chess.com/news/view/researc … wer-grades
I have my doubts as to there being any statistically significant correlation. That said, I do think that all children in all grades, K-12, should be given regular classes involving music, art (both doing and history of art), drama and chess.
As for chess, I would focus on social skills development with kids working together to solve problems as well as engaging in team events within their class, school and inter-school.
Contrary to popular misconceptions that are promoted by a segment of the media, the public schools are not centers of liberalism or even “critical thinking.” The foci of the schools are on keeping order, conformity, promoting conservative ideologies, and avoiding controversy. The content of history and civic education is scrutinized by political bodies and book publishers to make sure that the students are not given full information about the development of our country and its institutions. Textbooks are heavily biased to minimize controversial issues. History is scrubbed of discussion of countervailing forces. Efforts to enhance critical thinking are looked at carefully to make sure that the established order on the local, as well as the national level is not upset. More than a few alleged educators have been against teaching chess in the schools because it would make the kids more judgmental and resistant to authority. Having an educated society, according to Thomas Jefferson, was important to the health of the political system. Unfortunately, those who are in charge of education today do not believe in creating a society that is socially aware, active, and participatory.
Teaching chess to students in the schools can upset the social order. Children who learn chess also learn to question their elders about all sorts of things, to prove the truth of what they are being told. Chess enhances curiosity, a dangerous trait that can lead to thinking and judgment of values. Chess leads the kids to not accept rote learning as the only method of education. Chess leads to more and deeper exploration of topics as well as soaring away on creative tangents at times. Chess leads to pattern recognition, the use of precedent materials, and comparison of ideas. Chess is inherently revolutionary as it seeks to discover new paths of thinking which may be outside the box of normal assumptions. Chess can lead to an undermining of the prevailing value system, and is therefore suspect as a process of educating young minds. Once children learn to think for themselves, it is very difficult to make them conform or accept the surrounding political ethos without question. A society that values critical thinking is more difficult to persuade with propaganda, biased textbooks, or social media. The fact that chess is also fun also makes it suspicious as real learning is supposed to be difficult and really only for the few. The democratic, leveling nature of the game which places no barriers to play is a challenge to a society that is increasingly divided by wealth, political ideology, ethnicity, religion, and purported nationalism. When kids learn chess in the schools, they learn that no obstacle is too big to be overcome, that no lie may remain unchallenged. In the end, they find that winning and losing are only temporary phenomena that lead them to strive for higher quality in all things. That is dangerous for those who wish to maintain the status quo. So, if chess is enhancing critical thinking skills, be forewarned. 
Although there’s too much research that is garbage, peer-reviewed publications are getting better about checking the claimed statistical significance before publishing, and researchers are generally required to make their raw data (appropriately sanitized) available to other researchers who want to double-check the statistics.
That being said, the best anecdotal story I can relate is that back when I was teaching chess to 4th graders at a Lincoln elementary school, their teacher told me when I started one year that she really looked forward to the chess instruction, because “the students are so much better organized afterwards.”
The history of public education in the US derives from the need for people with the skills to be good worker bees in industry. Critical thinking not needed for that. Standardized testing with teachers teaching to the tests, does nothing for critical thinking skills. Would that the effects of chess on critical thinking in the classroom and in life were more profound that they are…