## 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 the bread, so it looks more realistic.

However, she said she don't know what to do for the quantity six case. I was puzzled, whey she couldn't. I asked her to write down what is the question in a normal sentence. She might not know what the question is. She didn't know what the question is. I told her that the question is I want to buy 2.9 Euro four pieces of bread as seen in Figure 2. She understood what this means, but she didn't understand why this is relevant to the problem.

I asked her all the related words. ``What is black bread?'' She knows it. 2.9 Euro? OK. ``What is quantity? (Was ist die Anzahl?)'' It took a while, but she answered ``I don't know. (Ich weiss night.)'' I see!  I told her, ``I think this means `how many', but, actually I also don't know this German word, so let's ask other teacher.'' My guess was correct, she said, ``Ah, you mean how many pieces (Wie viel Stück).''

Then, she solved 6 and 8 cases so easy. I always have fun to find what they don't understand. They usually don't know what they don't understand themselves. I asked her what is her mother tongue. She talks her father with German and her mother with Turkish. However, I don't see so much problem in that case.

When I was a high school student, the students are classified with literature course and science course. Japanese and English were important in the literature course and Mathematics and Science were important in the science course. I didn't understand this classification because to learn Mathematics and Science I needed Japanese and English. Without Japanese and English I can not learn any Mathematics and Science. I can not think anything without language. So, my favorite classes were Japanese and Mathematics. I am not sure there is still this classification in Japan. I am more confident now that the language is so essential. I learn the word Anzahl in math teaching last week with S.

By the way, in Japan the price of four pieces of bread should be calculated as 2.9 x 4 and 4 x 2.9 sometimes is not correct.  (Asahi.com's article) In Japanese, ``I bought bread four times (パンを4つ買いました)'' is natural saying order, so maybe this is reflected. But, in English or in German, four pieces of bread (vier Stück) is also natural. In Japanese, we can also say in the same order (4つのパンを買いました). I think this too much restriction harms later because: 1. later a student learns algebra, then constant factor multiplication of x is ax instead of xa. 2.9 x 4 becomes wrong without any reason explained, 2. ax is the international standard in math. In this global time, teaching the international standard is wrong sounds not so good idea. These two reasons, I think we should not make the 4 x 2.9 wrong.