Schools buy Pearson because Pearson has strong marketing and is the biggest obvious choice, and because Pearson helps schols exploit their students by selling customized (not nationally standard!) hard-to-resell textbooks. Not the only well-developed choice.

> "Common Core approved"

This does not exist, so you have no problem with Common Core. Common Core is an open standard, not a certification. Any book can be Common Core aligned, and any school board can choose to certify a book within their jurisdiction.

Common Core is a standard (checklist of abilities students should learn), not a curriculum.

Ilustrated Mathematics ( https://illustrativemathematics.org/math-curriculum/ ) is a curriculum, which is Common Core aligned, that fills in details of the standard. IM certified partners who each publish books. (Both open and proprietary partners exist) This is closest to the accredidation you mention.

Yes, accreditation is an expense. That expense can be paid by donations, grants, or government payments.

There are currently 3 textbook publishing partners of Illustrated Mathematics.

https://openupresources.org/frequently-asked-questions/open-...

Cengage and McGraw-Hill are major textbook publishers, who are not Pearson.

Illustrative Mathematics's publishing partners are also not Pearson.

Illustrative Mathematics (used in various school districts) is based on Open Up Resources, a Creative Commons -licensed curriculum https://openupresources.org/math-curriculum/ that includes teacher/home/student lesson plans (not exactly a text book, but K-12 school curriculum is designed to be led by a teacher, not self-studied.)