Monday, February 23, 2009

M4N7 - Order of operations and writing math sentences

M4N7. Students will explain and use properties of the four arithmetic operations to solve and check problems.
b. Compute using the order of operations, including parentheses.

As I look back on my own school experiences in Japan, I really don't remember explicitly learning about the order of operations. As I examine the current Japanese elementary school mathematics textbooks, there is really no unit titled "Order of Operations." Of course, that does not mean that Japanese children do not learn the order of operations. What is important to notice is that how they study order of operations is more from the perspective of writing mathematical expressions.

For example, in Grade 4, students are given the following problem:
Makoto had a 1000-yen bill. He bought a 460-yen notebook and a 140-yen pair of scissors. How much change did he get back.However, the focus here is not on solving this problem as the solution is actually given on the textbook page. In fact, there are two possible solutions given. Here is Naoko's method:
1000 - 140 = 860
860 - 460 = 400
Makoto, on the other hand, solved the problem this way:
140 + 460 = 600
1000 - 600 = 400
The task given to students is to think about how these sentences may be combined into one math sentence. They are also given a math sentence with words:
[Money Paid] - [Total Price] = ChangeFrom here, students are expected to understand that Makoto's 140 + 160 is one quantity, namely [Total Price] in the math sentence with words. Therefore, they learn that the two sentences may be combined into: 1000 - (140 + 460). Since (140 + 460) represents one quantity, it has to be calculated first.

A little later in the unit, students are given the following problem:
Let's make one math sentence for each of the following problems, then find the answers.
a) You have a 100-yen coin. You buy 3 sheets of paper, and each sheet costs 25 yen. How much change will you get?
b) You buy a 500-yen pencil case and a half dozen pencils. A dozen of pencils cost 480 yen. How much will you pay?
The textbook simply tells students that, in math sentences, multiplication and division (i.e., products and quotients) can be considered as one quantity and no parentheses is needed. Thus (a) can be represented by the math sentence, 100 - 25 x 3, while (b) can be represented as 500 + 480 ÷ 2.
Finally, after a few practice problems, the textbook summarizes the rules about the order of operations:

1. Generally goes from left to right.
2. If there are any parentheses, we calculate inside the parentheses first.
3. Multiplication and division are performed before addition and subtraction.
Clearly, knowing the order of operations is important. However, what I observe in these pages from the Japanese textbooks is that they place just as much, if not more, emphasis on writing and interpreting mathematical symbols/expressions/equations. Mathematical symbols/expressions/equations are the language of mathematics, and it is important for students to be able to communicate their ideas using the language of mathematics effectively. Furthermore, it is important for children to understand that we can create compound "sentences," too. Mathematical sentences allow us to concisely represent quantitative relationships and our own thinking processes. As we teach the order of operations, perhaps we should also keep in mind that it isn't just about calculation students are learning. We should consider including questions such as writing and interpreting compound sentences as well.