There are two kinds of medrash (which should technically be called “midrash” to be grammatically correct). Midrashei Aggada are non-halakhic statements, those of mussar, Jewish thought, Qabbalah, and the like. The thought is usually connected to the text through details added to the narrative, or other stories intended as metaphor (whether in addition to being historical or not).I want to discuss here Midrashei Halakhah, which derive laws from the text. Most often, through the rules of derashah. Hillel made a science of derashah, and reduced it to 7 rules. R’ Yishma’el and R’ Aqiva, broke down those rules into subcategories. Because of the differences in approach, R’ Yishma’el’s exposition yeilded 13 laws, R’ Aqiva’s, 19.

Derashah could be understood in 2 ways: Either as applied to the semantics, the meaning of the clauses of the verses, or as applied to the syntax — that particular words have coded meaning.R’ Yishma’el’s school believed the former. “The Torah is written in human idiom”. Therefore, derashos apply to the meaning of clauses, not individual word choice — if it’s normal idiom or metaphoric description. This also lead R’ Yishmael to view derashah as a means of getting what the Torah is telling us, such as “shomei’ah ani” (I hear).R’ Aqiva learned “mounds of halakhos from the tags and serrifs on the letters”. He understood derashah to be about the text itself. Doubled words (e.g. “aseir ta’aseir — you shall tithe” is also taken to mean “aseir bishvil sheti’asheir — tithe so that you may become wealthy”), or the presence of limiting or inclusive keywords (akh – except; raq – only) are grounds for derashah. R’ Aqiva’s language is more one of finding truths, “yachol”, it could be that… Being less related to the plain meaning of the verse, he understands a suggested derashah as less compelling than R’ Yishma’el would.

By their day, these rules of derashah were descriptive only. While Hillel and Shammai may have had the power to make new derashos (there is debate on this point), R’s Aqiva and Yishma’el generation certainly didn’t beyond qal vachomer (deriving from the less obvious case to the more).

Also, none of this necessarily means they invented the rules of derashah or even disagreed over fundamentals. The debate between the two schools of medrash were not over the creation of new laws of derashah. For that matter, it is clear that Hillel’s laws were known to the previous heads of the Sanhedrin, the Benei Beseira. The discussion is over taxonomy; how to understand derashah as being the product of a few clear rules. They could well have simply divided the existing derashos into existing categories, and categorized differently. In fact, we find R’ Yishma’el using ribui umi’ut (a principle of R’ Aqiva’s list) and R’ Aqiva using kelal uperat.

The two series of medrashei halakhah are:

	R’ Aqiva’s school	R’ Yishma’el’s school
Shemos	Mekhilta deRabbi Shim’on bar Yochai	Mekhilta (a/k/a Mekhilta deRabbi Yeshima’el).
Vayiqra	Sifra (a/k/a Toras Kohanim and Sifra deVei Rav)	Sifrei (lost sometime during the late geonim or early rishonim)
Bamidbar	Sifrei Zutah (“Small Sifrei”)	Sifrei (the remaining portion)
Devarim	Sifrei	Mekhilta Devarim (largely lost; some portions were recovered from citations including some only found in the Cairo genizah)

The texts seem to have been redacted in the 3rd and 4th centuries.

The traditional publication of the medrashei halakhah includes four books, mixing the two schools: Mekhilta, Sifra, Sifrei (on Bamidbar) and Sifrei (on Devarim). In fact, the two Sifrei’s often get published as a single volume, despite the difference in style that makes their different origin obvious (once you know to look for it).

A more complete publication would have all seven books, typically published in the order: Mekhilta, Mehilta deR’ Shim’on bar Yochai, Sifra, Sifrei (Bamidbar), Sifrei Zuta, Sifrei (Devarim), Mekhilta Devarim.

The word “mekhilta” is Aramaic, and means “measure” or “rule”. The words “sifra” and “sifrei” are conjugations of the root /spr/, meaning “book” or “writing a book”. Sometimes the word “sifrei” is used to refer to all 4 books.

After Rabbi Yehudah haNasi compiled the Mishnah, organizing halakhah by topic rather than verse, the notion of composing Medrashei Halakhah fell out of use. However, as he was from R’ Aqiva’s school (a student of R’ Aqiva’s student, R’ Meir), that school ended up making greater impact on the final law.