It is customary for Kaddish to be recited after a group reads Tehillim or learns Torah. If only verses from Tanach were read, then Kaddish Yeheh Shelama Rabba is recited. If the learning included Torah She'be'al Peh, then Kaddish Al Yisrael – which includes a prayer for the Sages of Israel, who developed the Torah She'be'al Peh – is recited. Of course, the recitation of Kaddish requires the presence of a Minyan. Sometimes, when fewer than ten men were learning, and one wishes to recite Kaddish, they will bring in men to complete the Minyan to allow for the recitation of Kaddish. May Kaddish be recited immediately in such a case, or must another verse be read, or some more material learned, so that the newcomers read or learn something before Kaddish is recited? This issue is subject to a debate among the Poskim. The Taz (Rav David Segal, Poland, d. 1667) maintained that the newcomers do not need to participate in the learning, and so Kaddish may be recited immediately upon their arrival. The Lebush (Rav Mordechai Yoffe, Prague, d. 1612), by contrast, maintained that the men who are brought in must, in fact, learn something before Kaddish may be recited. He reached this conclusion on the basis of the ruling of the Rama (Rav Moshe Isserles, Cracow, d. 1572) that at Minha, the congregation should not recite Ashreh without a Minyan. The Lebush understood that if a Minyan is not present for Ashreh, then even if the tenth man arrives immediately afterward, the half-Kaddish cannot then be recited, since there was no Minyan during the recitation of Ashreh. This indicates that a Minyan must be present not only for Kaddish, but for the learning or reading preceding the Kaddish. The Taz disagrees with this understanding of the Rama's ruling, and explains that Kaddish must be recited immediately after Ashreh, without any interruption, and so the congregation should not recite Ashreh without a Minyan, as they might then need to wait after Ashreh for a Minyan to arrive before saying Kaddish. The Magen Abraham (Rav Abraham Gombiner, Poland, 1633-1682) concurred with the ruling of the Lebush, explaining that Kaddish is recited after Torah learning because public Torah study, in the presence of ten men or more, creates a Kiddush Hashem (a glorification of Hashem's Name). After creating such a Kiddush Hashem, we declare, "Yitgadal Ve'yitkadash Shemeh Rabba" – that G-d's great name shall continue to be honored and glorified. As this is the reason for reciting Kaddish after learning, it follows that Kaddish is recited only if a Minyan was present during the learning. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) concludes that it is proper to abide by the stringent opinion, and thus if a tenth man arrives after the reading of Tehillim or Torah study, another chapter should be read, or another piece of Torah She'ba'al Peh should be learned, before the recitation of Kaddish. It is for this reason that the custom developed to recite the Mishna of "Rabbi Hananya Ben Akashya" before the recitation of Kaddish Al Yisrael following the study of Torah. In principle, this is not necessary, as the learning itself suffices to warrant the recitation of Kaddish. However, since sometimes the tenth man arrives only after the learning, it became customary to always add this brief Mishna just before Kaddish to ensure that a Minyan is present for the learning and not only for Kaddish. There is some discussion in the Poskim as to how many people must read a passage before Kaddish once the tenth man arrives. The general consensus seems to be that it suffices for just one person to read a passage, and this is, indeed, the common practice. Even someone who did not actually learn or read any verses may then recite Kaddish. Summary: If a group consisting of fewer than ten men read Tehillim or learned Torah, and then a Minyan arrived for Kaddish, someone must read another passage in the presence of the Minyan before the recitation of Kaddish. It is for this reason that the custom developed to recite the Mishna of "Rabbi Hananya Ben Akashya" before the recitation of Kaddish Al Yisrael after Torah study.