To get a different perspective on this divide, I reached out to John Chubb, president of the National Association of Independent Schools, a nonprofit organization that represents more than 1,400 private schools in the United States. "I think there is some merit to the criticism of independent schools not paying adequate attention to teacher training," he told me. "But teaching any kind of student really well is hard—advantaged, disadvantaged, average, or gifted."

And while many private-school teachers could flourish in inner-city public-school settings, many would struggle, Chubb said. I would likely be one of those teachers, but I’m not at all convinced that this would make me less of a teacher—just a different kind of teacher. I make a difference in my private-school classroom, and I work with students who need and depend on me. The same holds true for my talented and dedicated colleagues who work in every other type of school, public or otherwise.

For their part, I hear many private-school educators lament that public-school educators have little choice when it comes to what they teach and have to focus on preparing students for testing rather than providing meaningful instruction.

A quick Google search shows that the news media have covered the overtesting issue extensively. In writing this article, for example, I came across a PBS NewsHour story, "Is there too much testing in the public schools?" In the segment, Alberto M. Carvalho, the superintendent of Miami-Dade County Public Schools, was quoted as saying, "We need to recalibrate the necessity of so many of these tests, conduct a thorough analysis of the duplicate nature of some of these exams, and have a rapid regression back to reason in terms of what’s appropriate for students and teachers alike."

And private-school educators staunchly object to critics who question their professionalism, acknowledging that though they do have fewer students, they use any extra time to offer their pupils higher-quality feedback and instruction.

I recently reconnected with Mark Barnes, a colleague who recently wrote a book arguing against grading, curious to hear from a celebrated education author and public-school teacher. "My private-school colleagues say they can do things public-school teachers can only dream of," he said, referencing benefits such as more in-depth feedback on their work, fewer lectures, and access to state-of-the-art technology. "Not many public schools do this because of constraints of the Common Core and high-stakes testing."

I can’t help but wonder how much the media have intensified animosity among all types of school personnel—and not just between those in public and private institutions. In 2013, Slate posted a blistering condemnation by one of its senior editors, Allison Benedikt, "If You Send Your Kid to Private School, You Are a Bad Person." With over 70,000 "likes" on Facebook, its bold title also leaves nothing to the imagination: "You are a bad person if you send your children to private school. Not bad like murderer bad—but bad like ruining-one-of-our-nation’s-most-essential-institutions-in-order-to-get-what’s-best-for-your-kid bad. So, pretty bad," Benedikt wrote.