mathematics is a lot of beautiful qualities, not the capital of elitist arrogance
2024-08-30
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there are countless stereotypes surrounding mathematics, and there are also many misunderstandings about how to learn mathematics well. chinese-american mathematician francis su believes that learning mathematics well is actually human nature, but many people are buried in it.
francis su is the first non-white president of the mathematical association of america (maa) and the author of the power of mathematics. he is not only a mathematician but also a mathematics educator. in 2013, francis received the 2013 heimer award for outstanding mathematics teaching from the mathematical association of america.
mathematics is an important basic subject, a widely used tool and ability, but in the popular context it has become a rigid indicator for measuring iq. many mathematicians are portrayed as eccentric geniuses. in 2017, francis sue gave a speech when he stepped down as president of the american mathematical association, which was later compiled into the book "the power of mathematics". he used sincere language to emphasize the humanistic feelings in mathematics and remind people of the beautiful qualities contained in mathematics. many people's experience of learning mathematics is inferiority and exclusion. in fact, real mathematical research attaches great importance to collaboration and communication. mathematical achievements are not one-dimensional. the fun of mathematics lies in the process rather than the achievement. he does not shy away from the discrimination and injustice in the field of mathematics, but these need to be changed and have nothing to do with mathematics itself.
this article is excerpted from chapter 12 of "the power of mathematics". the subheadings are added by the editor and the notes can be found in the original book.
"the power of mathematics: making us better people", written by francis su [usa], translated by shen jier and han xiaoxiao, citic publishing group, june 2022.
there are only two things in mathematics that can really bring people satisfaction: learning from others and sharing with others. there are actually very few things that each person can fully understand, and most things we only know the surface of, without going into depth.—william thurston
studying mathematics is not a solitary endeavor
ricardo gutierrez, a new yorker born in a working-class neighborhood to immigrant parents whose father did not finish high school and whose mother dropped out of school in eighth grade, wrote to me in 2017 after reading a transcript of a talk i had given on “how to thrive in mathematics” in a report.
the letter mentioned that he showed a talent for mathematics at a very young age, but no one could give him the right guidance. so he changed his interests during college and devoted all his time and energy to audio engineering in the past 19 years, becoming a very successful audio engineer - although it is not a real technical job, but as he said: "music is everything in my work, i will find ways to make them more beautiful and more pleasant." he loves his career - in fact, a project he participated in was nominated for a grammy award - but he still feels that something is missing in his life.
i just feel like i'm still longing for something else, without which my life would not be complete. i want to learn more, and experience the logic of thinking in subjects like mathematics and computer science. perhaps it's because the feeling of logical thinking that goes deeper and deeper attracts me and makes me feel the joy of thinking; perhaps it's because my job can no longer satisfy my thirst for knowledge, and mathematics and programming just fill my appetite... perhaps, to be more precise, i'm too proficient in my current job, so proficient that i feel like i'm just "punching in" every day, with nothing new.
he then took a brave step and returned to school at the age of 40 to participate in a teaching program for "non-traditional students." he said:
this rigorous academic environment brought me unimaginable pressure and hardship, and even made me feel a little difficult to accept - especially for someone who has not been exposed to high-intensity study life for a long time - but sometimes, i think it is not these pressures that make me sad, but the feeling that "i don't belong here at all" that i have had since childhood in math and computer classes. the reason why i have this feeling may have a lot to do with my bad childhood experiences. the neighborhood environment and living conditions at that time were so harsh that every dream i had would be immediately crushed by the ruthless reality. no one stood up to guide me and help me correct those deeply rooted misconceptions. the words "i don't belong here at all" kept echoing in my mind, gradually distorting the direction of my life and becoming an indelible shadow in my life.
the feeling of "i don't belong here" can be very damaging to people, so the community is very important to individuals because it allows us to feel a sense of belonging. parker palmer said: "teaching means creating an environment for everyone to learn together and let everyone experience the power of the community in practice." if this is true, when others cannot see the truth because of their own limitations, we have an obligation to take the initiative to stand up and expose the lies for them. we can help them find a sense of belonging.
no one can thrive without the community (people who share our pain, feel our joy, see our hopes, and understand our fears). the community helps us understand that struggle is normal and that “i am not alone in my struggle.”
the collective is a desire that exists deep in everyone's heart. whether in leisure and entertainment, education and learning, or in career and family life, the collective plays the role of a "guide", helping us to enter the door of mathematics and guide us to keep moving forward on the road of mathematics.
screenshot from the documentary "the story of mathematics".
the concept of "math group" has been mentioned many times in this book. now everyone should understand that any group of people who gather together because of common mathematical experience and mathematical knowledge can be considered a math group. when you share math jokes with your family, show your love for math to your family, make some geometric objects with your family, read math-related articles with your family, or even cook with your family (add ingredients according to the instructions given in the recipe and discuss the amount of seasoning with your family), you have actually established a math group at home. when you walk into a math classroom or participate in a strategy game, you have actually entered a math group.
for most people, the word "collective" has nothing to do with mathematics. on the contrary, everyone thinks that the typical image of a mathematician is "a person who works alone for decades on a problem." yes, several famous mathematical problems have been solved in recent years, and these examples do fit this concept.
for example, in 1993, andrew wiles gave a proof of fermat's last theorem (with some flaws at the time), ending this problem that had remained unsolved for more than 350 years. in fact, the theorem is very simple: when n > 2, the equation xn + yn = zn does not have integer solutions. but it took andrew wiles seven years and a lot of effort to find a proof for such a simple sentence.
for example, in 2003, grigori perelman gave a proof of the poincare conjecture, which put a perfect end to this century-old problem in the field of topology. in general, we can understand the poincare conjecture in this way: every closed three-dimensional object without holes is topologically equivalent to a three-dimensional sphere. before giving the proof, no one knew that perelman was studying the famous poincare conjecture.
for example, in 2013, when yitang zhang successfully proved the bounded gap between prime numbers (a major breakthrough for the twin prime conjecture), no one in the field had ever heard of the name zhang.the above examples have created a superstition that mathematicians should work alone and silently.but these examples become news precisely because they are unique; they are not actually representative.
mathematics is full of collaboration, and people come together spontaneously for all kinds of mathematical projects—studying together, reading together, playing together, researching together. as william thurston said, learning and sharing together is the essence of mathematics (a statement made in response to those who often worry that they can't produce any original work). that's why we spend time enjoying mathematics with others.
from a professional perspective, mathematics is becoming more collaborative than it has been in the past: a 2002 study showed that the percentage of mathematicians involved in collaborative research projects had risen from 28% in the 1940s to 81% in the 1990s.
in 2009, mathematician timothy gowers even became famous for calling on the internet to work together to find a proof of the hales-jewett theorem. (roughly speaking, the theorem says that for a higher-dimensional version of tic-tac-toe, no matter how many players participate, there will always be a winner.)
in addition, more and more mathematics teachers are beginning to encourage students to adopt active learning methods and use class time to allow students to participate in interaction and collaboration. with the rise of social media, mathematics teachers are also trying more new teaching methods and setting up more interest groups to better spread ideas and share opinions. teamwork is the core way for today's mathematical explorers to interact with each other, and it is also a necessary skill for talents in business, industry, government and other fields.
the importance of the collective is self-evident. it can bring more people together to explore mathematics together and cultivate various excellent qualities in a mutually supportive environment. a successful mathematics project always focuses on the team, helps participants get together, and allows everyone, whether children, teachers, or researchers, to benefit from the team.
however, some learning disabilities are further amplified in a group environment, so building a math team means not only bringing together math enthusiasts, but also being able to identify problems in a timely manner and clear obstacles for everyone.
mathematical achievement is not a one-dimensional
the math community often focuses too much on what one person has achieved—usually a narrow definition of achievement. when we mentally assess who is “better at math,” we often rank them based on a single “ability,” which only serves to reinforce the stratification. we often convey the idea that there is only one way to succeed in math, like forcing kids to solve problems quickly or giving them calculus in high school or telling mathematicians that they are not “real mathematicians” if they don’t do research.
screenshot from the documentary "the man chosen by mathematics".
in fact, there are many ways to succeed. mathematical achievement is not one-dimensional. we must correct our misconceptions. we always think of mathematics as a pole stuck in the ground, thinking that the vine has only one direction to grow and can only keep climbing up the pole. but in fact, mathematics is more like a trellis: as a vine, you can find a random starting point where the trellis connects to the ground, and then climb in multiple directions at the same time.
therefore, those who wish to participate in the mathematical community must find ways to change their one-dimensional perspective. whether at home or in the classroom, we should appreciate the good qualities that others have developed in the process of learning mathematics and remind everyone that these qualities are also an integral part of mathematics.
perseverance, curiosity, good at summarizing, inclination to beautiful things, desire for in-depth exploration, and the qualities i mentioned in each chapter are all manifestations of your growth and gain in mathematics. in high school and college, we should provide more guidance to help students enter the door of mathematics instead of forcing everyone to learn calculus.we should make math club a place for fun rather than a place for elites to show off.
at the professional level, we should value the diversity of ways that math teachers and researchers come up with to improve our understanding of math, and we should also provide a variety of math role models to show us that math can be fun and interesting, not just something to plod along in solitude.
fun and achievement don't always equal
mathematical groups can sometimes be quite hierarchical, even if they don’t want to be. in the hiking club i joined, everyone was united by their love of hiking. before each trip, we were divided into groups based on individual ability, and each group went at different speeds. i openly told everyone that i was a slow hiker, and was assigned to the beginner group, but i didn’t feel ashamed because i knew that the joy of hiking—the scenery along the way, the friendships made, the quiet space for thinking—had nothing to do with hiking ability. the same is true for activities such as piano concerts and basketball games. watching is also fun, and this enjoyment is not affected by your personal ability in the activity.
mathematics is a little different. the joy of mathematics often requires certain abilities. for example, if i go to a math class, i will never experience the joy of mathematics unless i can understand what the teacher is saying. in addition, the joy of telling others about mathematical knowledge is not only about your knowledge of relevant theorems, but also about your ability to give a clear and coherent proof. unfortunately, the proof process is difficult for the audience to digest and absorb in a short period of time, and usually few people will take the initiative to ask the narrator to find a way to adjust the way of telling so that everyone can understand. although i have no idea about some topics now and i can't understand what others are talking about, i have long been accustomed to this frustration, and i also know that this is a normal phenomenon, but for beginners, this frustration is still easy to make people discouraged from mathematics.
similarly, in the classroom, in this collective learning environment, teaching mathematical skills is the core of teaching, so many people will face many challenges when learning. if the group cooperation is not arranged properly, then those students who take longer to think will feel frustrated when facing students with quick thinking.
although we must sometimes pay attention to individual ability, if we only focus on individual ability, it will lead to people blindly worshipping those who are widely praised for their outstanding personal abilities, thus creating an unnecessary class division within the mathematical community. simone weil once said in despair:"it is painful to think that i shall be completely excluded from that kingdom of excellence into which only the truly great can enter."many people are suffering from similar pain as her, and i have met quite a few of them alone.
therefore, those who have expectations for the mathematics community must develop a habit of hospitality, provide good teaching and guidance for new friends, and give them some encouragement and support from time to time. as a hospitable mathematical explorer, we must also put down our airs and be approachable, so that newcomers believe that no matter how high or low their previous level is, there will be an open door for them here. we must also take the initiative to show newcomers the "secret menu" of mathematics, let them see the less popular content - of course, even those knowledge that even experienced veterans find difficult to figure out in a short period of time - patiently guide them to master various mathematical skills, such as "how to put the content of the textbook into your own knowledge framework."
we also need to learn to recognize their excellence and tell everyone that they are fully capable of learning mathematics. those who are respected in the mathematics community should also remember that they have an unshirkable responsibility in how to standardize the welcoming system. on the other hand, in order to be a hospitable mathematical explorer, we must also strive to transform ourselves into excellent mathematics teachers so that beginners can also experience the fun of mathematics. as for how to provide a good education, there are too many examples in this regard. we should make good use of them and lead everyone into this outstanding kingdom through pleasant exchanges and communication.
screenshot from the documentary "the man chosen by mathematics".
the leaders of the mathematics group must play their role and adjust the group's management strategy at any time according to the students' specific performance, personality differences, and ability levels. experienced teachers are very clear about this. they know that in order to make the way they get along with each other more standardized, it is necessary to establish corresponding norms; they also know that if someone in the group is arbitrary, the team will become inefficient. if everyone in the group cannot find their own meaning in team work, it will bring serious negative emotions to everyone.
therefore, in order to make participants gain something, mathematics educators attach great importance to the design and arrangement of team work. they will set up multiple important roles for team work and assign some tailored tasks to each member to ensure that everyone can successfully complete the work only when they work together. a conscientious teacher must know how to encourage students to actively communicate and share ideas, and how to reduce the probability of social risks brought to participants by team activities in an appropriate way.
if you want to build a mathematics community, you have to find ways to improve everyone's ability to collaborate and try to eliminate class divisions.only when members are inclusive of each other and everyone can benefit from the "a hundred schools of thought" environment can it be considered a successful collaboration. we must remember that collaboration is more than just a simple division of labor. true mathematical collaboration is highly collaborative. through a lot of preparatory work, it ensures that every participant can grow in a mutually reinforcing environment and gain a deeper understanding of knowledge in a healthy competitive atmosphere.
implicit discrimination in the mathematics community
mathematics, like any other group, is prone to all kinds of implicit discrimination: we all have unconscious stereotypes. we make false assumptions about others, which limits their chances of being heard. on campus, we must remind ourselves: who hasn’t spoken up? whose efforts and contributions are often overlooked? in the professional world, we must also be aware that biases sometimes lead us to make decisions and behaviors that are detrimental to the group.
for example, when women and men publish papers together, few people recognize the women’s contributions—people assume that the work was done by men. a 2016 study in a field similar to economics showed that women were twice as likely to be rejected for tenure as men, even though they published as many papers as men, unless they published alone (in which case, there was really no difference in rejection rates between men and women).
therefore, those who want to establish a mathematical community must constantly reflect on themselves to see if they are unintentionally showing any bias. in addition, we must also establish appropriate rules and regulations in the community and enforce them ourselves. only in this way can we minimize the occurrence of bias.
many math groups suffer from a lack of belonging. this can manifest in many ways: i don’t want to be seen as someone who doesn’t know enough (underlying: i don’t feel like i deserve to be here); everyone else is different (underlying: no one really understands my situation); i’ll never be good enough (underlying: i’ll never be as good as the people i admire). this feeling can be exacerbated by the rigid hierarchy of many groups.
as a college student who is already over 40 years old, it is difficult for ricardo to avoid similar experiences. he has encountered all of the above experiences to a greater or lesser extent. whether from the perspective of race or social class, ricardo is in a relatively disadvantaged position. moreover, he has not been in contact with campus life for a long time, and it is difficult for him to re-adapt to this high-intensity learning environment. what happened in the past is also constantly eroding his perseverance and determination, so he always feels that "i should not have returned to school in the first place."
in fact, many of us have similar feelings for one reason or another. for example, i often feel lonely in the mathematics community, even though i have established myself in the field of mathematics. this feeling has not disappeared. in the middle of my career, i changed my research field and came to a new scientific research institution. i spent a semester trying to get along with everyone and integrate into this group, but unfortunately, the result was very limited. i still often felt that i was isolated from the group.
because i knew very little about this new field, and my previous institute was a little different - other institutes focused on scientific research and academics, but it seemed that only our institute focused on teaching and educating people. people didn't know me very well, and they didn't invite me to participate in group activities. they preferred to get together with people they were familiar with. but to be honest, if they could understand my feelings at the time, i believe they would definitely be willing to lend a hand to me and help me out of the predicament. so i said before that only by caring about others from time to time can we truly accept others.
screenshot from the documentary "the man chosen by mathematics".
so, for those of us who value the mathematical community, in addition to being hospitable, we must also care about others. this means that we must see others correctly, especially those who are young, those who are new, and those who are overlooked. it means that we must put aside the identities and backgrounds of others and only understand them from the purest mathematical perspective. you must do this even if you are just a newcomer.
when i was ignored before, i thought about it seriously, and then i suddenly realized that there might be many people who are experiencing similar experiences as me, because this scientific research institution generally only provides short-term exchange programs, and it can be said that each of us is a newcomer. but then again, even if you are a newcomer, you can take the initiative to care about those friends around you who feel like you are a stranger to the new place and welcome them.
no matter which math group you come from, as long as you are in a leadership position, you should actively cultivate your empathy, be good at discovering other people's difficulties, and understand other people's situations. as a leader, only by setting an example and actively sharing your past experiences and the difficulties you encountered in your academic journey can you achieve a top-down effect. as a teacher, only by taking the lead and actively sharing your "mathematical resume" - what you see, hear, and feel in the process of learning mathematics, can you make students happy to imitate.
an empathetic leader can comfort others and help them overcome their frustrations. as karen uhlenbeck, winner of the abel prize (known as the nobel prize in mathematics), said: "it is not easy to set an example for others...you have to understand that your most important task is to make students realize that successful people are not perfect people, they also have many flaws and weaknesses.”
creating a healthy mathematics community
i always have a lot of fun discussing with others how to thrive in mathematics, as they often share with me the profound experiences they have had.
this reminds me of an anecdote that mathematician professor erin mcnicholas shared with me. she was in great pain over something that had nothing to do with academics, and by chance, she experienced a moment of self-forgetfulness with several students and another professor:
it is hard for you to imagine the pain i was going through at that time. anxiety, worry, fear, anger and other negative emotions intertwined and swept my entire brain like a tide. i felt that i was about to collapse. ... however, a boy i met by chance brought me a turning point. at that time, he was studying real variable functions in another professor's class. i was discussing this week's real variable function homework with one of my female students. the student told me that she found a loophole in the problem solving process, but we analyzed it for a long time and couldn't figure out how to solve it. so i asked the boy if he had solved the problem. although he said he had calculated the answer, we took a closer look and found that his problem-solving steps were the same as my female student's, but he didn't notice the loophole, so i had to point out the problem to him.
in this way, within just 20 minutes, a total of 7 people gathered in the classroom, including 5 students of real variable function theory, another professor and me. everyone expressed their opinions and discussed the solution to the problem together. however, just when the problem was about to be solved, we encountered another new problem. however, with concerted efforts, we finally solved this problem.at that moment, everyone's face was filled with the joy of victory. the student who was responsible for recording the discussion process on the blackboard even danced a little to celebrate after quickly writing the last stroke. infected by him, we all laughed, and the whole classroom was filled with the relaxed and cheerful atmosphere after a great victory.
she also said that it was during that moment of laughter that she realized that in the 30 minutes of solving problems together with everyone, she had completely forgotten her worries and sorrows. in this spontaneously formed mathematics group, mathematics has become a spiritual harbor where she can laugh and dance to her heart's content, and no longer have to worry about the storms and troubles outside.
from her story, we can see how much benefit a healthy mathematics community can bring to people. there is no hierarchy, everyone is thinking about the same problem, and professors can also use practical actions to show everyone that they also have a lot of things they don't understand, and they also have a lot of problems they want to work hard to solve. frustration is a very normal phenomenon, and to some extent it can even make people a little excited and excited.
everyone came together because of a common interest: even though the students knew that the problem was not easy for the professor to solve and they would not lose points if they could not solve it, they still wanted to do their best to explore the truth and find the answer, just like the professors. in the process of brainstorming, they saw the same ray of hope; at the moment of success, they tasted the same taste of victory. when looking back on this experience, erin expressed the following emotion:
although each of us has devoted our own sweat to find the answer, i still can't help but want to praise my student, because it was because she initially discovered the flaws in the problem-solving process that we have achieved what we have now. on the other hand, although i have discovered her beautiful character of rigor and prudence, this character is easily underestimated or ignored by most professionals because they tend to value creativity and mathematical intuition more. although she also has these advantages, because she is humble and understands that there are people who are better than her, these advantages are not easy to be noticed in a group environment.if the challenge of mathematics is likened to a river that must be crossed, some mathematicians will choose to start from the shore immediately, jumping from stone to stone, thinking about the next step; while others will choose to wait and see on the shore, and then find a way to cross the river, calculate the speed of the water and the probability of falling, and use google maps to search both upstream and downstream to see if they can bypass it by bridges. it is easy to be impressed by the courage of those brave warriors who are brave enough to move forward and jump from stone to stone. but in fact, when these people are trapped in the middle of the river, it is often those diligent and rigorous people on the shore who come to rescue them.i think both the professors and the students overlooked the careful, methodical part of the collective work that led to our ultimate victory. i was deeply moved by the fact that two university professors with doctoral degrees and several senior students with outstanding professional abilities did not see the flaw in the solution process, but instead let an unknown student whose abilities were often not recognized by others discover the problem.
this is a truly thriving and flourishing mathematics group: all members come together because of common exploration directions and interests, actively communicate, learn from each other's strengths and weaknesses, and respect each other's work results. in this process of unity and cooperation, everyone will cheer for the breakthroughs made by the team, and every excellent character has been perfectly interpreted.
original author/francis su
excerpt/lotus
editor/wang han
introduction proofreading/lu xi
