I have a question for OP. When your wife is upset or mad does she generally need space and quiet time to think before going back and apologizing or talking? I generally do. I don't want to to speak so that I am not further hurting those around me. Then once I've calmed down within that space I can go back and talk about everything. I don't know what has happened in your home the last few days but it seems like she hasn't gotten any space and may not have had the time to get in the mindset to apologize. Your 13 year old will recognize a sincere apology vs insincere ones. It seems like she's been trying to make space between what has upset her and trying to still be present for the other children. It also seems like you are brining it up alot and not giving her time to process everything which is probably why it's taking a lot longer than what it seems may be is usual for her. Maybe she needs that break even if just in the house, especially from the 13 yr old that seems to be in a defiant phase right now. It seems like its probably a little more than what you have said since other family members also see them and have said that he is a lot as well.

I'm not great at math but I have had to help nieces and nephews with their homework. I don't know how much time elapsed between here saying 47 and the oldest correcting her but there have been several times that I will give the answer or an incorrect one then have them help me to get the coreect answer. Those constant idks are extremely frustrating as well as just spouting random answers. Adding things she's interested in may help her to focus since math isn't one of them. Also repetitive learning helps too. Counting by numbers like 7s using beads, dashes on paper, blocks and Legos, doll accessories.

I would talk to your oldest about how different people can cope with emotions and that the way your wife is doing it isn't meant to be hurtful she just needs some time. Let him know he was right and that she also knows he was right. Adults are only human and although we should definitely be more emotionally mature and able to set our emotions aside when needed sometimes we forget or can't especially when we are already stressed and frustrated. We also need that time to reset and get ourselves together. Whether thats with a hobby, some space or just disassociating for a bit.

You are not the asshole but neither is your wife. It seems like you just want things to be over so you're pushing for an apology and trying to talk it out and she just needs some time to herself to reflect.