It's not that the poor don't have the drive to do well in high school or shoot for a college education. Or rather, that is not the only reason. For some, it's cultural; for others, it's because of family influence; for others, it's because they have disabilities, such as mental illness; for yet others, it is because of regional differences in industries.
And yes, we do give PreK - High school education paid for by taxpayers. And we do that because we have an economic interest in providing opportunities for education to our people. Originally, public schools was formed so that children would be able to do basic reading, writing, and arithmetic so that they would be able to read directions on the use of such things as fertilizers and calculate how much of which components they would need to apply the fertilizer to their fields.
We started public school in order to have an educated populace that is easier to train for occupations necessary for our well-being. As our society has advanced, so has the requirements for basic occupations. For example, plumbers no longer learn how to put pipes together. They also have to learn the environmental impacts of sewage. This requires even basic tradesmen to have a higher quality of education than previous.
I'd also like to point out a report I read one time. They took different factors of death row inmates and calculated common factors between them. The number one common factors between criminals sentenced for death was a lack of education.
So providing easier and better opportunities for education will not only help reduce criminals (since most criminals commit crimes because it is an easier way they can make an income, and thus there's an economic factor to crime) it will also ensure that we have an educated and trained populace that can pursue occupations and thus earn a living.
Basically what I'm saying is that the best way to get people off of food stamps is to ensure they get professional occupational training and education so they have the opportunity to get hired. Considering this will increase the economic benefits of the entire country for having an educated and skill populace, and it will reduce the crime rate, I think that's tax money well spent.