Well, you already know that the answer to that. Desirable employers don't screen job applicants based on which childcare facility and high school they attended. It's always college that's the focal point. Seems inevitable that the reactionary conversation then gravitates towards fixing college access rather than K-12 quality. (Therefore, it doesn't matter if K-12 is actually the more important thing to fix.) As an example, James Clark of Netscape dropped out of high school[1] but his future employers didn't care because he had a college degree. The college degree overshadows the stigma of being a high school dropout.
If McKinsey Consulting, Goldman Sachs, Google Inc, etc filtered out candidates based on what kind of public libraries the candidate had access to while he grew up, I suppose we'd be having political conversations about "building better public libraries!" so children can compete for jobs.
[1]https://en.wikipedia.org/wiki/James_H._Clark#Early_life_and_...
I'm not arguing against college (I think there are arguments against college as an opportunity equalizer, but I'm not making them). I'm arguing against free tuition for everyone as a vector for getting more low-income people into college.
If you want more low-income people to attend college, fix childcare first, so high-school age students aren't sacrificing class rank to take care of younger siblings and graduating high school into circumstances that demand they immediately find a wage.
Or, if you're dead set on applying pressure at college itself: means-test the benefit, so at least you're not giving billions of dollars away to families making six-figure incomes while families making less than $40,000 are sending their 17-18 year olds to 5 years of sporadic dead-end service industry employment. Because: "college is too expensive" is not the only reason that happens. There are plenty of extremely inexpensive community colleges that are vast improvements over 9 months working at the Sprint Cell Phone Store. But that's where low-income teenagers often end up.
Free childcare for working families (particularly single mothers) would do so much to lift kids out of poverty.
Free college tuition is a start, but you're right that even more investment is needed. College students should be paid by the state to invest in their education and enjoy their carefree student days.
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Mortgage interest deductions are obviously and inexcusably evil policy, of course. They make a bad example for comparison, like holding up someone's politics to Naziism.
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Also, we can't just fix K-12 or child care first and leave everything else aside. Those are complicated issues that will be controversial forever. We do have to improve other things in the meanwhile.
It's also the case that the pipeline to college attendance is constructed to reward children brought up under middle class circumstances. Unless you think it's the case that food and shelter insecurity, the requirement to care for and sometimes transport younger siblings, and sometimes total lack of supervision doesn't contribute to poorer school performance.
In that sense, proposals for free college tuition aren't that much different than the (totally, unbelievably, egregiously regressive) tax favoritism we accord home ownership through the mortgage income deduction.
How about we fix childcare and K-12 first?