Andrei Hodoroaga's Key Ideas from A Mathematician's Lament
by Paul Lockhart

Mathematics should not be seen as a practical subject, where you are taught only stuff that will become useful later (without seeing exactly how).

Instead of plugging some variables into memorized formulas, we should do mathematics for fun, without thinking about its practical applications. The author argues that teaching mathematics in its natural state will completely change how students see the subject.

Just as music or painting are not done with a sensible purpose in mind, but rather as food for the soul, mathematics should be treated in the same way.

Pure mathematics exists only in our imagination, so starting from a problem you can continue to explore countless other implications. There is also a lot to discover, and you feel exalted when you suddenly discover something that was there all along.

The way to teach mathematics should be to encourage playing and having fun with ideas, only introducing new concepts and definitions when they are necessary.

The question that arises in this case is whether it is a problem that students could end up finishing school and not knowing formulas such as the quadratic equation. The author makes the case that most adults don’t actually remember it, and don’t even need it anyway.

Even if you consider its applications in science and engineering, teaching creative and independent thinking in mathematics will help students more than memorizing formulas.