《9加9除3=？》

說穿了，還是英語惹的禍！

2024年8月25日，台視報導了一則有趣的新聞：9加9除3=？

對於1960年代四年級以上的學生來說，這個問題並不難。那時候，算術老師教我們：9 ÷ 3 = 3，讀作「9 除以 3 等於 3」（從左到右）；或「3 除 9 等於 3」（從右到左）。

因此，9加9除3=？，如果寫成算術式，應該是：9 + 3 ÷ 9 =？。顯然，答案是9又三分之一，而不是9 + 9 ÷ 3 = 12。

要徹底解答這個問題，1960年代的算術老師會教學生兩個規則：

1. 四則運算，先乘除後加減（用背的）。
2. 「a ÷ b」讀作「a 除以 b」或「b 除 a」（也是用背的）。

在學習過程中，凡是老師沒有解釋的（或問了也沒有用的），只能暫時背起來，直接烙印腦海。直到高中時，由於當家教的關係，才開始認真去想這兩條規則：

1. 為什麼四則運算要先乘除後加減？

2. 「a ÷ b」讀作「a 除以 b」或「b 除 a」，這種讀法是從英文翻譯過來的。在英文中，「a ÷ b」讀作「a is divided by b」或「b divides a; using b to divide a」，意思是「用b來除a」。但是，後者並不是如台視新聞中那位老師所說的「to divide a with b」（using 和 with 用法有別）。

此外，「除以」的「以」是「用」的意思，亦即英文的「using」。因此，「a ÷ b」讀作「a 除以 b」，是由英文的「a is divided by b」翻譯而來的。

「橘越淮而為枳」— 我們的算術學者們將「舶來品」「a is divided by b」翻譯為「a 除以 b」，或將「b divides a」，「using b to divide a」翻譯為「b 除 a」，真是用心良苦。然而，後者在美國並不會教，因為不太符合算術式的直接含義。這也是為什麼當你問ChatGPT時，它會回答「9加9除3=12」的主要原因。

由此可見，要徹底解答並解釋9加9除3=？這個問題，除了上述兩條規則，還需要加上第三點：

3. 加強英文閱讀與翻譯的能力。

說穿了，還是英語惹的禍。《9加9除3=？》這個算術式，凸顯了英文閱讀與翻譯在數學教學中的重要性與挑戰。

English version for international students

《9加9除3=？》

On August 25, 2024, TTV (Taiwan Television) reported an interesting news story: 9加9除3=？(9 + 9 divides 3 = ?)

For students in the fourth grade and above during the 1960s, this problem should have been very simple. At that time, arithmetic teachers taught us: 9 ÷ 3 = 3, which is read as “9 除以 3 等於 3” (from left to right); or “3 除 9 等於 3” (from right to left).

Therefore, 9加9除3=？(9 + 9 divides 3 =,?) if written as an arithmetic expression, should be: 9 + 3 ÷ 9 = ?. Clearly, the answer is 9 and one-third, not 9 + 9 ÷ 3 = 12.

To thoroughly answer this question, arithmetic teachers in the 1960s would teach students two rules:

1. In the four arithmetic operations, multiplication and division should be performed before addition and subtraction (this must be memorized).
2. “a ÷ b” is read as “a 除以 b” (a divided by b) or “b 除 a” ( b divides a) (this must also be memorized).

Throughout my schooling, I memorized anything that the teachers didn’t explain (or anything that was pointless to ask about), and these rules became deeply ingrained in my mind. It wasn’t until high school, when I started tutoring, that I seriously tried to understand these two rules:

1. Why should multiplication and division be performed before addition and subtraction?
2. “a ÷ b” is read as “a 除以 b” or “b 除 a” entirely due to the influence of English. It turns out that this method of reading is translated from English. In English, “a ÷ b” is read as “a is divided by b” or “b divides a,” meaning “using b to divide a.” Besides, it is not as the teacher in the TTV news said, “to divide a with b.”

In addition, in traditional Chinese, the “以” in “除以” can be interpreted as “用,” meaning “using” in English. Therefore, “a ÷ b” is read in Chinese as “a 除以 b,” which is translated from the English “a is divided by b.”

The phrase “橘越淮而為枳” (literally “an orange tree transplanted north of the Huai River turns into a different fruit”) reflects the thoughtful efforts of our arithmetic scholars in translating the “imported” concept “a is divided by b” as “a 除以 b,” or translating “b divides a” and “using b to divide a” as “b 除 a.” However, the latter is not taught in the United States because it contradicts the direct meaning of the arithmetic expression. This is also the main reason why, when you ask ChatGPT, it answers “9加9除3=12.” (9 + 9 divides 3 = 12.)

From this, it can be seen that to thoroughly answer and explain the question 9加9除3=？, in addition to the above two rules, a third point must be added:

3. Strengthen English reading and translation skills.

The truth is, it still comes down to English. The arithmetic expression《9加9除3=？》highlights the importance and challenges of English reading and translation in mathematics education.