Understand the concepts in Math and Physics
By Alfred Tang
(Math & Physics Instructor at The Edge Learning Center)
Math and physics have a reputation of being hard to understand. Whenever I tell people that I am a physicist, their first response is always something like “physics is hard” or “I did not do well in physics”. Similarly, the mere mention of the word “calculus” melts the heart. Of course, young Einstein would not know what I am talking about. Even though math and physics are so intimidating, many students still choose to take the challenge because they intuitively know that the process is good for them. At the same time, no one wants to fail. It is why students seek help to turn challenge into success. Private tutoring and tailor-made courses at The Edge provide these helps.
There are basically 3 main teaching styles in math and physics: (1) Terse and intelligent short answers, (2) constant massive drills and (3) self-teaching and self-learning. Teaching style #3 is the most hated by students. The motivation behind this teaching method is to force students to develop independence and inductive reasoning. The job of the teacher is simply to facilitate learning by providing materials and an environment for the students to discover knowledge for themselves. Teachers may give no more instruction than to tell students to “think like a scientist”. This approach works well for graduate students and a few highly gifted teenagers. Most students find this teaching style frustrating and completely useless. Why do they need school if school is not teaching them anything? In developmental theory, students grow through stages. In the secondary school stage, method #3 is not effective because the students are not intellectually ready for it. However a small dose of self-learning may push the students to the next level of their development. Method #1 is characteristic of highly intelligent teachers who are experts of their subject areas. A famous professor of Christian philosophy J. P. Moreland once said that every graduate student should be able to summarize a book in one sentence. The idea is that, if you understand something, you should be able to explain it briefly. Sometimes when a student has a simple question, they just want a simple answer. If the teacher beats around the bush too much, the student becomes annoyed and confused and thinks that the teacher is inapt. On the other extreme, bright people who are terse as a habit of thought expect the same thing from other people. In the opposite direction of the highway of communication, if a student is fumbling around too much and never be able to get to the point, a very smart teacher may get impatient. An impatient teacher stresses out a student. At the worst of method #1, the teacher’s answer is too terse and the student does not get enough explanation. It is why a teacher needs to constantly observe the student’s micro-expressions to gauge how much explanation he needs to give. Teaching style #2 is typical of Asian teachers. When I was a kid taught in Chinese school, I did so much homework that my middle finger developed a huge callus that still stays with me today. Most of our students go to international schools and do not understand what it really means to have a lot of homework. It is true that too much homework is counter-productive. But a certain amount of drills and practice are necessary to help the students internalize the knowledge and develop the mental muscle memory needed to do well in tests. For the teachers, a lot of homework for the students means a lot of homework for them to grade. It is why lazy teachers do not give a lot of homework. But then, these teachers are also not doing their job.
Another teaching style not yet mentioned so far but is becoming more and more sought after in the multi-media age is the ability to speak well in public and be dramatic or “interesting” (I call it a pseudo method). In the TV generation, teenagers are used to sound bites and having all their senses stimulated all at once in multi-media presentations. In addition to sight and sound, technocrats are now designing gadgets that generate smell as well. Soon, when you see a peach on the computer screen, you will be able to smell peach as well. It is the multi media world that our students live in. Actors are experts in engaging all the senses and emotions. Actors do not get work by being boring. Teachers on other hand can get work even if they are boring. If math and physics are inherently interesting, it is a crime to bore the students with them. All the dynamic math and physics teachers I know have personality, are first rate public speakers and actors in addition to being experts in their fields. For instance, physics Nobel laureate Richard Feynman was called the world’s smartest man by New York Times. He was notorious for being a joker. People who did not know him did not know that he was one of the most brilliant minds in history. He taught freshman physics at Caltech. His lectures were hard to understand because he was too smart. His lecture notes become a classic 3 volumes set and food for fodder even to the experts. Regardless of the challenge, Feynman got the students’ interest because he had something to say. His dynamic personality helped a lot too. The phenomenon of combining information and personality in teaching is rapidly becoming a norm in a market driven media world.
Most of our students follow MYP, IB Diploma Programs and AP courses. MYP is traditionally internally assessed—meaning that the teachers decide the students’ grades. Because of that, teachers have some freedom to tailor-make the curriculum to their taste. As a result, I have seen MYP students learning things way beyond their levels and subsequently struggle. MYP is just beginning to implement computer based external assessment this year (2016). But most schools are still slow to adapt. Until MYP curriculum is standardized by external assessment, we will continue to see some students struggle. IB and AP levels math and physics strike fear in the heart of students as well—especially calculus and calculus based physics. Math and physics are different than other subjects in that understanding is lauded. Rote memorization gets a student only so far. Critical thinking and problem solving skills are key. These skills are highly sought after in today’s work place because of the complexity and the constantly evolving nature of real life problems. It is why companies like to hire math and physics graduates even though their day-to-day operations have little or nothing to do with the subjects. Students who do well in math and physics tend to do well in everything else. Hence math and physics are good education even though the students do not plan to pursue careers in these fields.
Math and physics can be very abstract. Students sometimes have trouble understanding certain concepts not because they do not understand the math per se but because they do not believe the concepts. For this reason, it is especially important for teachers to relate abstract mathematical formalisms to everyday life as much as possible. It is a misunderstanding that brilliant people are so heavenly minded that they are of no earthly good. The truth is the opposite. Some of the best teachers are geniuses. Richard Feynman as mentioned before is the best example. He invented quantum electrodynamics that revolutionizes particle physics and won himself the Nobel Prize. On January 28, 1986, space shuttle Challenger exploded 73 seconds after liftoff. Feynman was asked by the Rogers Commission to investigate the disaster. Feynman quickly understood the source of the problem as the O-rings losing their elasticity in cold weather. He demonstrated this simple physics idea before the panel by using a piece of O-ring, a C-clamp, a Styrofoam cup and some iced water. He showed that it did not have to be fancy to explain physics. Take for example: In my teaching experience, some students have trouble understand the concept of momentum. The definition of momentum is very straightforward—i.e. momentum is mass times velocity. But students cannot visualize it and do not know how to relate it to everyday phenomena. So they do not believe it. Following the spirit of Feynman, I used a portable fan, some plastic bowls and aluminum tubes to demonstrate the concept of momentum as shown in the video. Some people think that expensive tools are indispensable for good teaching. Many schools spend a fortune on their teaching labs but the students still do not understand the physics concepts any better. Tools are important. But intelligence is much more important. Students do not care about expensive equipment unless it helps them understand better.
As another example, the concept of limit in the beginning of a calculus course sometimes seems mysterious to students. Teachers often do not give the students a simple answer but instead go through a long drawn out mathematical process that make the explanation unnecessarily complicated. As I said before, “If you understand something, you should be able to explain it in one sentence.” OK, I lie. I may need more than one sentence. If I were asked by a student to explain the concept of limit, I will simply say that it is based on the concepts of continuity and infinity. If I have a collection of coins and keep dividing it in halves and keep half the coins each time, eventually I will always end up with just one coin. The process of division stops when it reaches the smallest indivisible unit—i. e. a coin. Hence coins are discrete and not continuous. Real number on the other hand is continuous because it does not have any smallest indivisible unit. For any two numbers, there is always another number between them. If I divide a number by 2, I get a number halfway between 0 and the original number. I can do it again…and again…and again…until forever. No matter how many times I divide, the process never ends. Here is the concept of infinity—i. e. something going on forever. The answer of my real number division problem will get closer and closer to 0 but is never exactly 0. We can either say that the answer approaches 0 or simply say that the limit is 0 for the economy of words. Hence “limit” is just a word mathematicians invented to facilitate communication. Once the students understand the motivation behind the concept, I will teach them how to do calculations with it. If a teacher gives a straight answer to a simple question as simply as possible, the students tend to understand it more readily. If a student is hanged up on a basic concept from the gecko, he will always be bothered by it even though he can do the calculations in the later stages. Therefore my note to myself is KISS—i.e. Keep It Simple, Stupid. Once the students are unhooked from simple things on the lower level, they are comfortable to receive more complicated things on the higher level. Math and physics are built on layers of concepts. Students need to master one layer before they move on to the next. It is why a strong foundation is so important for a student to move on.
Thinking well is key to speaking and writing well. What I heard from Readers’ Digest ages ago still sticks to my mind: In terms of effective communication, the keys words are clarity, brevity and conciseness. I cannot always be brief, as evidenced by this 2000 words essay. At least I can be clear and concise. At the end of the day, what I want to model for students is good thinking by communicating math and physics concepts well to give them understanding and a strong foundation.