25 Oct. 2017
Perspective
Climbing a mountain may seem hard for normal people, but by following professional instructions and using the right method, it may be easy, or even become an interest. Learning math is the same. No-one knows how to climb a mountain as they born, so is understanding math. No-one understands math better than others at the beginning, but as some people have an insight to the way of learning math, they get better and better.
Many people have come to me and ask about how do I learn and understand math. There is no one right way to do the math, but there are always patterns of the way to achieve success. Examining different kinds of learning style can give us a briefest view of student’s mathematical performance. Here are a few of examples.
“Nothing is more important than monitoring Chinese economy.”
Obvious as it is, some students do not read the book and practice problems inside or outside of class. They do not necessary care little about their grades and mathematical ability, though they might, they care more about other things. One of my higher grade classmates, seemingly knowledgable according to his various experience, did not learn in his math class. When I approached him and asked him what was he doing, he started to lecture me about what he just found out about the release documents and statistics about Chinese economy from the government website. Well, that didn’t make him any wiser to me. If he know about economics, he should know that many economical concepts are based on mathematics, involving tremendous amount of calculation and modeling, rather than sitting there and predict the economic trend of next year blindly.
Most students who don’t read textbook and practice believe that the learning of math is a waste of time, they could live fine even if they know little about math. Sure they will live fine, biological systems do not just break down because of their lack of understanding on math. But will they live fine? Will they be comfortable encountering basic mathematics in their lives in which they should have learned in class? Will they have difficulties applying to college when the admission office finds out the poor performance in math? Will they have a difficulty on calculating profits when starting a business? Will they be comfortable when their kids ask them to explain some mathematic concepts? Will they really find an easier life without math?
“This is a Math class, not an English class.”
In my sophomore year, I was always surrounded by students who needs help on math questions. Although the problems were not easy, I admit, as I ask them whether or not they have looked through the textbook, they were puzzled, looking as if the textbook only provided questions but didn’t give the essential knowledge to solve them. I have no suggestions other than just take some time and read the book word by word.
Students who don’t read the book but practice are surely capable of doing math, but they lack sufficient knowledge that should be acquired by reading textbook––they need the assistance of their textbook. The textbook is not only a place where homework problems are presented, it is also a place where conceptual knowledge is taught in great detail. Beyond that, textbooks usually offers useful study strategies. James Stewart, the author of Single Variable Calculus Early Transcendentals, for example, suggests the importance of reading the book before practicing in the preface of his book:
“Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises.”
A comprehensive understanding of concepts is crucial for solving problems and applying to real life. These students are often too eager to practice the problems or wants to rush through their work, paying less attention to the more fundamental element of learning. They skip the step of learn and directly jump to apply, no wonder why they get confused.
“I understand all the math in the book, there is no reason why I can’t do great.”
Students who don’t practice but read the book are rare, but there exists such kind student who are too confident on applying the knowledge they just learned. I was one of the member of this kind of students before. When I was in sixth grade, I often took some time to go over the textbook briefly. This made me feel that I learned a lot, however, I did not. As the teacher went over the concept and posed some question, I have no clue about how to solve those questions. This confuses me: how could I be knowing the concept and unable to solve the problems at the same time? It took days to figure it out.
Learning the concept is not the same as understanding the concept, nor does it represent the capability of applying the concept. The one and the most important one way to strengthen the understanding of the concept is to apply the concept. Without the practice of the application of the concept, the understanding will not be full. The complementary relationship between understanding and application of the concept forms a positive feedback loop, in which the understanding provides a basis for application, the accumulation of experience strengthens the understanding. The revelation on learning method marks my shift from a student who doesn’t practice but read the book to a student who reads the book and practices.
“Genius is one percent inspiration, ninety-nine percent perspiration.” –– Thomas Edison
Successful math class students tend to fall in this category––students who read the book and practice. They participate in both aspect of the positive feedback loop, consolidating their knowledge whenever they read, practice, reread, apply, think, teach. One crucial element of the loop is the order, that is, understanding first, application second. Without a solid foundation of knowledge, application will not strengthen the understanding, but rather deepen the confusion and resentment to math.
I have seen a person who has brought the method to a next level. He not only thoroughly read the textbook prior to the class to have some background information when the teacher is lecturing, he also practices intensely after class, perfecting his skills. He further uses the opportunity of teaching other students to gain even more thorough understanding of the concept. He applies math to multiple subject areas and starts a research on physics, reinforcing his physics knowledge, using math as a tool to interpret the world. I hope I can become someone like him one day. His effort on math strikes me every time I intend to give up.
Unfortunately, some students who have mediocre in math class may also fall in this category. They have the will to learn and understand math. However, their passion are usually extinguished by the complexness of math. This is not their fault. Understanding math is a process. It is not something that one can come up with in an instant; it requires careful examination, classification, and analysis of pattern.
“Math harder!”––Ben Trey
There are always voices ask me why I am so good on math, I can’t think of any advice other than the suggestion that my math teacher mentions every class––math harder. There are no shortcuts one can find in the process of learning math. Perhaps one can cut the amount of practice into half the original, but sorry to mention, it also cuts the understanding into half. Make progress one step at a time, practice and apply one concept at a time, then the success is reachable. Learn math, know math, apply math, feel math, appreciate math, may the force of math be with you.