Hao Zhuang

Learning Math Without Tears – Is It Possible?

When people are asked to recall their days in college or high school, math classes are often among the most unpleasant experiences. This happens even in countries with high average scores of math.

John von Neumann said, “In mathematics you don’t understand things. You just get used to them”.

So, do current pedagogies have serious drawbacks? Can we find a way to learn math without tears?

Those highly used pedagogies include motivating students to explore concepts and theorems during the class, providing students with a large amount of examples, putting stories or problems with real background into lectures and worksheets, etc.

Overall, these pedagogies follow the way how humans learn and how math theories are naturally developed. For example, graph theory starts from an interesting story called Königsberg seven bridge problem. So, it is reasonable to put such a story at the beginning of a lecture on graph theory.

So what is the crux? I got the answer after a meeting with my advisor. I showed him a very long proof of the local expression of an operator. Then, my advisor showed me a much more concise proof. I was surprised and confused because I never saw such a technique before. I even doubted the correctness of his proof. However, just two weeks later, I saw a more intrinsic definition of that operator in another book. Using that, I could gave the same concise proof as my advisor did.

My advisor is much more experienced than I am. so he views each concept in a much deeper way. Similarly, students are much less experienced than instructors. They try to get accustomed to a new definition overnight, but usually end up with exhaustion. The straightforward meaning of some concepts pop up when students are sufficiently experienced, or, when they have learned advanced topics in a systematic manner. This procedure requires their diligence and patience, and cannot be completed within a 45-min class.

Also, all the pedagogies on math teaching must require students’ active participation since students learn how to do math, not read math. Appropriate pedagogies get students motivated, but cannot avoid struggles.

Even if math stories arouse students’ interest, even if instructors tell students the intuitive idea behind a concept and show them sufficiently many examples, it is still students themselves who need to struggle with homework.

Therefore, we have to admit that there is no way to learn math without tears. We can imagine it as a marathon. Even with the best coaches, it is athletes themselves who run the 42.195 km race.

Although tears are inevitable, they can actually be wiped away. As I said, using all the current pedagogies correctly can get students motivated, and willing to face any challenge.

Instructors can tell students not to try to comprehend concepts overnight and guide them to accumulate their experiences so that they are gradually approaching the destination. In addition, when students know their instructors are always there to help, their anxiety relieves greatly.

For students, they should get access to plenty of information related to lecture subjects. There is no time to go through different textbooks in a one-semester course. However, it is good for students to read several other textbooks. This must be a time-consuming procedure, but at last, they become more experienced. It is also necessary to develop a habit of “trying every calculation yourselves”. As Paul Halmos said, “Don’t just read it; fight it!”

Look back on von Neumann’s words, now we can say, “get used to them” actually means, our comprehension on abstract concepts gradually becomes better as we become more and more experienced, but not overnight.