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 yielded 13 rules, 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 held the former. R Yishma’el famously said “Diberah Torah belashon benei adam — The Torah is written in human idiom”. He therefor saw derashos as applying to the meaning of clauses, not individual word choice. He wouldn’t finds grounds for a derashah if the wording is a normal idiom or metaphoric description. His rules of kelal uperat, how to understand various combinations of inclusive and exclusive clauses, was about just that — clauses. This understanding of the nature of the Torah’s language also lead R’ Yishma’el to view derashah as a means of finding what the Torah is telling us, such as “shomei’ah ani — I hear”.

R’ Aqiva learned “mounds of halakhos from the tags and serifs 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”). And instead of lelal uperat, his rules of derashah  or the presence of inclusive or limiting keywords, ribui umi’ut. Such terms as akh – except; raq – only; or kol – all) are grounds for derashah, even if they fit gramatically, or if the phrase as a whole doesn’t talk about excluding or including anything. And in terms of describing what dersahah does, R’ Aqiva’s language is more one of finding truths, “yakhol — it could be that…” Perhaps because his work is 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 later debate on this point), in the generation of Rabbis Aqiva and Yishma’el rabbis certainly did not venture new derashos beyond qal vachomer (deriving from the less obvious case to the more, almost a rule of logic rather than derashah).

None of this necessarily means they invented the rules of derashah or even that they disagreed over fundamentals. The debate between the two schools of medrash were not over the creation of new rules of derashah. For that matter, it is clear that Hillel’s rules were known to the previous heads of the Sanhedrin, the Benei Beseira. The discussion is over taxonomy; how to understand a given individual 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:

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.