I think there are practical reasons as well. A crowd of 5-6 year olds is much more homogeneous than a crowd of 3-4 year olds. Up until about five, kids grow in such spurts, physically, emotionally, mentally, and verbally. Some seem like big toddlers still, and others are little adults. But they all slow down/catch up in that 4-5 range, and by 5-6, they are much more uniform. It makes having an organized, unified class (instead of educating a lot of kids in the same room, but each differently) a lot more feasible. It all falls apart again in middle school, when they start changing in spurts again.

I just want to point out the dynamic here. You told him there was no right or wrong answer, but you sigh when he picks his sister's name. On some level, that disappointed you. He likely picks up on that disappointment, which makes him even more nervous about disappointing you in the future.

My husband quit work when our son was born, and the plan is that he will homeschool, though we very much think of it as a mutual project. There were a lot of reasons why this made sense for us: I love my job: he just went to work. He speaks a second language, and if our son is going to be bilingual, he needs more exposure to not-English. He's also got a better personality for it: he's more patient and much less prone to frustration than I am. I am too much of a control freak: at work, I can spread it around. If I were to direct all that desire to be In Charge on one little person, it really wouldn't be fair.

It sounds to me that some of your frustration is because of your pretty arbitrary goal of how long it should take. Perhaps you could readjust your own expectations to the idea that it just takes a half hour? Then, just be cheerful and patient and persistent: they can drag it out, but nothing else happens until handwriting is done. It's like someone who thinks they and the kids "should" be able to get out the door in 15 minutes, so they plan for that, but Every Single Time it takes longer, so they end up running late and getting mad at the kids. The kids don't know how to be faster/more efficient because kid brains don't work that way, so they resent mom for being mad at them and never improve. Better to just budget in half an hour and be pleasantly surprised if it all comes together more quickly.

You might try going ahead and putting the answer at the bottom, to make it more clear that the assignment is to record the steps, not get the answer. Right now she may be trying to impress you that she doesn't "need" the steps to get the answer. Also, if she is like most kids and likes to ask "why" about everything, you might turn it around with "Yes, it's three. But WHY?" You might even try scribing her verbal explanation and then showing her how the mathematical notation says the same thing.

Making something a priority means putting it ahead of other things. It took me like 30 years to learn that. I always thought making something a priority meant "Get all emotional about it and beat yourself up inside when it doesn't work out". So if you really want to be organized, you've got to do it first. That means accepting that there will be less of something else going on. You've got to accept covering less material in a day or something. "Organize time" has to be like "math time" or "reading time", something that happens virtually every day. You wouldn't expect to add a foreign language without scheduling time for it, so don't expect to add organization without the same thing. I teach public school, and one of my big challenges is getting papers back to kids. I am a pretty quick grader, but the handing back seems like such a waste of time. However, 85% of them won't pick up their own papers--and those are the ones that need immediate feedback. For ten years, I told myself every year "This year I will be better" but it never happened because I didn't define "better" as anything but "will feel guilty when I don't". This year I sat down and really thought about it: what would help my kids more: timely feedback or an extra 10 minutes of instruction a week? And I decided that they really needed the feedback, even though there is a real cost in total instructional time. So this year I have really prioritized handing back papers, but it's the absolute truth, there are things I'd like to teach that I am not getting to. I think it's worth it: more kids are doing their work, and doing it better, but it is a tradeoff. I regret the instructional time I've lost. So I think you have to decide if being organized is worth anything, really, beyond a feeling that you "should be". Will it add to your children's education? What will it add? What are you doing now that you could give up to make the time to do it? I have no idea what the answer is to that question: I'm really, really early in my own homeschooling process. But I know these things work better when they are an informed choice, not just an emotional reaction.

That's a terrible thing to let a child hear, and in terms of things like Christmas gifts, of course one should be equal, but I have to admit: if one of my sibling's kids needed a kidney, my reaction would be very different than if it were one of my husband's siblings' kids. One of my "blood" relations? I wouldn't even think about it: obviously, if my kidney works, take it. One of his? I honestly have no idea. Probably. But I wouldn't have that automatic response.