Halakhah for Kiddushin 61:3

"A person is forbidden to walk four Amot standing upright, as it is written: 'The whole land is filled with [God's] glory'" (Kidushin 31a). "And anyone that walks standing upright — it is as if he is pushing the legs of the Omnipresence" (Brachot 43b). And the Gemara's intention is not that one should walk hunched over, but rather that one shouldn't walk too straight with his neck outstretched like people with too much pride. And these are the words of the Rambam in the fifth chapter from Hilchot Da'ot, Law 8: "A Talmid Chacham should not walk straight and with an outstrectched neck, like it is written, 'And they went with their throats outstretched.' Also, one should not walk with his heel to his big toe, like women and haughty people, as it is written: 'The women would go with their feet close together'. And one shouldn't run in public areas like the crazies, and shouldn't stoop over like people who suffer from a hunched back, but rather look down, like one who is standing in prayer. And one who is walking in the market should look like he is busy with his business. Also from the stride of a person can it be recognized whether he is wise and full of knowledge or if he is dim-witted. And so said Shlomo in his wisdom: "And also in the way that a fool walks his heart is empty, and he is telling everyone that he is a fool.'"

In order to assess whether women are required to cover their heads when praying or reciting berakhot, we must first clarify the law regarding men. Originally, a few eminent sages practiced the extra pious custom of not walking four amot bareheaded. Rav Huna b. R. Yehoshua is praised for not having walked four amot without a head covering (Shabbat 118b). He said, “Indeed the Shekhina is above my head, so how can I go around with it uncovered?” (Kiddushin 31a). Similarly, the Talmud teaches (Shabbat 156b) that after it was made known to Rav Naḥman bar Yitzḥak’s mother that he would become a thief according to his astrological sign, she was extremely meticulous about keeping his head covered constantly, enabling him to grow in Torah and mitzvot. Once, when his head covering fell, the evil inclination attacked him and his temptation to steal dates from the palm tree overtook him. He then understood his mother’s strictness. In time, this extra pious act became accepted in all of Israel until it became an obligatory custom (SA 2:6).