Is there a connection of definition between the two words used for "tribe" - מטה and שבט which both mean "stick" or "rod"? If so, why is the term "rod" used to refer to a tribe?
שבט - Shemot 21:20
מטה - Shemot 4:2
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Rabbi Shelomo Wolbe, in his book on parenting, Zeri'ah uVinyan beChinukh (Title of English translation: Planting & Building in Education: Raising a Jewish Child), discusses the verse (Mishlei 13:24):
חוֹשֵׂ֣ךְ שִׁ֭בְטוֹ שׂוֹנֵ֣א בְנ֑וֹ וְ֝אֹהֲב֗וֹ שִֽׁחֲר֥וֹ מוּסָֽר׃
One who spares his sheivet hates his son; but the one who loves him, disciplines him early.
The source of the English idiom, "Spare the rod, spoil the child."
Rabbi Wolbe's thesis in this book is that children are to be planted and/or built. All too often we try raising them by "pruning" unwanted aspects from the child. We should instead look to build up the positives.
His explanation of this verse is that "sheivet" doesn't mean the kind of stick one would punish that child with. Rather, it's the staff of a leader. Like in Yaaqov's blessing of Yehudah. He phrases the fact that David's royal line will come from Yehudah with the words, "לֹֽא־יָס֥וּר שֵׁ֨בֶט֙ מִֽיהוּדָ֔ה -- the sheivet will never leave Yehudah..." (Bereishis 49:10) Note the phonetic relationship to sharvit - scepter. (E.g. Esther 4:11)
Similarly, I would note that a "mateh" is a shepherd's crook. (Yehudah has one, Bereishis 38:18; and it's the term for Moshe's staff -- Shemos 4:20, etc...)
The rod the translation presumes Mishlei is talking about would have been a "maqeil".
So it makes sense that a unit of the leadership hierarchy of the Benei Yisrael would be a "sheivet" or a "mateh".
In this essay, I offered three approaches to this question:
Citing an explanation from Rabbi Shmuel Dovid Luzzatto (1800-1865), Rabbi Yaakov Tzvi Mecklenburg (1785-1865) writes that the words shevet and mateh actually have different meanings, but because they are conceptually related they became interchangeable. He explains that the word shevet originally referred to the leader of a tribe, while mateh originally referred to the other members of a tribe. The leader of a tribe is similar to a mast upon which a flag is mounted because all of the members of the tribe rally around the leader. In this way the word for a leader of a tribe is homonymous with the word for stick. Since all the members of a tribe are united behind their leader, references to them can be subsumed under the word used for the leader. Thus, the word shevet also came to mean members of a tribe because the leader of the tribe embodies the entire tribe itself. In terms of sticks, he argues that shevet and mateh can both refer to the exact same type of stick, but they refer to different parts of the stick. The word shevet refers to the top of the stick (just as the leader sits atop the hierarchal structure of a tribe), while the word mateh — seemingly related to the Hebrew word lematah (underneath) — refers to its bottom (just as the other members of the tribe live under the leadership of their clan chief).
Malbim (to Gen. 49:28) writes that the word shevet does not literally mean “stick”, rather it means “branch,” which was the most common item used as a stick. Based on this, he explains that shevet means branch and tribe because each of the Tribes of Israel is simply a branch of the greater family tree of Jacob’s descendants.
Using the synonyms in question as a springboard for discussion, Rabbi Zev HaKohen Hoberman (1930-2012) sums up the dual role assumed by the Tribes of Israel. On the one hand, they are called shevatim because G-d has granted them certain forms of authority and responsibility over the nations of the world. Yet, paradoxically, they are also called matot (which he equates with the words mishenet and chutra) because they serve as Jacob’s “cane” in death, upon whom he “leans” to carry on his legacy. In that way they are not autonomous, but simply follow the tradition of their illustrious forefather.
SOURCE: What's in a Word?, "Tribesmen Stick Together"
Radak in Sefer HaShorashim (p.245 0f 280) on the word 'שבט' provides an answer.
He says that the line of connection that joins a family / tribe together is like a stick. In other words, if you imagine a family tree, the branch that links generations is symbolic of a stick