someone once described the different areas of study as a wheel. You've got

The Arts -> The Humanities -> Life Sciences -> Physical Sciences -> Mathematics -> The Arts

We often view The Arts and Mathematics on opposite ends of the spectrum, but as I've gone deeper into mathematics, I've learned that an elegant proof is far more similar to Poetry than to the sciences.

If you approach mathematics from an engineering/applied science angle, it seems like doing it by hand is useless.

Another great analogy I've heard is that the language of mathematics requires about 12 years of "spelling and grammar tests". Most students then stop, considering themselves savvy enough. After those twelve years, those who go on experience the poetry of proof, where the basic building blocks are twisted and interwoven in new and creative ways. Math at this level is often more like creative writing, exploring new areas and explaining exciting things.

I really feel that most students get so turned off by the calculations that they never get to see how creative and elegant math can be.