There are algorithms which involve advanced mathematics on classical computers, too. You don't have to understand them to understand how classical computers work. I've never bothered to learn the general number field sieve, and similarly I've never bothered to learn Shor's.

I say if you understand gates as unitary matrix multiplication, representing multiple qbits with the tensor product, entanglement, and projective measurement, you basically understand quantum computing. Throw in an algorithm or two to convince yourself of the benefits.