Kant: a priori vs a posteriori

Kant devotes a lot of time to drawing distinctions between different kinds of judgments that we make. The first distinction that he makes is rather simple, and that is the distinction between a priori and a posteriori judgment. The distinction comes from the origin of the information that we obtain. A priori (literally "from what comes before") information is determined by reason alone and can be found out without any particular experience. This makes a priori information universal tautologies. A posteriori (literally "from what comes later") information, on the other hand, requires a certain experience in order to obtain that knowledge. The information obtained, therefore, is limited and subject to interpretation. Kant, who blends rationalist and empiricist theories, says "that all our knowledge begins with experience there can be no doubt... But... it by no means follows that all arises out of experience." I agree with Kant on this point, because we are not born with knowledge about mathematics, colors, or really anything else. But, with simple lessons here and there through experience, we can learn how to rationalize and reason through problems we have never encountered. So, for example, I can figure out what 29381818291301 plus 3181029138104 equals, even though I have never experienced that problem before. However, I need the prerequisites of knowing how to count and add in order to reason myself through the problem.