Doing mathematics is thinking about abstract objects and writing proofs or proof sketches. For example proving that there are an infinite amount of prime numbers. Doing a computation is following some algorithm, performing set steps usually resulting in an object such as a number. For example computing 43 + 19. These two are fundamentally different.
Wikipedia (03/05/23) seems to be of the same opinion as it states:
Arithmetic … is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers
Although in 2026 it seems to be more vague, I like the 2023 definition more.
en.wikipedia.org/wiki/Arithmetic
The Dutch education system and language also make this distinction. In elementary school you learn to add, subtract, etc. They call it “rekenen”, literally translated “calculate”. In secondary education you learn things algebra, trigonometry, etc. They call this “wiskunde”, translated “mathematics”. The activities of “calculating” and “maths” are fundamentally different. Mathematics is not just a harder form of calculating.
In the English speaking world it’s all just called maths. Which is unfortunate in my opinion. On the internet I often read titles or headings like “the math of [something]” or similar. Usually it ends up not having any theorems or proofs and I am left disappointed. Another thing is the question “are you good at maths?”, which is very confusing. What is that supposed to mean? If you are good at proving theorems in a certain field? Or whether or not you can perform some additions and multiplications quickly and accurately?