There is nothing wrong, as far as Islam is concerned, in paying a portion of the zakah money to help establish educational institutions or upkeep Islamic schools.
Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states the following: Zakah, the third pillar of Islam, although primarily intended to help the poor and the needy, is also instituted to cater to other beneficial services and activities that are of general benefit for the entire community.

The categories of recipients of zakah, as enumerated in the Qur’an, are as follow: (Charities are (meant) only for the poor and the needy, and those who are charged with collecting them, and those whose hearts are to be won over, and for the freeing of human beings from bondage, and (for) those who are overburdened with debts, and (for those who strive ) in Allah’s cause (fi sabil Allah), and (for) the wayfarer: (this is) an ordinance from Allah–and Allah is All-Knowing and All-Wise.) (At-Tawbah 9: 60)
It is clear from the above list that one of the recipients of zakah is fi sabil Allah. Traditionally, it has been interpreted by the majority of scholars as those who are engaged in jihad, more specifically the actual warriors, as well as those who are devoted to the task of defending the frontiers, etc. However, there has always been a minority of scholars, both among the commentators of the Qur’an (mufassirun) as well as the jurists (fuqaha’), who have used the term fi sabil Allah in a far wider sense, thus extending it to include all beneficial works and projects that are of common benefit to the Ummah. They have thus cited examples such as funeral arrangements, building and taking care of schools and mosques, establishing hospitals, building bridges, etc. in this category. So they have always considered educational institutions as worthy of receiving assistance from zakah funds.
Although it may be going too far to extend the term fi sabil Allah to include any virtuous deed or project, there is no need for us to stretch the term too far to say that establishing educational institutions and initiating projects to promote the cause of Allah will be naturally included under the term fi sabil Allah (in the cause of Allah).
To put it differently, the view held by the majority of scholars that fi sabil Allah is intended for jihad is more in line with the Qur’anic usage of the term, which is always precise and specific in its terminology. At the same time, however, we need to stress most emphatically that there is no justification whatsoever, either in the Qur’an or Sunnah, to limit the term jihad and struggle for the cause of Allah to warfare or military enterprise.
Jihad and struggle in the Qur’anic vocabulary embrace all efforts and endeavors that are specifically intended to raise the word of Allah. The Qur’an and the Sunnah are replete with the uses of the term for intellectual efforts to disseminate the message of Islam. For instance in the Qur’an (Al-Furqan 25: 52), the faithful are reminded to make supreme jihad (i.e., intellectual jihad) through the medium of the Qur’an.
In the modern world, especially for Muslims living in the west in general and in North America in particular, jihad is primarily intellectual. There is no greater jihad for us than establishing viable institutions that uphold the cause of Islam and help make Islam a living reality in the life of Muslims.
It is, therefore, perfectly Islamic, and in perfect conformity with the true understanding of the higher objective of zakah as well as its definition of recipients, to say that Muslims in North America can legitimately set aside and give at least part of their zakah towards the establishment and building or upkeep of Islamic schools and educational institutions. Our foremost challenge today is to prepare a new generation of Muslims who will be imbued with sound knowledge of Islam from its pristine sources and committed to live its message and pass it on effectively to posterity. The above ruling has been forcefully upheld by scholars and jurists such as Imam Rashid Rida, Sheikh Qaradawi, and Sheikh `Abd Al-Kareem Zidan, to name only a few scholars.
In conclusion, the ruling that part of zakah (up to one third) can be given to establishing educational institutions that serve the cause of Allah is legitimate and is perfectly grounded in sound fiqh tradition.