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 ...

D is a bit noisy. She sometimes made a story and talked out laud in the class. However, I respect a person who can make a new story. So, I don't want to discourage her to do that. On the other hand, she disturbs other children. So, I try to ask her to concentrate the problem. Last time, D practiced plus calculation with Zahlenhaus. In the beginning, Zahlenhaus has plus and minus examples. Figure 1 shows a Zahlenhaus example. Zahlenhaus: 7 = 4 + 3 A practice example is the following: \begin{eqnarray*}  4 + 3 &=&  \\  3 + 4 &=&  \\  7 - 3 &=&  \\  7 - 4 &=&  \\ \end{eqnarray*} Another typical practice example is the following: \begin{eqnarray*}  1 + 6 &=&  \\  6 + 1 &=&  \\  7 - 1 &=&  \\  7 - 6 &=&  \\ \end{eqnarray*} This practice book tries to explain that the plus calculation is commutative, i.e., we can add numbers ...

I first met C three months ago. I teach her one digit plus, like 3+4. One digit plus is somewhat OK for her, but, two digits were hard to her.  If I taught slowly, she could do it. However, next week she glanced at the same problem, then she said ``Geht's nicht (Can't be!)' and she immediately gave up. The whole month was like that. I thought, ``I see, she is a tough one.'' I told her every time, ``Mathematics is a language. It is a reflection of human mind. Maybe this doesn't make sense for you, but, it's a language. Which means there is a meaning of all of them. So what does it mean 3+4?'' One day, I used a drawing, the other day, I used a block to explain the numbers and plus. I try to show that we can touch the numbers. However, what she wanted know was the answers of her homework. Once she cried that she needed to fill the homework until tomorrow. I felt sorry a bit, but, I said ``The answer is not important. Your understanding is important....

L is not a so diligent girl. But she sometimes finishes the exercise very quickly. Although I found she repeatedly mistook the same problem. For example, in one exercise, 4+7 was always 12. I asked what is equal to 4 + 7. She used fingers, took a little while, but she answered correctly at the end. Then I noticed, ``Wait, if she solve the problem with this speed, how she can finish the exercise so quickly?'' So, I sat down next to her and watched how she solved the problem. Surprise! She doesn't calculate at all. She just looks up the last pages and copies the answers. What I was impressed was she was so fast. She was faster than me to look up the questions. I understood why she made the same mistakes again and again. I learned a new German sentence, ``Bitte nicht angucken!'' This means, ``Don't cheat by looking.'' though it's hard to translate to English. Her speed of cheating is impressive, this might be her talent. I don't know that I sha...

A boy (T.) is working on 7. The question is what two numbers make 7. The material using here is shown in Figure 1. This is called ``Zahlenhaus''. One house has two rooms and each room has some people. How many people are in the house? is the question and the students think about addition. Figure 1. Zahlenhaus: 7 = 4 + 3 Figure 2. Zahlenhaus: Questions Beside the house figure, there is a equation (Figure 2), usually missing one part. For example, 7 = 3 + ? 7 = 4 + ?. The leaner fills in ?. T. has no problem to fill in those questions. However, he doesn't understand the following. 7 = 0 + ? By the way, this is a easy case. Since he understands what he doesn't understand. I, myself, have a problem to find when I lost in a mathematics book. If I know where I lost, it is rather easy to return to the track. But, in many cases, I didn't know where I lost. Then I first need to find out where to return in the book. OK, let's back to the Zahlenh...

What I first asked to children is Kannst du ,,gleich'' sagen? (Can you say ``equal to''?) Because many children say Eins plus zwei ist drei. (One plus two is three.) Of course it is no problem if the children know the difference. But when a child use ``ist (is)'' I am not sure they know what the 1 + 2 = 3 means. Therefore, I asked often children to say, Eins plus zwei (ist) gleich drei. (One plus two equals to three.) But if I force this, children might hate mathematics, so I only ask this sometimes. I am afraid they hate mathematics. If they hate, it doesn't matter what is correct. By the way, I just thought that does usual grown up know the difference? I asked some of colleagues, though, they earn money by doing mathematics, so they knew it. (Maybe wrong samples.) The problem is how to explain this to 8-11 years old. I explain as the following. All the children said ``I see''. But still some children forget to say ``equal to'...

I think mathematics and computer science are fun. I can explain some example cases, but, why do I think these are fun in general?  The same as why am I interested in novels, movies, climbing, or teaching? I don't have a clear answer. How do children think about mathematics? If I could understand a part of that, maybe I could help them. Therefore, I would like to learn how the children think rather than teaching. I learn how the people think about mathematics. I am interested in artificial intelligence. Therefore, I am also interested in how the people's intelligence develop. Children learn speaking, counting numbers, logical thinking, then develop their intelligence further. I should have experienced these process, but, I totally forgot how to learn my native language, how to learn counting numbers. Because of that, I would like to ask children to teach me them. Though, I found it is not easy. Children hardly teach me how they think. I am not sure, but maybe children thems...

I started leaning how to teach mathematics at Hasenschule ( http://www.hasenschule.de/ ). I attend twice to four times a week, one hour lesson each. This is a voluntary activity and I don't get any money. (Well, sometimes I got sweets or a piece of cake.) I would like to write about this activity. Briefly speaking, Hasenschule is a special school for children who could not catch up with their school. This school teach how to read, write, and calculate. I am interested in teaching math and physics to children and want to learn how to teach that. With help with my friends, I found the following web pages. Gute-tat is a kind of portal organization, they introduced me Hasenschule. There are many activity of this kind in Berlin, if you are interested in, these pages are good start. My lessen is taught in German, but my German is not well. Hasenschule provides a rechtschreiben (reading/writing) course for 8-10 years old children, I maybe should take the course. http://www....