度更像一门艺术或体育，大脑在无数次的思考练习之后才能内化、推进。即使是足球名将梅西，也需要得到正确的指导，日夜不断地训练，才能在赛场上发挥出色。
Paulo Tirao教授与学生进行足球比赛
我来自阿根廷，很多中国的师生都向我表达了对阿根廷足球的向往和热爱。
但我更想强调的是，阿根廷足球能取得优秀的成绩源于我们的足球文化和培养体系，而人才培养体系是能够模仿和迁移的。
希望在广东以色列理工学院，我们也能够把以色列理工学院对于创新人才的培养体系、把数学家的培养体系带到这里。
Faculty Feature Series:
Meet Professor Paulo Tirao
Teacher is like a lighthouse, enlightening all the passing generations. In Guangdong Technion – Institute of Technology, language is not a barrier between teachers and students. Teachers devoted themselves to teaching and research, winning many students' respect and love.
From this article, we will present a series of interviews with teachers, who will share their insights on education, so that more people can get close to GTIIT's classes and teachers and know about its characteristics and educational philosophy.
Before the interview, we have heard from many students about how much they love Prof. Paulo Tirao's class. Professor's enthusiasm about mathematics and education is like the sun-drenched land of Argentina, where he comes from. He not only teaches mathematics, but also philosophy.
What comes to you first when thinking about mathematics? The difficult math quiz, the unproven Goldbach's conjecture, or the widely-used financial pricing model? “Math is like invention,” Professor told us, “like art and sports, we need to explore it with enthusiasm.”
Below are Professor's words.
1
Doing math in GTIIT
I started teaching in this beautiful coastal city Shantou in 2020.
My biggest feeling is that it is a great place to study math. Thinking about math problems is like planting trees. It takes generations, time, and patience. And here, perhaps because of Chaoshan's tea culture, people don't live a fast pace of life, and are more willing to wait for the flower to bloom. Ideas flash into my mind while walking, chatting and drinking tea, and the atmosphere allows us to settle down and do valuable research. Such life is perfect for a mathematician.
Professor Paulo Tirao(second from left)
with collegues and students
For students, they can not only enjoy the academic atmosphere here, but also learn what they truly need. Same as in Technion, math classes in GTIIT are divided into different types. Every program provides math classes with similar course names, but they are actually different. Students in other programs like Materials Science and Engineering (MSE) and Chemical Engineering (CE) are learning “math for engineer”, in which formulas and languages are more like tools to solve problems in their field. But the students of Math program are taking “math for mathematician”, which offers a very rigorous and systematic training, requiring them to master the formulas while understanding how these were invented. This sounds hard, but each step is a foundation stone for the road toward a mathematician.
2
What is mathematics?
Every time a student asks me this question, I will always answer: math is a very special subject.
Math is totally different from any other subjects in many ways. Math is the only subject invented by human beings. Biology and chemistry are studying nature, and economics is studying the society. They all have a specific target, while math is more like developing a game. Chess does not exist before it is invented. Only after the first person set and popularized the rules did chess become part of our life. So it is hard for laymen to imagine what will happen – even the mathematicians have no idea about the future in math world.
So I always tell my students to be patient, especially when they would like to continue their academic road, because undergraduate study only helps you complete the basic math language training in this complicated unknown field.
3
What can a mathematician do?
Professor Paulo Tirao in class
Whether you want to pursue a career in academia or not, training in a mathematics department will equip you with three skills that will last a lifetime: the ability to learn, the ability to think precisely and the ability to innovate.
If you go to the library and find the math books on the shelves, most of them contain only 100 or 200 pages. Compared with the thick textbooks of biology, chemistry, law and other subjects, math books are so thin. That's because mathematics is a very, very condensed subject. In my classes, we give a one-year-long course with 100 pages, but each page may take a week or more to fully understand. Therefore, the important part of learning mathematics in GTIIT is: learning how to learn from the condensed content, in addition to specific knowledge of algebra and computer science.
In GTIIT, the most relevant skill for each student is not knowledge per se, but the ability to learn new things. I often tell students that when you learn math, you're not just learning algebra. You're learning "how to learn Math". Here's a skill you may not need in other subjects: knowing how to learn. This transferable learning ability allows students to handle other knowledge frameworks as well. The skill may sound abstract, but in fact, every mathematician has been trained to learn new things in a very profound way. If a good mathematician enters the auto industry, even knowing barely anything at first, he will be able to use his learning ability to quickly master the tools of a completely new environment within half a year.
Professor Paulo Tirao Introduces GTIIT Program
Precise thinking is also an important skill for mathematical research. Mathematical statements are different from daily words: it happens "under all circumstances" or "under some circumstances"; if there is a number that does not meet the criteria, the description is not “real”. We use precise languages and statements. Precise thinking, which means asking the right questions, making precise statements, and giving precise arguments and reasons, is the core and unique ability of mathematicians. With the rapid development of science and technology, supercomputers can process a large amount of data, but there are still many open problems in mathematics that remain unsolved, which indicates that the thinking framework of mathematical system cannot be simply replaced by machines. At the same time, this framework can help solve many practical problems, such as fixed priorities, strategic design, organizational relations and so on, and can be widely used in medicine, stocks, industry, chemistry, physics and other fields. That's why we always emphasize precisely thinking to students.
Innovation is another ability required by mathematics departments. Students learn proof of all results already proved, hoping to find a new solution or discovery. This is the only way before becoming researchers and mathematicians. However, as I have mentioned above, mathematics is a subject invented by man, so it is extremely difficult to design new proof methods or make breakthroughs. And every person who studies mathematics is also cultivating his or her ability to innovate during the process, and on someday, accidentally discovered a result that has not been done before. Therefore, in mathematics education, we continue to emphasize the ability to innovate, which is the tradition of Technion-Israel Institute of Technology and what we pursue in GTIIT.
4
Fight against frustration,
to enjoy the art of math
Many students may feel unready when first coming to GTIIT, unaccustomed to the English teaching environment as well as the new teaching methods. In GTIIT, taking classes is not simply listening to the professor, and mathematics is not equal to calculation. Students are required to precisely prove the formulas that were only taught to use in exam-oriented education.
I have taught students from different countries and found that Chinese children are always anxious compared to their foreign peers. They always want to get answers in two minutes and in this way evaluate their learning based on the calculations, but that's not the way to learn math. The math world requires a more open mind and tolerance for frustration and anxiety. A very simple formula may take you one day, three days, or even a month to prove. However, most Chinese kids cannot accept that.
I tell my students every day that there are a lot of open questions in mathematics. Human beings have been waiting for over 2,000 years, however, even until the invention of computers, they still have no answers. This is normal, and more importantly, is part of the beauty of mathematics. Unlike economics and law, where you can type in a search bar and get an answer, math is in some ways more like art or sport, where the brain can truly understand after countless thought exercises and meditation. Even Lionel Messi, the great soccer player, needs the right guidance and training day and night to perform well in the match.
Professor Paulo Tirao plays football with students
I come from Argentina. Many Chinese teachers and students have expressed their yearning and love for Argentine football. But what I want to emphasize is that Argentine football has achieved excellent results because of our football culture and training system, and the system can be copied and transferred. I hope we can also bring the training system of innovative talents and mathematicians in Technion to GTIIT.
文/图：GTIIT传媒与公共事务部
Text/Photos: GTIIT News & Public Affairs