I’m just figuring all of this out myself, but I *think* I can answer your question to some extent. Please, someone else jump in if you have wisdom to offer!
Ray’s New Arithmetic is one of the eclectic/classic curricula sets like the McGuffey readers or Spencerian penmanship or Harvey’s Grammar. It’s what the one-room school houses used in the US throughout the 1800’s and into the early 1900’s. Ray’s is very self-led once a child is reading on their own. One set of Ray’s math books should get you through 8th grade, and then there are other Ray’s books in the public domain for higher level math. In fact, all of the Ray’s books are in the public domain, so if you’re strapped for cash, it’s a valid option for teaching math virtually for free. Ray’s relies heavily on word problems, so you read the problems out loud to your student, or once the student is able, he/she reads the problems himself/herself, and then the student works through them at their own pace. Because it relies heavily on word problems, and there are no fancy manipulatives to buy to go along with it (you just use whatever you have around the house), it’s supposed to help your child develop an excellent ability to do math in his/her head. All of the problems are based on real-life situations, so instead of doing worksheet after worksheet of problems represented by numerals and symbols, your child is seeing how math would be applied in real life. In that aspect, I think it has something in common with Life of Fred, though I have not actually looked through a Life of Fred book myself yet.
I have JUST started looking through the Manual of Methods linked above. Basically, it looks like a teacher’s guide to all of the eclectic books (Ray’s, McGuffey’s, Spencerian, Harvey’s, etc.). I’m also using the McGuffey readers and was feeling a little lost as to how to teach phonics as opposed to just sight reading, but already, just glancing through the Manual of Methods has provided some great ideas for me. It takes it lesson by lesson and tells me how the lesson should be taught. I wish I’d known about this manual before I started working through McGuffey with DD! I also skimmed through a little of the chapter on Ray’s, and I’m still a little unclear on oral vs written. I *think* what this means is that when you introduce your child to math, you should do so without representing numbers with numerals and symbols so your child develops an intrinsic understanding of the value of numbers before being presented with abstract symbols. Does that make sense? For example, you would use math in everyday life . . . at the grocery store, count apples into a bag. “How many do we have? Oh! We have one too many, let’s take one out! How many do we have now? Hmm . . . you know, now that I think about it, we may want extra apples to share with Grandma and Grandpa when they visit this weekend. Let’s add two. How many do we have now?” I was actually inspired while skimming through the Manual of Methods and reading Ruth Beechick’s “An Easy Start In Arithmetic” last night. I’ve decided to take math outside. I’ve been trying to figure out how to incorporate nature study into our school days, and I think I’ve found my solution. Take school outside! We’ll count, add, and subtract birds, bugs, flowers, clouds, whatever strikes our fancy. Science will take care of itself, we can do our reading outside, which will cover most other subjects, and I can let the kids draw pictures of what they see. That mostly covers it there. 🙂 I’m excited to get started! It helps that we live in an area where this is possible right now too though.
I don’t know what you are already doing with regards to teaching your child to count, but if you haven’t yet, I guess just spend a year counting everything with them sort of like my example above. If you have already been playing counting games like that, then you could probably just jump right in to “year 2” and start introducing those abstract numerals and symbols starting next year. Like I said though, I’m still reading through all of this stuff, so I’m not absolutely positive that’s the way it works. I haven’t looked much into Math-U-See, but from what I hear, Life of Fred can either stand on its own or serve as a nice supplement to most anything.