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.
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.
算数 allows for only the operations add, subtract, multiply, and divide (足す引く掛け割る) which are specific algorithms. 算数 also usually imposes the constraint of using only natural numbers with a standard log scale. Considering this, the operations in 算数 traditionally allow for memorized expected results with possible aid of tools (such as multiplication tables).
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).
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