Short answer: Generally speaking, accreditation is neither required nor necessary.
Longer answer: While accreditation is generally neither required nor necessary, each parent should determine what sort of program best meets the needs of their student for their future plans and complies with their state’s homeschooling statute, if applicable.
What is accreditation?
Accreditation typically means that all teachers and administrators are state certified or credentialed, and the entire school program has been scrutinized over a period of years by a particular accrediting agency. As you can see, accreditation is not feasible for most home-education programs. Curriculum itself is not accredited; the program or school offering it would be the one accredited, and this would really only apply to your program if you were using their online academy or correspondence/distance learning program. In many cases, this can also limit your flexibility in how you administer your homeschool, but you could ask each vendor about any specific requirements or what amount of leeway you have in choices within their programs.
Why does this matter?
Most concerns about accreditation are usually expressed in the context of readmission to public school or acceptance to college. You are basically re-joining their game, so you have to play by their rules. They get to make those rules, and it helps to know in advance what those rules are because your why will likely influence your what or how of curriculum choices.
When re-enrolling in public school before high school, most schools generally admit the student to his chronological grade level, and some may administer their own placement test, but the curriculum you used is generally of little interest to them; you can call the school in advance and ask their readmission requirements, if this is a concern.
When enrolling in a conventional school system during high school (after some high school credit course work has been completed), the school holds all of the power regarding what credits, if any, they will accept for transfer student. It would be advisable to contact your local school to learn what they require of transfer students so you will be prepared if that’s the case.
As far as college: Most colleges are not concerned with accreditation, but instead will have their own admission criteria and will be primarily looking at your transcript and college admission tests; it is advisable to contact the school as early as possible (before high school would be ideal) to find out what their admission requirements are for your homeschooled student, so you can plan accordingly.
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If you do decide that you really want an accredited program, keep in mind that it will generally cost considerably more than other options and you may have limited flexibility in scope, sequence, and time to complete. You’ll want to ask the provider for specific requirements and options. Here are just a few of the many accredited elementary programs available to homeschoolers:
Abeka Online Academy
Calvert Academy (gr 3 and up)
Christian Liberty Academy
Bridgewater Academy
Bridgeway Academy (preschool and K)
Ignite Christian Academy (AOP online academy)
BJU Press Academy of Home Education
Acellus Academy
Keystone School Online
Enlighteum Academy
Forest Trail Academy
Laurel Springs School
Oak Meadow School
Liberty Online Academy
Northstar Academy (gr 4 and up)
Morning Star Academy
Sycamore Academy (secular or faith based options in texts)
Veritas Academy
Here is another list of online schools of various grade levels (although it is compiled by the state of Tennessee, the schools are not specific to Tennessee).