We already do. Most charter schools, especially those run by private (usually for-profit) companies do not have unionized teachers. Parents do not have to pay for their children to attend these schools as they are funded by taxpayer money.
Would you like me to dig up the research comparing charter vs. traditional public schools?
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As for the unions being “private” orgnaizations, the bulk of that money goes to the teachers — the ones who do the actual work. With private schools, especially the for-profit charters, the teachers tend to be underpaid while the administrators and “owners” (who don’t do any of the work and, all too often, have no educational background) take a disproportionate share of the money.
I am opposed to the concentration of wealth, and I firmly believe that the people who do the work and produce the value should be the ones who benefit the most from their work.