He sees her notes- she's learning some English words, okay, but... he needs to give her a vocabulary she can use to explain how she got here. What words take priority, which can be most easily communicated... how to get an intelligible explanation from her in under an hour?
Movement words, place words, prepositions...
Or take a more primitive approach. Cave paintings. Drawing. He grabs a piece of paper, draws... a circle, with a crudely-rendered America, next to a parallelogram, a plane. On the plane, he places a stick figure with curly hair, and atop the globe he places an unadorned stick figure. He then draws a curved arrow from the girl to America, with a question mark.
Actually, she probably won't understand the question mark. Maybe not the stick figures, either? He leaves the question mark, and draws tiny smiley faces on the stick figures.
Whether it's the quality of the drawing or some unexamined assumption, she doesn't have an answer for him.
He draws another stick Pyay at the end of the arrow, then points to the first Pyay with his pencil. "Pyay Kweengow..." he traces the arrow to the other Pyay, "then Pyay Earth?"
Or, wait, maybe that was an explanation. Just, a brief and vague one, asking for more information from him? He could...
"...I draw Earth." He flips the paper over and draws a crude globe. "I draw Pyay." He adds a- not a stick figure this time, a more detailed rendering. His sense of proportion is all wrong, but it's recognizable.
He hands the paper and pencil to Pyay. "Pyay draw 'tlaa'?"
She makes a dubious face, but makes a drawing of - a monster. A snake with a big eyeless face. Chasing a stick figure Pyay.
This connects even less to the whole materializing/no English/mice/flat world situation.
Ohhhhhh. She's crazy. She's- no, wait, the appearing, that's not a thing that her being some escaped mental patient would explain. That's not a thing anything explains. There's got to be connections, between something, and something else, anything else... the tlaa is a snake, snakes eat mice, what is with the mice...
He points at the mouse cage. "...Mice?"
She looks at her mice. "Mais," she repeats obligingly, and she writes this down.
He is interrupted by a humming noise. It's 6:45- the hallway lights click on in unison, and the air conditioning turns on. Various quiet whirrs and hums sound from all directions.
The front doors downstairs ought to be open, now. It'll be a while yet until things start, but they could be interrupted at any time. He's going to have to find a way to accelerate this, or else cut class to deal with this mystery girl.
"Do you have any frie-" No, not words, drawings.
He takes her drawing of the "tlaa" and adds a few more stick figures running away alongside her. Were there others with her?
She crosses out his stick figures but draws one of her own, a taller, blonder stick figure, farther away from the snake with the door-sized face. "Praip ruum Fwaysyee."
"And... is he here?" And then he remembers words, again- how does he keep forgetting this, moment to moment?- and lays this stick figure's drawing on top of the piece of paper with Earth on it, and draws an arrow between "Praip room fwaysee" and Earth, to match the arrow he used to ask how she got here. He traces the movement of the arrow with his finger a few times, to emphasize the movement.
So... he wants to know if she has a co-conspirator who could have helped her make that appearance. She's confirmed that someone else was involved. (The idea that she was literally teleported here from another world by a snake monster being clearly a lie or insanity, he must identify how this person helped or manipulated her into faking sudden teleportation.)
So... he wants to find out where this person is, in hopes that they'll be more knowledgeable or at least more cooperative. Her gesture implies that they couldn't be here, as they're not immediately here in the room. The information she needs... is that this person's lack of immediate presence doesn't imply a lack of involvement- they could be hiding, or not personally present for the trick.
That's a complicated sentiment... he needs to explain "imply", and "involvement" and various vague ideas that can't be drawn on paper.
How about...
He gestures around at the school building, and draws a small circle on a fresh sheet of paper. He then draws a large outline of the US around that circle, and puts several additional circles on the map nearby. He puts stick figures for himself and Pyay in the first circle, and stick figure for the accomplice in another circle. And then... a parallelogram with a black spot for Kweengow, with another circle. And a snake in that circle, and then two arrows- one pointing from the snake to the school circle, and another to the accomplice's circle.
He stares at his handiwork in disgust- there's got to be a thousand ways she could misinterpret this- but hands her the drawing anyway.
She seems pretty bewildered. She looks at it for a while, then starts a new drawing: snake monster, herself, tall blond stick figure. Solid arrow from snake monster to herself; arrow from herself to a doodle of the United States. Dotted arrow from snake monster to blond stick figure, and a much more sparsely dotted arrow from the blond stick figure to an empty, blank circle.
So, okay, most likely she is in on whatever this is, considering how unhelpful that was. She has no idea where her accomplice is, or where they would be under whatever circumstances? If she'd been being manipulated, she'd have some idea of what was going on.
Max has had enough of this. He's going to take her down to Gloria's classroom, and stick her with the English as a Second Language kids, and tell security about some trespasser talking in gibberish. Too old to be a student sneaking in to change their grades, but... he'll let them figure it out.
...except she materialized out of thin air. That'll eat him up all day if he lets it go unanswered.
"Y'know, if you're just making up this don't-speak-English crap to distract me, you're being a real jackass."
(If she can't understand him, the language won't matter, and if she can, she deserves it.)
Then she draws an arrow from the end of her own chain-of-travel, to another empty circle.
He draws a circle right next to a circle containing them, with a shorter, wider stick figure with curly hair for the English teacher, and draws an arrow between them. He walks a ways down the hall and opens a door to a staircase, making a beckoning motion.
Does she even have the same general sign language? Would she understand shaking her head for no and nodding for- oh, as if it matters, she's probably faking it anyway.
There's a picture of Gloria with her class- he shows it to her and points to the stick figure drawing he made previously.
But how to explain that she teaches language...?
He points at her papers with the columns of translations- pointing rapidly between pairs of words, and at the picture. "Gloria... teach... English."
Her stomach grumbles. She rubs her tummy and looks at him inquisitively.
He goes through his pockets. There is... an apple core in a ziploc bag he forget to throw away, rapidly going brown... half of a chip... he sets this aside on the desk and starts looking through his bag. It's a mess in there, there must be a granola bar he forgot about, or something...