What Is The Midrash?
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Midrash (Hebrew: מדרש) is ancient rabbinic interpretation of scripture. Aggadah (Hebrew: אגדה) is rabbinic narrative.
The two terms are, however, often used interchangeably to refer to those many aspects of rabbinic literature that
are not related to Jewish behavior or law (Hebrew: הלכה)...
Burton Visotzky, Benjamin Haber Kamine - Oxford Bibliographies
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The word Midrash is derived from the verb דרש which means to demand, seek out, or investigate. For example ותלך לדרש את ה' – she [Rivka]
sought out God’s word (Gen. 25:22); כי יבוא אלי העם לדרש אלקים – the people come to me to seek [the word of] God (Ex. 18:15); ודרשת וחקרת ושאלת היטב –
you shall seek out, investigate, and question (Deut. 13:15). In that sense, the word Midrash originally meant seeking out the meaning
of the text and referred to the exegetical process.
Rabbi Haim Ovadia, Sefaria.org
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A beit midrash is a Jewish house of study and discussion; in ancient times it was a school of higher learning.
The Pluralism Project, Harvard University