Challah, Chapter 4, Mishnah 4. The Mishnah deals with someone who has before him a kav of new flour and a kav of old flour that have become joined to one another. First a point of background: a kav is not the amount that obligates one in challah; the required amount is five quarters of a kav, that is, a kav and a quarter. The new kav is from this year's crop, and the old kav is from last year's crop. "Shenashchu zeh bazeh" - that they became attached and stuck to one another, so that when they are separated each one takes a bit from the other.
The halachic question:
The rule is that a species combines with its own species. Here, indeed, both kavim are of the same species - wheat, for example - but one is from one year and the other from another year. Regarding Terumah (and likewise ma'aser) the halachah is that one does not take Terumah from one year's produce for another year's produce. The question, then, is: what is the law regarding challah?
"Rabbi Yishmael omer: yitol min ha'emtza" - Rabbi Yishmael says he should take from the middle, from the place where the two are joined to each other, so that he takes a bit from each one. The Yerushalmi points out that even though by taking from the middle he is in fact taking from the old for the new and from the new for the old, with challah this is permitted. For if with wheat and spelt, which are two different species and nevertheless we said that because of the similarity between them one may take from one for the other, then all the more so with new and old, which are a single species, and their law is not like that of Terumah and ma'aser. Therefore Rabbi Yishmael permits taking from the middle.
"Vachachamim osrin" - according to the Sages it is forbidden to take from the middle, because of a decree: the Rabbis forbade this lest an onlooker conclude that one may also take Terumah from the new for the old and from the old for the new. So what should he do? He should take challah from another dough that is obligated in challah, and separate from it for the dough before him.
One who takes challah from a kav:
Having discussed separating challah from a kav, which is not the amount that obligates, and knowing that separating from a kav does not amount to a dough that is obligated in challah, we must now clarify the law after the fact: if someone separated challah from a kav, was his separation effective?
"Rabbi Akiva omer: challah" - the separation is effective. Although one should not do so at the outset, after the fact the separation takes effect and this is challah.
"Vachachamim omrim: ein challah" - the separation is not considered challah at all.
It is explained that Rabbi Akiva was speaking about a particular case: not only did he take from a kav that does not contain the amount required for challah, but after the separation he added flour until he brought the dough up to the required amount. In such a case the separation takes effect retroactively according to Rabbi Akiva, while the Sages hold that even though he added afterwards, the addition does not help retroactively, and the separation does not take effect.
In summary: in this Mishnah we learned the law of a new kav and an old kav that became joined to one another - that Rabbi Yishmael permits taking from the middle, while the Sages forbid it because of the decree relating to Terumah and instruct him to take from another dough that is obligated. We also learned the dispute between Rabbi Akiva and the Sages regarding one who takes challah from a kav and afterwards added flour and brought it up to the required amount: according to Rabbi Akiva the separation takes effect retroactively, and according to the Sages it does not take effect.