Hasenschule: Was bedeutet das? Bitte erklären das mir. What does it mean? Please explain me that. (1)

When my students asked me a question, I usually answerd the following:

``Was bedeutet das? Bitte erklëren das mir. (What does it mean? Could you please explain me that?)''

I continue as:

``Mathe ist eine Sprache. Es gibt eine Bedeutung. (Math is a language. There is usually some meaning.)''

When I asked my students to explain the meaning of the question, they sometimes answer me, ``You are a teacher, you explain me.'' Well, that's true. But, I want to know they understand the question. I also want to teach them how to explain something. Therefore, I ask them, ``What does the question mean?'', ``Is it true?'', ``Please explain that why.''

Sometimes some students cried saying, ``You didn't teach me an answer.'' or ``You didn't help me. Help me, please.'' I was thinking, ``The answer is not so important. I want to you to learn how to learn by yourself. This is a practice. I wish soon you don't need my help. I want you to practice to solve a new problem. In the future, you will confront a total new problem that no human kind ever met, and you need to solve it. I want to help you to prepare that time because I can not help you that time.'' However, I am also still learning how to solve a new problem. I cannot really teach it since I still don't know it well. Therefore, I said, ``Please don't cry. You can understand if you think slowly, take time as much you need. If you can not do it today, there is a tomorrow. The answer is not so important. The understanding is the most important.'' By the way, in German the word ``correct'' is ``richtig'' and ``important'' is ``wichtig''. I can not say well that ``Richtig ist nicht so wichtig.'' since for Japanese, ``r'' and ``w'' are difficult to pronounce. This makes them laugh.

Five months ago, one student was always crying and her grade was always 5. She got last month 2 in math. I am so happy for her. (`1' is the highest grade in Germany.)

The following articles, I would like to talk about two stories.

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

Lx=d: SundayResearch

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