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There is no "canon" child for Professor Birch. I believe we make it obvious that Birch is May's father in the visual novel.
Also, in any school environment, teachers do not regularly interact with each other, especially high level schools. Teachers by nature are incredibly busy people. If they're not physically teaching a class, they're designing lesson plans, researching, or attending education meetings/conferences—especially so in college-esque settings like this one. When teachers do interact, it'd be during some rare downtime (not often).
And if you consider that this is the "first day" of classes, that adds even more weight to how busy the academy could be.
To top it all off, since May is obviously Birch's daughter and is well aware of her father's work, would it not be better for her to deliver the good news first? Certainly Oak and Birch can talk about it later on, but it's such a minor detail there's no reason to elaborate on it in the visual novel.
FYI it would be good feedback if you specifically point out how you're confused about Birch's child. Personally, I don't think it's even a question based on what's written in the visual novel. Leaf literally points out that Birch is May's father.
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