Progress in three months

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.'' One point I know her school demands or expects that Hasenshule helps her homework. I am a useless teacher who doesn't tell the homework answers. But I believe it doesn't matter that something is just written on the paper. I believe it is more important she can understand something. So, I continue to be useless.  She once left my class and asked to help the homework to another teacher.

She has already been in Hasenshule for two years. It seems she is always like that. However, recent two or three weeks, she has changed. Last week, I asked her what is equal to 802-456? She first said, 2 - 6 geht's nicht. But she still looked on the paper. Then she put 1 under the 5, then said 12 - 6 = 6. I thought, it doesn't work. But, I just watched how she continue. It looked like she did:
802 - 456 = 800 + 12 - (450 + 6).     ... (1)
This is not correct, it should be:
802 - 456 = 790 + 12 - (450 + 6).     ... (2)
So, I expected she made a mistake. But actually, she did:
802 - 456 = 800 + 12 - (450 + 6) - 10.  ... (3)
This is correct. I didn't understand what she did, so I asked her how to compute this. It was hard that she first could not explain it to me. But, a few minuted later she explained she computed like (3). I asked, ``What you did is completely correct. By the way, how do you think the method (1)?'' I explained the method (1). Unfortunately, she didn't get it. I saw the method (3)'s advantage. In this problem, when I calculated the 1's position, the 100's position was also changed. But, the method (3) didn't change the 100's position. The method seemed more complicated, but, I realized there was an advantage. She solved the rest of all the problems by her method.

I didn't know this method. I learned a new thing that day, she showed the great progress.

Yesterday, she said ``Geht's nicht'' again to the same problem she solved in the last week. However, now I know she can do it.

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