@Brandon9000,
my training in physics, first general physics then physical chemistry and then geophysics, gave me the ability to think in dimensional equivalents where we can solve one set of equations in terms of another item.
Like doing hydraulic equations in terms of drum head theory.
It made me become involved in analog and digital models. Analog models of fluid flow fields were solved in terms of capacitor /resistor networks where we substituted the electrical properties for chemical diffusion and hydraulics.
I didn't call it only a "new way" of thinking but a "confident" way of thinking about new stuff.
I would tell my later students that you could keep only a few field equations in mind and solve problems in everything else in terms of those few. To me, the physics preceded the delving into deeper math. (There became a logical reason for transforms and operators and differential equations)
That's just my experience. I was originally terrible in math (I was hyperactive as a young kid and it took some "help" to overcome or work with). Once I saw the relationship of physics to math , I followed the math train like an obsessed monk.
