Philosophers disagree profoundly about what the best way to do philosophy is. Disagreements of this sort are as old as the subject itself, which in the West dates back to the 6th Century BCE in Greece. Even then, some philosophers thought of what we now call philosophy as much the same sort of activity as natural science, while others thought of it as much more like religion. Disagreements of this sort have persisted to the present day.
The most common approach to philosophy, not only at the University of Michigan, but also in most other major universities in the English-speaking world, is what is known as analytic philosophy. In the first half of the twentieth century, analytic philosophy was a movement that drew on emerging developments in mathematics and logic to clarify philosophical problems. Some analytic philosophers believed that making philosophical questions precise would allow their definitive resolutions, others that it would at least make clear what earlier philosophers had been arguing about—and still others that it would show these questions to be ill-formed "pseudo-problems." Today "analytic" philosophy still looks to this tradition, valuing clarity and precision in formulating philosophical positions, and scrutinizing arguments carefully. Analyses of meanings are less central than they were half a century ago, and analytic philosophers are now more wide-ranging in their interests, writing on subjects as diverse, for instance, as law, aesthetics, feminism, and Marxism. Analytic philosophers continue to share a belief that philosophy has much to gain from close ties to the natural, social, and mathematical sciences. In the history of philosophy, analytic philosophers stress clear reconstructions of the positions and arguments of the philosophers under study. Analytic studies in ethics, language, thought, mind, knowledge, and the like stress careful formulation and argument, in hopes that clarifying issues and arguments will lead to progress with the problems. In the words of J.L. Austin, the approach is to make progress by asking, persistently, "What does it mean? How do you know?"
Another way of studying philosophy is via the careful interpretation and examination of classic texts in the history of philosophy, works by past philosophers who have proven to be of enduring interest to contemporary philosophers. Although philosophers often conceive of their discipline as like a science, insofar as they hope that it makes progress in solving problems and discovering truth, they typically devote more attention to the history of their discipline than would be common in the sciences. One reason for this is that, to the extent that philosophy does make progress, it does so by building on the work of past philosophers. Understanding contemporary discussions—understanding why philosophers ask the questions they ask, and consider the answers they consider—frequently requires understanding how we got to the point we are at in this continuing conversation and why the questions have so far resisted definitive solution. But another reason is that the greatest works of past philosophers are a continuing source of inspiration to contemporary philosophers, who often go back to them to mine them for ideas which haven't previously been given their due. And finally, some love to study the history of philosophy for the sheer challenge of trying to understand how the world looked to the best minds of other times and places.
The third major approach to philosophy in American universities is via continental philosophy. This designation originated in the English-speaking world as a way of referring to those European philosophers of the 19th and 20th centuries not directly involved with the analytic movement. The term denotes neither a single philosophical program nor even a single line of inquiry, but encompasses a number of quite distinct movements. Hegel, Marx, Nietzsche, Kierkegaard, Husserl, Heidegger, Sartre and Foucault are a few of the figures usually listed under the "continental" rubric. The much-discussed "analytic/continental" divide was an artifact of the conviction, held by many English and American philosophers into the 60's and 70's, that analysis was the only way of doing philosophy. As this conviction becomes less widely held, and as analytic philosophers expand their areas of interest, the distinction is becoming less and less significant -- with the result that even predominantly analytic departments like Michigan generally offer courses covering all the major "continental" figures.