Why was Anne so mortified when she tied with Gilbert on one of their assignments in Anne of Green Gables?

Gilbert Blythe is Anne's fiercest rival in class. Not only that, but there's a good deal of personal animosity between them. Gilbert teases Anne over her red hair, calling her "Carrots." Anne, for her part, once smashed Gilbert over the head with a slate. It's fair to say, then, that Anne doesn't—initially, at least—have much time for Gilbert, though he admires her and tries to gain her friendship.
But there's no time for friendship when it comes to completing assignments. Anne desperately wants to finish ahead of Gilbert, as this will show the whole class that she's so much smarter than him. Unfortunately, after one assignment, Anne is mortified to discover that she's tied for first place with Gilbert; both their names are written on the board by Mr. Philips. Anne is so upset at this outcome because she positively loathes Gilbert and genuinely believes herself to be smarter than him.