A short black man named Mr. Mantz stood in front of the classroom in a tight blue suit, with a small wooden ball in his hand, and his voice was high and muddy. He put the ball on a grooved slope and let it roll all the way to the end. Then he began to speak, setting the acceleration as A and the time as T. Suddenly, he began to write letters, numbers and equal signs on the blackboard, and my brain was shut.
At least in this semi-autobiographical retelling of Plath's life, Mr. Mantz once wrote a book with more than 400 pages. There are no pictures and photos in the book, only charts and formulas.
It's like trying to appreciate Plath's poems, but not reading them yourself, but listening to others. In Plath's version of the story, she is the only student who got an A in this course, but physics still scares her.
After all, mathematics is rational poetry, and poetry is the mathematics of the mind. -David Eugene Smith (American mathematician and educator)
Richard feynman, a physicist, gave a completely different introduction to physics. Feynman, the Nobel Prize winner, is a passionate person. He usually likes to play tambourine. He is more like a pragmatic taxi driver than a sober intellectual.
Feynman 1 1 years old, a casual conversation had a great influence on him. He told his friends that thinking was just talking to himself.
"Is that so?" Feynman's friend said, "You know that the shape of the automobile crankshaft is extremely complicated, right?"
"Yes, so what?"
"yes. Now tell me, how do you describe this shape when you are talking to yourself? "
It was this question that made Feynman realize that if ideas can be expressed in words, then they can be expressed in images.
Later, he wrote that as a student, he tried to imagine some concepts and visualize them, such as electromagnetic wave, which is an invisible energy flow that carries everything from sunlight to cell phone signals. But it is difficult for him to describe what he sees in his heart in words. If even one of the greatest scientists in the world can't imagine how to treat some (admittedly unimaginable) physics concepts, what should we ordinary people do?
We can find inspiration and inspiration in the country of poetry. Let's look at the lyrics of a song written by American singer and songwriter Jonathan kolton. This song is called "Mandleberg Collection" and it is about the famous mathematician Benoit Mandelbrot.
Mandelbrot in heaven.
He showed us hope in the chaos.
His geometry succeeded where others failed.
So, if you get lost, a butterfly will flap its wings.
Millions of kilometers away, a small miracle will send you home.
In Coulton's touching words, he captured the outstanding mathematical essence of Mandelbrot, from which we can form images in our minds-only the wings of butterflies flutter and spread gently, even affecting millions of kilometers away.
The new geometry created by Mandelbrot tells us that sometimes things that look rough and chaotic, such as clouds and coastlines, are regular to some extent. Visual complexity can be created by simple rules, just like the magical means in modern animated film making. Coulton's poems also allude to the concept contained in Mandelbrot's achievements-that small and subtle changes in one part of the universe will eventually have an impact on everything else.
The more you read Coulton's lyrics, the more you will find that this concept can be applied to all aspects of life-the deeper you understand Mandelbrot's achievements, the clearer these meanings will become.
Just as there are hidden meanings in poems and equations. If you are a novice at reading physical equations and no one teaches you how to read and understand the vivid connotation under symbols, then these equation expressions are dead silence to you. Only when you begin to learn and give the hidden content to the equation expression, its connotation begins to jump and flow, and finally it is presented to you like life.
In a classic paper, physicist Jeffrey prentiss compares the differences between novice physicists and mature physicists in looking at equations. When a novice looks at an equation, he just adds a memory to a huge number of unrelated equations in his memory. However, higher-level students and physicists can see the meaning behind the equation in their minds, see how it is placed in the macro background, and even feel the same way about part of the equation. Mathematicians need some poetic temperament to be worthy of the name.
-Karl Weierstrass (German educator)
When you think of the letter A as acceleration, you may feel like stepping on the accelerator in the car. Bang! You feel the accelerating thrust.
But is it necessary to evoke this feeling every time I see the letter A? Of course not. You don't want to arouse every little detail in your study, or you will go crazy. However, if you see the figure of A everywhere in the equation and try to analyze its meaning, the feeling of pushing back will linger in the background of your brain, just like a big piece of things ready to sneak into working memory.
Similarly, when you regard n as mass, you may feel the lazy inertia of a 50-pound boulder, and moving it is a big project. When you regard F as a force, you may see the force The Secret Behind through the eyes of your mind, as shown in the equation F=ma, and the force depends on mass and acceleration: ma. Maybe you can understand the mystery behind F. Force exerts a lifting momentum (acceleration) on the lazy mass of the boulder.
Let's add icing on the cake. The physical term for doing work means energy. When we push (force) something through a distance, we poeticize it: W=Fd. Once we see that W stands for doing work, we can imagine the meaning behind it through the eyes of the mind and even through the feelings of the body. Finally, we extracted a line of equation poems, like the following.
W
W=Fd
W=(ma)d
In other words, there is information behind symbols and equations-once people are more familiar with concepts, their meanings will be clearer. Although scientists don't express it this way, they often regard equations as a form of poetry, so that they can quickly write down what they try to see and understand with symbols. A good observer can realize the depth of a poem, which has many possible meanings. Similarly, students with increasingly mature knowledge can gradually learn to see the hidden meaning behind the equation through the eyes of the mind, and even form different interpretations by intuition. Not surprisingly, pictures, tables or other images also have hidden meanings-in the eyes of the mind, their connotations are even richer than those on paper.
We mentioned this point before, and now we have a deeper understanding of how to imagine the concept behind the equation, which deserves our attention. When studying mathematics and science hard, the most important thing we can do is to give life to abstract concepts in our minds.
For example, Santiago Ramon-Cahal treats the microscopic scene in front of him like a group of small creatures living in it, with hopes and dreams, just like human beings themselves. Cahal's colleague and friend, Sir Charles sherrington, the creator of the word synapse, told his friends that he had never seen any scientist make his work more energetic than Cahal. Sherrington guessed that this may be the key factor to Cahal's success.
Einstein's theory of relativity stems not from his mathematical ability (he often needs to cooperate with mathematicians to make progress), but from his ability to "pretend". He imagined himself as a photon moving at the speed of light, and then imagined what the second photon would think of him. What will the second photon see and feel?
Barbara mcclintock won the Nobel Prize for discovering the existence of gene transposon ("jumping gene" will change her position on the DNA chain). She once wrote how she imagined the corn plants she studied: "I can even see the internal components of chromosomes-in fact, everything is there." Because I almost feel like I'm in it, and chromosomes are my friends, which surprised me. "
Perhaps it is foolish to imagine these elements and biological mechanisms as living things to study and let them have their own feelings and thoughts under the gaze of the mind. But this method is very effective, it makes the research object come alive, and it can help you see and understand some phenomena that you can't intuitively feel just by looking at boring numbers and formulas.
Simplification is also important. In order to facilitate their own understanding, richard feynman, the physicist who plays the bongo drum mentioned in the previous part of this chapter, is famous for asking scientists and mathematicians to simply explain their concepts. Surprisingly, no matter how complicated the concept is, almost any concept can be simply explained. In order to brew a simple explanation, you break down a complex material into several key elements, and as a result, you have a deeper understanding of the material. Learning expert Scott Young put forward this concept, also known as Feynman method. This method requires people to find simple metaphors or analogies to help them understand the main idea of the concept.
The legendary Charles Darwin did the same thing. When he tries to explain a concept, he will imagine that someone has just walked into his study. He will put down his pen and try to explain it in the simplest language. This made him know how to describe the concept in words. Similarly, Reddit.com's website has a section called "Explain Like a 5-year-old", where anyone can post simple answers to complex topics.
You may feel that you need to know before you can give an explanation. But please pay attention to what happens when you talk about learning with people around you. When trying to explain to others and yourself, you are often surprised to find that understanding is often the product of explanation, not the understanding before explanation. This is why teachers often say that the first time they really understand textbooks is when they teach their students.
Nice to meet you!
Learning organic chemistry is no different from meeting some new people. Each element has its own unique personality. The more you know about their personalities, the more you can understand their situation and predict the outcome of their interaction.
-Katharine Norta (Ph.D., senior lecturer in chemistry, winner of the Golden Apple Award, which is awarded to outstanding teachers recognized by the University of Michigan).
It's your turn to try!
? Self-guidance and self-performance in the brain
Imagine yourself in the research content-you look at the world from the perspective of electrons in cells or even from the perspective of a mathematical concept. Try to perform a play with your new friends and imagine how they will feel and react.
Migration is the ability to apply what you have learned from one knowledge background to another. For example, maybe after learning a foreign language, you find it easier to learn a second foreign language than the first. This is because when you learn a first foreign language, you gain basic language learning skills, and at the same time you potentially learn similar new words and grammatical structures, which are transferred to your second foreign language learning.
Studying mathematics but applying it to a specific subject, such as accounting, engineering or economics, is a bit like deciding not to study another foreign language seriously-just stick to one language and learn more English words. Many mathematicians believe that studying mathematics for a specific subject will make it more difficult for you to use mathematics flexibly and creatively.
They think that if you learn mathematics in the way they teach you, that is, the essence of the concept of chunks formed around abstraction, and there is no specific application category in your mind, you can acquire the skills of easily transferring knowledge to various applications. In other words, acquiring such skills is just like acquiring basic language learning skills in language learning. For example, you may be a physics student, but by using your abstract mathematics knowledge, you can quickly understand how to apply mathematics to other very different fields, such as biology, finance and even psychology.
This is also part of the reason why mathematicians like to teach mathematics from an abstract point of view, because there is no need to narrow their horizons to specific applications immediately. They want you to see the essence of the concept, considering that it will be easier to transfer the concept to various problems. If we use linguistics as an analogy, it seems that they don't want you to learn to say "I ran away" in a specific language, whether it is Albanian, Lithuanian or Icelandic, but they expect you to understand more basic concepts, such as verbs, which need to be changed.
The dilemma is that if you apply concepts directly to specific problems, it is often easier to master a mathematical idea, even if it is more difficult to transfer concepts to new fields after doing so. As expected, we will eventually see that there is always an urgent discussion about concrete and abstract mathematics learning methods. So mathematicians step back and try to grasp the highland to ensure that the learning process revolves around abstract methods. On the contrary, many other majors, such as engineering and business, naturally tend to specialize in mathematics in a specific field to improve students' sense of participation and avoid students complaining, "When can this be used?" Specific applied mathematics is also troubled by this problem. Many "real-world" questions and answers in math textbooks are just put into practice by students in a thin disguise. Finally, we see that concrete methods and abstract methods have their own advantages and disadvantages.
The advantage of migration is that as the difficulty of a subject increases, migration can often make students learn more easily. As Jason Deschamps, a professor at the University of Pittsburgh, said, "I always tell my students that with the progress of their nursing programs, there will be less and less to learn, but they don't believe me. In fact, they learn more and more things every semester, and they are just better at integrating what they have learned. "
One of the most serious aspects of procrastination is constantly interrupting your attention to check your mobile phone text messages, emails or other updates, which will interfere with migration. Interrupting one's work from time to time will not only prevent students from studying deeply, but also prevent them from easily transferring what they have learned to other problems. You may think that you didn't stop learning when you checked the text messages, but the reality is that your brain can't form solidified nerve blocks because you don't have enough concentration time, but these blocks are the core of concept migration to other areas.
Concept transfer, the effect is good.
? I learned fishing skills in the Great Lakes. I tried it in Florida last year. Completely different fish, completely different bait, unused fishing method, but the effect is good. People think I'm very ill, but it's interesting that I show them that I can really catch fish in this way.
? -Patrick Skinkin (a senior majoring in history)
Summary of this chapter
Equations are just methods to abstract and simplify concepts. It can be seen that the deep meaning contained in the equation is similar to that in poetry.
Your "eye of the mind" is very important because it helps you rehearse in your mind and personalize what you have learned.
Migration is the ability to apply what you have learned from one knowledge background to another.
The key is to master the lexical essence of a mathematical concept, which will be beneficial to the transfer of the concept and its new application.
In the process of learning, multitasking makes learning impossible, which will limit your ability to transfer what you have learned.
Stop and review.
Close the book and look away. What is the main point of this chapter? Can you describe these concepts with symbols in your mind?
Learning promotion
1. Write an equation poem and show the connotation behind a standard equation in a few sentences.
2. Write how to guide and perform some concepts you are learning. What sense of reality do you think the characters in this play will have and what kind of interaction will take place?
Take out a mathematical concept you have learned and see how this concept is applied to a concrete example. Take a step back and look at this application example. Can you appreciate the abstract conceptual chunks behind it? Can you think of a completely different way to use this concept?
I'm Qi Xi Qiao Yue. Have a good night. ...