Such teachers, though very well paid by their students, might be as much disposed to neglect them, as those who are not paid by them at all, or who have no other recompence but their salary. (5.1.71)
Again, Smith says that teachers would perform better if they had more financial incentive. That's why he thinks they should lose their salaries and be paid based on how many students they attract to their classes.
Force and restraint may, no doubt, be in some degree requisite in order to oblige children, or very young boys, to attend to those parts of education which it is thought necessary for them to acquire during that early period of life. (5.1.73)
Smith is willing to admit that some really young children should be forced to go to school because they're not yet capable of knowing what's good for them. But once they hit thirteen, he's cool with letting them do whatever.
Those parts of education, it is to be observed, for the teaching of which there are no publick institutions, are generally the best taught. (5.1.74)
Smith believes that the things that are best taught in society are the things that are taught outside the public school system. That's because these things are taught in direct connection with how badly people want to learn about them. This way, people learn more useful stuff.
In modern times, the diligence of publick teachers is more or less corrupted by circumstances, which render them more or less independent of their success and reputation in their particular professions. (5.1.77)
Adam Smith thinks that schoolteachers would perform much better if their salaries were directly connected to the demand for their services. But public teachers are paid a salary and students are forced to be in their class, so there's no incentive for the teacher to improve the way she or he does things.
Were there no publick institutions for education, no system, no science would be taught for which there was not some demand. (5.1.78)
Adam Smith isn't sure what to think of public education. On the one hand, he's a fan of publicly accessible education. But on the other hand, he believes that education should be governed by supply and demand and that teachers should only teach what people want to learn about.
A private teacher could never find his account in teaching, either an exploded and antiquated system of a science acknowledged to be useful, or a science universally believed to be a mere useless and pedantic heap of sophistry and non-sense. (5.1.78)
If education were governed according to supply and demand, the world would have a lot fewer Latin teachers. That's because people would get to choose what they wanted to learn about instead of having the government tell them what to learn.
Every part of their [women's] education tends evidently to some useful purpose; either to improve the natural attractions of their person, or to form their mind to reserve, to modesty, to chastity, and to economy. (5.1.79)
Smith is a big fan of women's education during his time because he thinks that women are only taught things that are useful for their lives. In other words, women are taught how to sew, run a household, and resist sex until marriage.
An instructed and intelligent people besides are always more decent and orderly than an ignorant and stupid one. (5.1.93)
There are many reasons to make sure that the people in your country have a good education. One of the main ones is that educated people tend to behave better than uneducated ones.
They are more disposed to examine, and more capable of seeing through, the interested complaints of faction and sedition, and they are, upon that account, less apt to be misled into any wanton or unnecessary opposition to the measures of government. (5.1.93)
Smith believes that if people in a country are more educated, they'll be less likely to be taken in by fads and by radical politics. That's because they'll have the skills necessary to think for themselves and to see errors in other people's reasoning.
In free countries, where the safety of government depends very much upon the favourable judgment which the people may form of its conduct, it must surely be of the highest importance that they should not be disposed to judge rashly or capriciously concerning it. (5.1.93)
It's very important for people to be educated if a government is going to do its job properly. After all, these people need to be able to understand the government's decisions even when these decisions are unpopular.