Indeed, teaching mathematics without thinking of it as an embodied practice is how you end up with terrible math courses.
You need the intuition.
You need to know the motivation (usually applications in physics/chemistry/CS/business in lower-level mathematics, but even highly abstract mathematics without any known immediate applications usually has a motivation.)
And of course, you also need the abstraction and formality and rigor. The certitude of proof is, after all, the raison d'etre of modern mathematics!
> By thinking about mathematics as performance, we liberate it from the straightjacket of abstraction into which it has been too narrowly confined.
We can also think about this the other way around. By thinking about mathematical objects formally, we free ourselves from the straight-jacket of uncertainty into which the exclusively performative approach inevitably binds us. The advent of formal systems and the development of mathematical theories that grew too hairy for purely intuitive reasoning coincided for a reason.
The most significant and beautiful pieces of mathematics -- and many of the examples in the article -- were discovered and fully understood precisely because we hone our intuition with the rigor of mathematical abstraction.
The blog post linked below comes to mind.
https://terrytao.wordpress.com/career-advice/there%E2%80%99s...