I believe the best method is private education. But this can only be accomplished when people aren't being robbed to pay for state schools that fail entirely to successfully educate (I'm speaking specifically of America, as I haven't lived anywhere else thus far.) If we're going to talk about the best way to operate grade school, we have to include everything which causes public school (AFAIK, there's no country which doesn't want to indoctrinate their students with state-owned schools; if anyone knows of a country that doesn't, I'd like to know.)
So it works like this: lets say Bill owns 90% of the schools in a country. We can agree, when a sole entity dominates the market, it's essentially a monopoly, right? As the effects of a monopoly go, the price goes high, and the quality goes low. As Bill receives a protection fee from his fellow citizens, he can afford to keep these schools running, and he can also control what the schools teach; he can also create a rule which requires his fellow citizens to attend some form of school, thanks to the fact that he has a monopoly on the security business, as well. Because most people are already paying for the public schools, and most people cannot afford to pay for public schools and private schools, most people opt to send their children to the public schools; they also generally have no option to home school, as both parents are probably working to make ends meet, and also no option to keep their kids out of school, thanks to Bill's rule. This is generally how our public school system works in America.
How, then, can we improve such a system? Now that we have outlined the flaws of the American public school system (at its core, because the school system is a monopoly founded by coercion, the quality of service deteriorates dramatically, while parents have no choice but to use the public school system,) we can develop a system which will solve these flaws.
First, and foremost, we must raise the standard of living. This can be achieved when we allow the American citizen his every dollar earned; this, however, goes much deeper than taxation, which would require the government to lose enough power until it could stop allowing corporations to purchase it. Because the government currently has a lot of power, it becomes much easier for those with a lot of money to push this power in any given direction with the use of money. For example: nobody wants to pay $500 for a vacuum cleaner which doesn't suck very well. Likewise, a government with very little power will not help a corporation; to buy a government like that, you would also have to buy its citizens, who, because their government is weak, hold much more power individually. All this adds up to people having the ability to actually function as their own businesses; corporations can no longer play dirty, and smaller businesses will have much more room to prosper. This will help further distribute money from massive intercontinental corporate empires, to the average Joe across the street. So now, your mom, or your dad, either own a businesses, or work for a guy who likely pays well, lest his employees leave for better wages elsewhere (we're assuming people have the spines to stand up to shit wages, which brings me to my next point.)
Next, you abolish minimum wage. This
video will explain it better than I can. Long story short, if you need work, you will find it, until you can figure a better method of selling your time.
Anyway, this all adds up to a better living; you draw money out of the rich by stop giving them your money, and instead, buy from mom and pop. Everyone gets a little more prosperous. So lets assume, because we're doing so well, we, together, decide to no longer pay taxes; we support our own infrastructure and all that jazz. Now our public schools cannot be funded. So how do we make schools? Well, educators in every country don't simply vanish in this event; instead, they go forth, and create their own businesses, and compete with one another to create the best god damned educational institutions in the world. Since Bill's power over his country is now slashed, he can no longer legalize any monopolies, and these businesses known as schools are forever destined to compete with each other, whether it be grand institutions like Harvard or guitar prodigy Jim Smith teaching music theory at the rec center. Specifically, in the case of primary school, because parents are no longer forced to give up their hard-earned cash for protection money, nor do they have to send their children to school, or which school, or for how long they go to school, the parents can decide which school they would be able to best afford in their budget, for the best service, and could even decide to teach their children on their own, or perhaps hire a tutor; really, anyone can teach a child basic math and writing skills, you just need someone to do it. There will, specifically, no doubt, be schools appealing directly to children, or there could be schools which appeal to all ages; depends on what people want in any given area, and what the business owner would believe is best for his business.
Essentially, you take a monopoly out of the hands of the government, and give people the opportunity to own their own businesses, again. This will provide a much healthier environment for children to learn, in general, and give them better opportunities to grow up and be something. At which point, it is up to the parent to decide what is best for their children, and takes the matter out of the hands of the faceless entity known as government. Our children are not a public matter. Our children are the public. Alas, as we can see, now, this entire matter boils down to the parent; the very core of the issue, the true creator of future societies. But, having never been a parent, I believe there are others better suited to continue that argument.