# Why are there two words 算数 and 数学 representing different fields?

Why are different words 算数 and 数学 used for mathematics at and after elementary school? I understand that the quality is different in that, in the former, the method of solving an equation is usually not allowed, and special calculation methods that developed within the 和算 tradition are used, but is that different enough to be considered as different academic fields? I don't see any more difference between these two methods than the difference between number theory, algebra, geometry, calculus, combinatorics, etc. which are all subsumed under 数学.

• Personally, I think there's a lot of difference between the English math fields you mentioned... Especially since I'm really good at some and terrible at others. – atlantiza May 31 '12 at 2:10
• @atlantiza That is my point. They are different, but are all called math. – user458 May 31 '12 at 2:12
• I think it's a separation between what kind of mathematics one can perform daily without much thought 算数 and the realm of mathematics where theory and explanation are required to gain sufficient understanding 数学. Although, I'm not sure.. – Chris May 31 '12 at 2:25
• @Sawa I wonder if it is handled as an independent field only in Japanese.. – Chris May 31 '12 at 3:08
• “Why are there two words 算数 and 数学 representing different fields?” Isn’t this title illogical? If we accept that they represent different fields, then there is no question why there are two different words for them. The actual question is why different words are used for arguably the same field, namely mathematics, depending on whether it is taught at elementary school or not. – Tsuyoshi Ito Jun 8 '12 at 23:00

To preface, I think that this question involves something more than just the structure of mathematics in Japan. It might be related to how mathematics are organized generally.

The two fields should be characterized for clarification:

If 算数 refers to the branch of mathematics taught in elementary school which is arithmetic, then that should be referred to specifically as "elementary arithmetic". This is a subset of number theory, but with more constraints.

When being taught 算数, the goal or purpose in mind is accuracy (正確性). Whether or not an individual is able to use the given numbers and operations appropriately is what defines this field.

However, 数学 deals with the process of arriving to a mathematical answer (論理の正確性). It is irrelevant whether the numerical accuracy is correct or not because that is not the goal of what is being taught. (I'm not sure for Japan, but in America teachers give points only whether the logical thought process was correct in solving the problem regardless of the arithmetic)

As for why these two are treated like two independent fields, I would make a guess that in the daily use of arithmetic (日常的) such a strong emphasis is placed on whether the numerical operations are handled correctly. If you are a cashier, your thought process regarding theories of mathematics are irrelevant compared to whether you can perform basic operations with accuracy.

For this reason, 算数 (which I would refer to as elementary arithmetic) seems to be a separate taught field due to the intended goals in mind.

EDIT: It might be useful to note that while arithmetic can be a field of study, "the practical use of elementary arithmetic" probably does not have enough merit to be a field of study. It has also been stated that elementary arithmetic is a "calculation / computational discipline"

EDIT2: I'm not sure, but I found the term 暗算 (doing calculations mentally) when reading about abacus' or 算盤. I'm not certain, but if 算盤 was used in elementary school, it would allude to doing 暗算 being the focus of the course.

EDIT3: If 算数 as a course includes geometrical aspects (幾何学), then the a similar distinction can be made with analytical geometry (解析幾何学).

• I like this answer, but I think there is a historical component too. IIRC 算数 was taught in terakoya schools and considered a practical skill useful for business, etc.; 数学 comes from a more "academic" tradition, and was the vessel into which Western mathematics was eventually poured. But I haven't had time to look up proper references. – Matt May 31 '12 at 23:49
• 暗算 refers to mental arithmetic – as opposed to doing it with pen and paper. – Zhen Lin Jun 1 '12 at 7:45
• @Zhen Lin: How right you are! – Chris Jun 1 '12 at 15:06
• @Matt: I also saw some details regarding what you say; however, I haven't been able to find anything sufficient to make any sort of claim. I will keep looking. – Chris Jun 1 '12 at 16:21

I thought `算数` was just arithmetic (addition, subtraction, multiplication, division) and `数学` was the whole scope of mathematics.

According to this dictionary definition, arithmetic is "the branch of mathematics dealing with the properties and manipulation of numbers." But we all know that math is way more than just numbers. And the definition says it right there: the branch of mathematics. So `算数` would have to be a (likely small) subset of all `数学`.

• I think that question is why does it deserve to be treated as an independent field when it is actually a subset of mathematics. (Specifically number theory). – Chris May 31 '12 at 15:37
• I understand that the word 算数 looks like just arithmetic, but it is actually not. Some elementary geometry is also part of 算数. – Tsuyoshi Ito Jun 1 '12 at 0:02
• @Tsuyoshi Ito Are you referring to ユークリッド幾何学 (Euclidean Geometry)? – Chris Jun 1 '12 at 1:40
• @Chris: Yes, but since the notion of proofs is outside the scope of 算数 taught in elementary schools, it is quite limited. The part of geometry taught in elementary schools includes “What is a rectangle,” “What is area (how we compare the ‘largeness’ of two shapes),” “net of a 3-dimensional shape,” and so on. – Tsuyoshi Ito Jun 1 '12 at 2:06