I can't speak to Connections. Laurel Springs has two divisions: regular and gifted. BOTH offer self-paced curriculum (it's all self-paced; none of it except optional AP discussion/lecture sections are synchronous, and even those are taped and can be watched at other times) and BOTH divisions offer acceleration. Basically, the kids take what they are ready for. My 7th grade dc took high-school Honors humanities and just skipped all the middle school stuff. What a relief after grade-level public school! DC is now going into 9th grade and has accumulated two years of high-school credit. DC is doing AP work now as a 9th grader -- something not permitted in most schools, but again, at either branch of LS, you take what you can handle.
Keep in mind it IS homeschool, though. My dc can learn a LOT independently, but especially as the work gets more advanced, it's a better experience if I do some discussion/teaching and sprinkle in some thoughtful documentaries etc. Writing is especially instruction-heavy for us, but then, it's an area of strength for dc.
The Gifted division of Laurel Springs is useful for two things: (a) you get selected teachers who have some training in dealing with gifted kids (the teachers have all been quite good and are enthusiastic and have plenty of time -- they do not seem to be under a crushing workload), and you can continue with these teachers throughout high school, which is really nice, and (b) (this is huge) the Academy kids get 100% customization, meaning that if you ask for it, and if what you ask for is academically respectable, they will enrich/raise the level/change the content of any course. To take an example: DC had read a lot of the 18th and 19th century British literature on the syllabus. So the teacher had him read 20th and 21st century British literary novels. To take another example: DC was able to substitute (as a 7th grader) long-form essays, paintings with commentary, and research papers for more standard guided reading type short answers.
