Lx=d: SundayResearch

I have a fun to teach C. Six months ago, she had a problem of one digit plus and minus. She often cried in my class. But now she can calculate three digits plus and minus. One day, she was solving a question shown in Figure 1. Figure 1. The question. I expected the answer as shown in Figure 2. If she could do it, I would be happy. Figure 2. Expected answer. While I looked other students, she worked on the problem. I just walked next to her, and what I saw was, my god, a matrix! (Figure 3) Figure 3. A matrix is invented. I asked her, ``Wait a moment! Did you do this alone?'' she answered me ``Yes, I did. It is not correct?'' It doesn't matter. The correctness of the calculation doesn't matter. I was astonished that she organized the answer like this. I could see two vectors in the original question, but both numbers are written in the horizontal direction. She rearranged one of a set of horizontal numbers to the vertical direction and put the

Case A. A. was studying geometry. That time, A Rechtschreibung (spelling) teacher Ms M watched her. The question was how many cross point (Schnittpunkt) of the three lines (Gerade) in the figure. In the question figure, the cross points are emphasized, but, she could not answer the question. Ms M asked me to help her. As usual, I asked her (A.), ``Could you please explain me what is a line? (Bitte erklären mir was ist Gerade.)'' She answered me ``A line is a line. (Gerade ist Gerade.)'' Well, that's true, but there is no information. ``How the school taught you. A line has an end? Or a line has no end?'' ``A line has no end.'' I see, so I know she learned the difference between line, half line (ray), and segment at her school. ``OK, then what is Schnittpunkt?'' I actually didn't know what is a Schnittpunkt. She answered, ``I don't know.'' So we asked other teacher, what is a Schnittpunkt. It is a cross point of two lines.

Case S. S. was solving a multiplication problem. One piece of black bread costs 2.9 Euro. How much is the each of Anzahl (quantity) : 2, 4, 6, and 8? Figure 1 show the problem. Figure 1. Case S. question. She answered the first question of Anzahl (quantity) 2 case as 5.8 Euro. (As shown in the figure, some European countries including Germany use the comma as the decimal point. In this text I use period as the decimal point.) However, next question, she calculated 2.9 x 5.8 for the quantity 4 case. I asked her why she did it. (In Figure 2, you can see the trace of that.) She believe she should do that and she explained something. However, I didn't understand it. So, I said, I don't understand your explanation. It turns out she also doesn't know why she did that. So I wrote Figure 2, then I explained if we have four pieces of bread, 4 x 2.9 would be the answer. Figure 2. How to calculate the price? First she fixed my figure to put the shadow on the left side of

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