They do look like drumsticks. The badge has a bar below the upper most pin that probably held some kind of ribbon, as I have seen that feature on other badges from the late 1800s. For something that is unmarked (aside from the “W”) and is likely for a band/school participation badge, it is quite intricate. Not even a makers mark. It really goes to show how much things have changed in the last 100+ years as far as what kind of awards are given out and their craftsmanship.