If she devotes herself quite strongly to the lessons she can make what her tutor assures her is actually quite a good amount of progress! Especially if she reads things and writes down all the characters she doesn't understand to ask about later, as suggested. She can make progress through most books alone, now, though it's slow work. She can read signs, and write without making a huge mess, just a small one.

She even begins absorbing some imperial history, at least the basics- Cai's ascension was really not that long ago. And her daughter went to this very sect a decade ago. The Empire has been around for a long, long time and so very many things have happened. Cao Shan assures her thay having at least a basic grounding in history is a good thing. It might help her recognize cultivation things, if nothing else.

The arcane world of mathematics is nearly impenetrable though. Perhaps 'zero' and 'variables' and 'orders of magnitude' are something that only makes sense when you become a cultivator? How can a number be less than nothing?

Mathematics is awful. She does well where she can just memorise things, but Mei Cao fails at this point to realise that mathematics is a procedure and not a series of cases to be memorised one by one and then keep getting blind-sided by the teacher providing new ones. It isn't helping her that cultivators apparently use different numbers than shepherds to count and she has to relearn that from scratch. She thinks she has addition down, more or less, though. 

She prefers reading and history. She's very excited to walk into the footsteps of the Cai Heiress. She likes knowing facts about the world. Reading seems hypothetically useful and people keep saying it's very important but she hasn't found things which are, to her, important to read as of yet. 

Eventually, her tutor does notice the constant struggles with mathematics, and takes the time to interrupt the usual lessons and restart on math from first principle, explaining how a number is just a symbol, not a thing, handing her copper coins and counting them out, using debt to analogize negative numbers and purchases of multiple copies of things for multiplication and division. Cao Shan is quite good at teaching, always seeming to say things in ways that just make sense.

(She also succeeds in not giving off the impression that a small child should already know these things, even while explaining that, well, the nobles and upper class will expect her to already know these things. But really, it can be learned, it's all just fancy counting in the end-)

Hopefully this will get her past the bit of a roadblock in math, because the time remaining before official sect entrance is vanishing quite quickly. Only a few days left now.

She was already completely clear she was behind, but you know. She's being rewarded by being allowed to try at all? She's going to do her best, but it's only natural if the people who started this far up already will look down on someone only getting there because the Duchess decided to be nice to her. 

She eventually understands the procedures that are addition and subtraction and some of the related abstractions. 

When the day of entry comes, she is given another carriage ride up one of the mountains. And then further up. And then further up. The peaks make a landscape framed by cloud and sky, here, sprawled out before her eyes. The air feels alive, somehow. It's a beautiful place.

She's dropped off with a polite 'goodbye' and pointed towards a two story building standing alone, looking a bit more like a shrine than most of the buildings down from the sect village. Stately stone, surrounded by pretty little gardens and well maintained paths and flowering peach trees. She catches sight of another grey-robed new disciple heading into the building, and upon entering herself- It may well be prudent that she did learn to read. A short instructional paper that was delivered the previous day indicates that she is to go to Hall 2, and one of the signs, conveniently enough, indicates the way to Hall 2.

She doesn't quite trust the ambience. Studying pushed down the sensation that everything was fake; this increases it. 

These are the most allegorical peach trees she's ever seen. She's much more used to the ones which are used to produce (in her experience) brandy and arguments with your neighbours about who gets to have the best smoked meat next year. These ones feel like they were planted specifically to make one's students forget that this is real life and not a folk tale, at great expense. 

Carrying herself with all the dignity and composure afforded a teenaged collier, she will go to Hall 2. 

It's a large lecture hall, with comfy seating arranged with plenty of space. Light snacks and juice sit on a few tables here and there, kept at the correct temperature by subtle runes. There's seven other people in here already-

Four seem to have formed a sort of clique in the corner, animatedly discussing something and facing each other. Something to do with finance, or taxes, by the occasional overheard word. She might recognize one of them as from the Teng clan, a Viscount level clan from the south of the Emerald Seas, from her past lessons. Or perhaps not.

There's a dark-skinned girl with bright white hair, sitting utterly still, with the obvious bulge of some kind of weapon under her robe.

There's a tall and lanky boy with slicked down brown hair, fidgeting with a small carrying case. Inkstains on his fingers. He's nodding along to something the last person is saying, and locks eyes with her for a moment when she enters, but evidently she is not the one he's waiting for.

There's a broad-shouldered guy with messy black hair and a soldier's countenance and muscles, steadily milling through a plate of snacks and telling some kind of war story, from the tone and hand gestures.

She will pause a moment to see if anyone expects anything of her. But then she'll go introduce herself to her nominal peers. The two talking boys? Four people is scary and finance is scary (she does not identify the clan) and the single girl doesn't look like she wants to talk to people. Also,.she probably needs to learn about war eventually. She will be polite and formal as she does so.