I am a bit confused with these two examples, which is the correct one?
- I learn mathematics, or
- I learn about mathematics.
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First let's consider learn about vs learn with a different object, like piano.
If you were to learn the piano you would be learning to play the piano yourself. If you were to learn about the piano you might find out that the invention of the modern piano is credited to Bartolomeo Cristofori (1655–1731) of Padua, Italy.
In the same way, learning mathematics would likely involve learning about the relative magnitudes of integers and progress on to addition, subtraction etc. I.e., you would be learning how to perform mathematical calculations yourself.
Learning about mathematics you would find that Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change, and that more complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations.