The Tosafists--beginning with Rashi's family and students--continued the conversation with their glosses on the text of the Talmud.
Tosafot are "Additions" to the Babylonian Talmud; the glosses, now printed together with the text in practically all editions, produced by the French and German scholars during the 12th to the 14th centuries. The Tosafot activity, in which the Talmud was examined minutely and in such a manner as to further the debates and discussions found in that work, began among the members of the family and pupils of Rashi (a 12th-century commentator on both Bible and Talmud).
The Tosafists flourished in Northern France, England, and Germany, the two best-known of the 300 or so practitioners being Rashi's grandson, Rabbenu Tam, and the latter's nephew, Isaac of Dampierre.
There are various collections of Tosafot, the printers of the Talmud selecting from those they had to hand in order to incorporate these into the published work. There is also a collection of Tosafot to the Pentateuch, know as Daat Zekenim (Opinion of The Elders) but this work is far less influential than the glosses to the Talmud.
Unlike Rashi's, which is a running commentary to the Talmud, the Tosafot consist of glosses to particular topics on which the authors have something to add. Very frequently, the Tosafot take issue with Rashi's understanding of a particular passage. They often point out apparent contradictions between Talmudic passages which they then either resolve by casuistry or else admit that the passages are in conflict.
Time and again the whole of the Babylonian Talmud relevant to a particular discussion is surveyed by the Tosafot, hence the saying that the Tosafot treat the Talmud as a ball thrown from hand to hand.
No responsible student of the Talmud can afford to ignore the difficult Tosafot. It was these that gave zest to the study of the Talmud, which became known as the study of Gefat, an abbreviation of Gemara, Peirush ("Commentary" of Rashi), and Tosafot. Although the Tosafot are in the nature of pure commentary, they naturally refer to the practical conclusions to be drawn. These were later collected under the heading of Piskey Tosafot (Decisions of the Tosafot) and have had an influence on the codification of Jewish law.
Excerpted with permission from The Jewish Religion: A Companion, Oxford University Press.