What Is The Midrash?
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Midrash (מדרשׁ) is an interpretive act, seeking the answers to religious questions (both practical and theological)...
Midrash falls into two categories.When the subject is law and religious practice (halacha ),
it is called midrash halacha. Midrash aggadah, on the other hand, interprets biblical narrative,
exploring questions of ethics or theology, or creating homilies and parables based on the text.
(Aggadah means”telling”; any midrash which is not halakhic falls into this category.)
MyJewishLearning.com
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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