Indeed! There is a popular saying, "Arithmetic is to mathematics as spelling is to writing." Of course, a mathematician is often good at arithmetic just like a writer is often good at spelling. However, someone merely good at arithmetic is no more a mathematician than someone good at spelling is a writer.
> There is also a running joke that mathematicians tend to be bad at mental calculations. The story about Grothendieck prime comes to my mind: In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."
A related story from http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf quoted below:
> There is also a running joke that mathematicians tend to be bad at mental calculations. The story about Grothendieck prime comes to my mind: In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."