I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenomenon under study. This requires looking at examples, making a hypothesis more specific or sometimes more general, using formal arguments, geometrical arguments, studying algebraic structures, basically anything that leads to better understanding. Ideally, one understands a subject so well that notation basically doesn't matter. In a sense, the really key ingredient are the definitions because the objects are chosen carefully to be interesting but workable.

If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. What really helps is looking at objects and their relationships from multiple viewpoints. This is really what one does both in mathematics and physics.

Someone quoted von Neumann about getting used to mathematics. My interpretation always was that once is immersed in a topic, slowly it becomes natural enough that one can think about it without getting thrown off by relatively superficial strangeness. As a very simple example, someone might get thrown off the first time they learn about point-set topology. It might feel very abstract coming from analysis but after a standard semester course, almost everyone gets comfortable enough with the basic notions of topological spaces and homeomorphisms.

One thing mathematics education is really bad at is motivating the definitions. This is often done because progress is meandering and chaotic and exposing the full lineage of ideas would just take way too long. Physics education is generally far better at this. I don't know of a general solution except to pick up appropriate books that go over history (e.g. https://www.amazon.com/Genesis-Abstract-Group-Concept-Contri...)