Is math independent of our sensory experience?
Kants answer, in a sense, is that it is both dependent on sensory experience and also not. He claims that our intuition for space, through which we construct geometry, is a priori and thus independent of experience, but also synthetic, so that it is more than the rules of logic; he says that this is possible because our intuition for space is a neccessary condition to have any experience at all.
Frege, agreed with this for mathematics considered solely as geometry, but disputed arithmetic fits into what he calls Kants psychologism, perhaps a term that he picked up from Hume. Frege is a key figure in the early 20 Century project to reduce arithmetic to logic; and it is this thought that bypasses Kant, or so one supposes, because this would mean that arithmetic being solely based on logic cannot be synthetic, but must be analytic.
Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics?
Thus, Kants answer is no for both arithmetic and geometry; and Frege says yes for arithmetic, and no for geometry.
As for learning mathematics - The SEP says on the Kantian philosophy of mathematics:
In a series of papers, Charles Parsons has argued that the syntheticity of mathematical judgments depends on mathematical intuitions being fundamentally immediate, and he explains the immediacy of such representations in a perceptual way, as a direct, phenomenological presence to the mind.
That is the abstract '2' as distinct from, say a concrete '2 bottles' or '2 books' that we might look at and perceive, is not abstract to our sensibility, it has a 'phenomenological' presence.
The hard work of learning mathematics is to synthesise these concepts so that the abstract concept has this actual sensual presence in the mind. One might say the moment of 'clarity' or 'illumination' is a spark given of by this act of mental synthesis. This is the beginning, the process and becoming of the mathematical Subject - subject as in subjectivity, not as in topic.