Discussion on mathematics education

This semester (this article was posted in 2011/Editor's note) I've been teaching a course, Calculus ii. The object is non-mathematics physics major University student, most comes from business/liberal art/economics and so on profession. The difficulty is roughly equivalent to the domestic high number B or C. The textbooks used are the widely used Stewart calculus throughout the United States.

My current university is located in the middle of the United States, where graduates will generally enter the white-collar class, but it is difficult to become the best American elite. In other words, they are basically typical representatives of the middle class in America.

Let me say at the very outset: why is science declining in America? I can see it from my class. A typical future American middle-class student, in his college science curriculum, basically can not learn anything. After he became the pillar of society, how could he understand science and respect science?

This is certainly a glimpse of the leopard, because I have not, after all, been in the other sciences of the university curriculum. But even with calculus, which is the basis of all science disciplines, the rest of the picture seems conceivable.

Why do they learn so badly in calculus? Let me give you a few examples.

The first part after I took this course was the integration technique. Most of the time, the students are practicing this topic:

The following indefinite integrals are calculated:

Then the differential equation. After the introduction of the basic definition, the students will encounter this topic in the exam:

Try to solve the following differential equations:

And then the progression. Although students are not required to master the Ε-δ language, they have to learn all sorts of theorems that determine whether the progression is convergent. The work and exams are like this:

Determine if the following series are convergent:

I can't understand how a non-math or physics student can get shred lessons from this kind of education. How could he not hate the course from the heart, and then forget all the contents within one hours of the test? Like the triangular substitution of this kind of integration skills, do not say that a general psychology or economics of the students in the life of the student can not use, even I have a lifetime to use. Even in extremely rare situations where such problems need to be solved, it is entirely possible to resort to wolframalpha.com or similar tools. In my opinion, in 21st century, it was also required that an average student hand count points, as if a car driving school student must learn from horseback.

To be realistic, Stewart's textbook is not bad, and has tried to include some of the content of the application of mathematics in other fields and a lot of interesting reading materials. The problem, however, is that it is still a textbook based on a mathematician's way of thinking, which begins with the most basic definitions and theorems on every subject, until it exceeds the level that the textbook may cover. For example, why do students need to learn the solution of variable-separated ordinary differential equations? Because it is the most easily solvable (and the only one that the student at this level can learn to solve) is the ordinary differential equation. That is, students learn this type of equation (and do a lot of practice) not because it is important to students, but because it is the highest position a student can climb in a math building, that's all.

But why is a student going to climb this building? Or in other words, what does a mathematician require that all ordinary people learn maths in the way that mathematicians do? It has no use in addition to throwing the students to the black and black. Every student has to learn seven or eight ways to judge whether a progression is convergent, but he does not even get the answer to a simple question: "Why should I study progression?" The problem is meaningless to mathematicians, and mathematicians always need to be series.

But it has meaning for others. Every time I walk into the classroom and start talking about the difference between the first class of anomalous integrals and the second kind of anomalous integrals or absolute convergence and relative convergence, I can't help imagining the students ' moods. Will they feel like I'm a fool? I feel like myself anyway.

I often run into someone who asks seriously, "What is the use of mathematics?" "The question is not difficult to answer, but I can cite countless examples to illustrate why the most modern mathematical tools are indispensable in every corner of society, but I understand why people ask that question." For them, math is the problem they have learned in college. And these exercises for 99% of college students do not have a little use, and even "exercise logical thinking ability" This is not the most illusory use.

If I were to write a college math textbook, I would try to get every student who has ever read a math class in college to answer the question: why can people accurately predict the solar eclipse after a few decades, but not accurately predict tomorrow's weather; why people can safely browse the web via https without being monitored Why global warming is faster than a single limit becomes irreversible; why compress a text file into a zip volume is much less, while the mp3 file compresses to a zip size but almost unchanged; Should people's livelihood statistics be averaged or median? What does it mean when people say that the sound of two instruments is the same pitch and the timbre is different? This is not "fun math", this is math. Basic, important, profound, beautiful mathematics.

In my vision, this is the task that college basic mathematics education should achieve. Not to cultivate a non-mathematical professional in the field of mathematics of modern professional quality (which is no matter how successful), but to allow a person to the non-professional premise to the fullest extent of the real useful modern mathematics knowledge, understanding how mathematicians work at all levels and social interaction, And how the community's investments in this sector have been rewarded. The same should be true for basic education in other scientific disciplines.

More importantly, anyone who has received a university science education, no matter what his occupation, should be able to clearly understand the following: Why in history there have been a few amateur scientists who have made significant contributions in the case of non-attention, Today's scientific community still rejects the participation of amateur researchers as a whole and opposes the use of social resources to encourage amateur research; since scientific conclusions are possible and are in fact repeatedly overturned, even Newton's mechanics will be replaced by Einstein's theory of relativity and the theory of relativity can continue to be amended, The scientific assertions made by contemporary scientists are, in any sense, worth believing (or not at all); science is not the same as political problems, there is no so-called right answer, and each position actually has its existence significance and value When a scientific professional problem has political clout at the same time (such as global warming, stem cell research or GM crop promotion), what kind of voice should the public have without a professional background. Let each modern people in the university education to hear the scientists answer to these questions, should be the University science education unavoidable task.

There is no doubt that this is not the situation.

I do not have the experience of teaching basic mathematics in Chinese universities, but as far as I know, the situation is similar to that in the United States. I've heard more than once about the pain that I've been given to them in high-class lessons, and I can imagine understanding it. This is not a question of China or the United States, but a problem that prevails in times.

The relationship between science and society has never been alienated, as the power of science has been sweeping the whole society in an unprecedented way. This is really too dangerous.

Discussion on mathematics education (turn)