The Imperative of a Jewish Day School Education?
שמאי אומר, עשה תורתך קבע, אמור מעט ועשה הרבה, והוי מקבל את כל האדם בסבר פנים יפות.
Shammai used to say: Make your Torah [study a] fixed [habit]. Speak little, and do much. And receive every person with a pleasant countenance.

I. Personal Obligation in Torah Study
כל איש מישראל חייב בתלמוד תורה בין עני בין עשיר בין שלם בגופו בין בעל יסורין בין בחור בין שהיה זקן גדול שתשש כחו אפילו היה עני המתפרנס מן הצדקה ומחזר על הפתחים ואפילו בעל אשה ובנים חייב לקבוע לו זמן לתלמוד תורה ביום ובלילה שנאמר והגית בו יומם ולילה.
Each Jewish man is obligated in Torah study, whether he is poor or rich, whether he is physically healthy or physically unwell, whether he is a young man or very elderly, whose strength has been diminished. Even if he was poor and sustains himself through charity and begs on the street, and even if he has a wife and children, he must establish times for study in the day and in the night, as it says, "and you shall examine it day and night."
אף על פי שחכם גדול אתה אינו דומה לומד מעצמו ללומד מרבו
Even if you are a great scholar, learning on your own is not the same as learning from one's teacher.
Rabbi Menachem Mendel of Kotzk

If you truly wish your children to study Torah, study it yourself in their presence. They will follow your example. Otherwise, they will not themselves study Torah, but will simply instruct their children to do so.

II. Parental Obligation to Teach One's Children
חֲנֹךְ לַנַּעַר עַל פִּי דַרְכּוֹ גַּם כִּי יַזְקִין לֹא יָסוּר מִמֶּנָּה.
Train up a child in the way he should go, And even when he is old, he will not depart from it.
רַק הִשָּׁמֶר לְךָ וּשְׁמֹר נַפְשְׁךָ מְאֹד פֶּן תִּשְׁכַּח אֶת הַדְּבָרִים אֲשֶׁר רָאוּ עֵינֶיךָ וּפֶן יָסוּרוּ מִלְּבָבְךָ כֹּל יְמֵי חַיֶּיךָ וְהוֹדַעְתָּם לְבָנֶיךָ וְלִבְנֵי בָנֶיךָ.
Only take heed to yourself, and keep your soul diligently, lest you forget the things which your eyes have seen, and lest they depart from your heart all the days of your life; but make them known unto your children and your children’s children.
וְשִׁנַּנְתָּם לְבָנֶיךָ וְדִבַּרְתָּ בָּם בְּשִׁבְתְּךָ בְּבֵיתֶךָ וּבְלֶכְתְּךָ בַדֶּרֶךְ וּבְשָׁכְבְּךָ וּבְקוּמֶךָ.
And you shall teach them diligently to your children, you shall speak of them when you sit in your home, and when you walk along the way, and when you lie down, and when you rise up.
Kiddushin 29a
דת"ר האב חייב בבנו למולו ולפדותו וללמדו תורה ולהשיאו אשה וללמדו אומנות וי"א אף להשיטו במים.
As it was taught in a baraita: the father is obligated regarding his son: to circumcise him, to redeem him, to teach him Torah, to marry him off, and to teach him a craft. Some say also to teach him to swim.
נשים ועבדים וקטנים פטורים מתלמוד תורה אבל קטן אביו חייב ללמדו תורה שנאמר ולמדתם אותם את בניכם לדבר בם ואין האשה חייבת ללמד את בנה שכל החייב ללמוד חייב ללמד.
Women and slaves are exempt from studying Torah. But a male minor - his father is obligated to teach him Torah, as it says, "And you shall teach them to your children, to speak in them" (Deut 11:19). Women are not obligated to teach their sons, for [only] all who are obligated to study are obligated to teach.
כשם שחייב אדם ללמד את בנו כך הוא חייב ללמד את בן בנו שנאמר והודעתם לבניך ולבני בניך ולא בנו ובן בנו בלבד אלא מצוה על כל חכם וחכם מישראל ללמד את כל התלמידים אף על פי שאינן בניו שנאמר ושננתם לבניך מפי השמועה למדו בניך אלו תלמידיך שהתלמידים קרויין בנים שנאמר ויצאו בני הנביאים אם כן למה נצטוה על בנו ועל בן בנו להקדים בנו לבן בנו ובן בנו לבן חבירו.
Just as a person is obligated to teach his son, so too he is obligated to teach his grandson, as it says, "And make them known to your children, and to your children's children" (Deut 4:9). And not just his son and his grandson: rather, there is a command upon each and every wise-man [heb. chacham] in Israel to teach all students, even though they are not his children. As it says, "And you shall tutor them to your children" (Deut 6:7) - from an [Oral] Tradition they learned that "your children" [here] refers to your students, for students can be called children, as in "And the children of the prophets left" (I Kings 2:3).

III. Communal Obligation to Educate Children
דאמר רב יהודה אמר רב ברם זכור אותו האיש לטוב ויהושע בן גמלא שמו שאלמלא הוא נשתכח תורה מישראל שבתחלה מי שיש לו אב מלמדו תורה מי. שאין לו אב לא היה למד תורה מאי דרוש (דברים יא, יט) ולמדתם אותם ולמדתם אתם התקינו שיהו מושיבין מלמדי תינוקות בירושלים מאי דרוש (ישעיהו ב, ג) כי מציון תצא תורה ועדיין מי. שיש לו אב היה מעלו ומלמדו מי שאין לו אב לא היה עולה ולמד התקינו שיהו מושיבין בכל פלך ופלך ומכניסין אותן כבן ט"ז כבן י"ז ומי שהיה רבו כועס עליו מבעיט בו ויצא עד שבא יהושע בן גמלא ותיקן שיהו מושיבין מלמדי תינוקות בכל מדינה ומדינה ובכל עיר ועיר ומכניסין אותן כבן שש כבן שבע
Rav Yehudah said that Rav said: this man should be remembered for good and Yehoshua ben Gamla is his name, for without him Torah would have been forgotten from Israel. At first, whoever had a father, he would teach him Torah, and whoever did not have a father would not learn Torah. What is the source for this? "And you shall teach them." Meaning that you shall teach them. They established an ordinance that they would appoint teachers for children in Jerusalem. What is the source for this? "For from Zion Torah shall come out." Meaning that from Zion Torah would come out. Still, whoever had a father would come up and learn, and whoever did not have a father would not come up and learn. They established that teachers would be appointed in every area and they would enter at age 16 or 17, but whoever's Rabbi would get mad at him would kick him out. Until Yehoshua ben Gamla came and established that they should appoint teachers for children in every city and every town and enter them at ages 6 or 7.
מושיבין מלמדי תינוקות בכל מדינה ומדינה ובכל פלך ופלך ובכל עיר ועיר וכל עיר שאין בה תינוקות של בית רבן מחרימין את אנשי העיר עד שמושיבין מלמדי תינוקות ואם לא הושיבו מחרימין את העיר שאין העולם מתקיים אלא בהבל פיהם של תינוקות של בית רבן.
They appoint teachers of children in every region and in every area and in every town. As for any town that does not have children who study at the Rabbi's house, they excommunicate the members of that town until they appoint teachers for the children, and if they do not appoint them, they excommunicate the town, for the world only exists for the breath of children who study at the Rabbi's house.
Sefer HaHinukh Mitzvah 419

וכן מה שאמרו [ב"ב כ"א ע"א] שחייב הציבור שבכל מקום ומקום להושיב מלמדי תינוקות, ועיר שאין בה תינוקות של בית רבן תחרב, וכ"ה תינוקות מושיבין אצל מלמד אחד.

As they said (Baba Batra 21a): Communities everywhere have an obligation to appoint teachers for their children. A city without school children will be destroyed. 25 children should be taught by one teacher.

ותניא כל עיר שאין בה עשרה דברים הללו אין תלמיד חכם רשאי לדור בתוכה בית דין מכין ועונשין וקופה של צדקה נגבית בשנים ומתחלקת בשלשה ובית הכנסת ובית המרחץ וביהכ"ס רופא ואומן ולבלר (וטבח) ומלמד תינוקות:
It was taught: a scholar is not permitted to live in a town that does not have these ten things: a court that metes out lashes and punishments; a charity fund that collects with two people distributes to three; a synagogue; a bathhouse; a bathroom; a doctor; a craftsperson; a blood-letter (a butcher); and a teacher of children.
The Chosen Few: How Education Shaped Jewish History, 70-1492 / Botticini and Eckstein

The difference between the provision of education in the Land of Israel and Mesopotamia is consistent with the divegent economic conditions in the two locations. In the Land of Israel, the provision of education was publicly organized by the community, educational taxes were levied on everyone, and the education of poor and orphan children was subsidized. In contrast, children's education in Mesopotamia was provided mainly by prvate teachers for individuals or groups of households, so that almost every family had to bear the entire cost of teaching itself. (p. 101)

A Jewish man who obeys the religious norm set by Judaism and sends his sons from the age of six or seven to the synagogue or the primary school to learn to read and study the Torah in Hebrew greatly enhances the boys' ability to learn to read and study any other text written in Hebrew (and then in other languages as well). Moreover, learning to read helps people learn to write. It also helps develop numeracy and the ability to compute prices, costs, interest rates, and exchange rates and thus to keep account books. These skills are useless for farmers in the agrarian economies of the first millennium for the reasons mentioned earlier. They are key assets for craftsmen, traders, and money lenders. (p. 127)


The Observant Life / "Torah Study"
In theory, all of us should personally teach Torah to our own children. In reality, most of us do not have the time, knowledge, or skill to do so. Consequently, we can discharge our obligation by employing others. Our responsibility does not end at the schoolhouse door, however. The primary obligation remains ours; our children's teachers are merely the agents we authorize to discharge our duty. This means that we must play an active role to make sure that our children are being educated properly. (p. 93)

...we should remember that the obligation to provide Jewish education is communal as well as parental. The Talmud attributes an ordinance to Joshua ben Gamla, one of the last High Priests of the Second Temple period, requiring each community to provide for the religious education of all its male children from the age of six or seven. Jewish communities throughout the ages have shouldered the responsibility of providing Jewish education to their youth. Consequently, federations and other communal institutions should make it their top priority to increase the availability and affordability of Jewish education for all Jewish children regardless of gender. After all, it is Jewish education that is the most important guarantor of Jewish continuity; more crucially, it is only through education that we can create a Jewish community worth perpetuating. (pp.94-95)