(My DS likes to do his math in reverse, too - gets the answer in head, and then grudgingly goes back to the "show your work" part.)
OP - You don't have to be able to homeschool to use what she talked about. Take a look at some things your DS likes to do and does well, and notice how he learned those things. Did you read to him about it, did he look at pictures or watch a program about it on television? Did he see someone do it, or try it himself and just get better by practice? Did he learn with things he could touch? Now when he has trouble with homework, have him do those things with the school information. - For example, if he learned about dinosaurs by playing with toy ones, give he counters he can move around to do his math (and you don't have to buy things - legos, goldfish crackers, whatever is already in your house works fine.) If he's good at basketball and learned by doing it over and over, chalk a number line on the driveway and have him do his math problems by jumping. Go with his strengths....And if the new teacher comments on good homework, etc., make sure to tell her how you worked with him! If she's the kind of teacher DS has had, she may incorporate some of it in class to help other students.
when my ds was in 1st grade, the teacher called me in for a conference, to tell me that ds was getting the correct answers for the math work (I believe it was adding double digit numbers w/ carrying-over), but that he wasn't doing it the "right way" and wasn't showing how he got the answer and this wasn't OK. She said "I don't know how he's getting the answer and it's going to not be OK very soon and they'll be marked wrong even if they're right"... I understood where she was coming from - she needed to "see" him carrying over to be sure he understood it - this is what school usually has to be because there's 1 teacher w/ 20 - 30 kids, she cannot stand over ds and just watch that he got the correct answer. I get it... but this is when ds's learning starting slipping. Incomplete classwork every day, because his brain was getting stuck on the "showing" of the work.
I want to recommend another book (to whoever - just want to throw it out there)... "the way they learn" by cynthia ulrich tobias. THIS book showed me this particular part of my ds. He's a "big picture" kind of learner, and his learning is random as opposed to sequential. (this one made me LOL - the book say that for his learning style, these people will ask when learning something "how much of this is really necessary?" This is so much my ds. School was definitely not a good fit for him. Hsing is perfect.
I also used sidewalk chalk w/ him on the driveway to learn his vowels, spell out words, etc. Usually I wrote out the alphabet, and he'd jump on the letters. Way better for him than writing something on paper.
so many options w/ hsing.