* This was originally posted on my old blog on December 5, 2010.
* 筆者の以前のブログに2010年12月5日付で投稿された記事の再掲です。
Today’s post is about a topic that recently provoked some controversy in some Japanese internet communities.
In an arithmetic class for second graders of an elementary school, the pupils were given a problem like this: “There are 3 apples on each plate. If there are 5 plates, how many apples are there altogether?” One pupil wrote his or her answer on the examination paper as below.
Formula: 5×3 = 15
Answer: 15 apples
The answer is correct, of course, but the pupil did not get any points because the teacher decided that the formula was inadequate. Usually, in Japan, teachers tell their pupils to interpret multiplication as “groups” when they teach it to the children for the first time. For example, children are told to think 3×5 as “five groups of three.”
3×5 = 3+3+3+3+3
In the case of 5×3, there are three groups of five.
5×3 = 5+5+5
(number in each group) x (number of groups) = (total number), this is multiplication. Yeah!
Let’s take a look at the problem once again. “There are 3 apples on each plate. If there are 5 plates, how many apples are there altogether?” Here, the apples consist of groups of three each while each plate exists individually. Therefore, the formula must be 3×5 = 15, and cannot be 5×3. Most adults know the commutative rule of multiplication: AxB = BxA, but in this case, the pupils have not been taught this rule. Children at this educational stage are not supposed to know this yet, and some teachers feel uncomfortable when their pupils use something they have not taught them yet.
Mathematically, however, there is nothing wrong if teachers teach that the formula should be written in the order of “(number of groups) x (number in each group) = (total number).” There is no rule. I imagine many people from English speaking countries interpret multiplication in that way (eg, 3×5 = 5+5+5) considering that you often read “3×5 = 15” as “three TIMES five equals fifteen” verbally.
All things considered, the pupil above did not get any points on the examination not because of any mathematical mistakes, but only because s/he did not follow what the teacher said. People who are against this teacher think it was outrageous that the academic truth was prevailed by what was comfortable for the teacher. Even if the pupils have not been taught the commutative rule at school, some of them may have learned it by themselves. How can a teacher deny this intellectual curiosity of the children? Apparently, most mathematics researchers and scholars insist that the teacher should have considered the pupil’s answer as entirely correct.
However, other people think that there are some practical reasons that teachers sometimes have to do such things. Some children in the first several grades of elementary school tend to ignore the context of math problems and just put all the numbers they see together using random mathematical symbols. For example, it is no surprise if some of those children answer that the total number of apples is 8 because they see 3 and 5 in the problem. If teachers tell their pupils that the order of the numbers in the formula does not matter and allow them to write either “5×3” or “3×5,” it becomes very difficult to tell whether the children really understand the concept of interpreting multiplication as “groups of” or they just randomly connect 5 and 3 using the multiplication sign.
Let’s imagine a hypothetical future. Many years from now, should I ever have a son, he might come to me and say, “Dad, I learned multiplication at school today!” Will there be anything I can do for him to make his understanding clearer? I’m going to give him this quiz: “There are 3 plates on each apple. If there are 5 apples, how many plates are there altogether?” I’ll start practicing placing plates on top of an apple now.
